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Opt. Pura Apl. 40 (3) 243-248 (2007) - 243 - © Sociedad Española de Óptica
Diseño de láminas de cristales fotónicos bidimensionales basadas en
agujeros triangulares: estudio paramétrico
Triangular air-hole based two-dimensional photonic crystal slabs design: a
parametrical study
Gonzalo Santoro(*), Iván Prieto-González, Juan B. González-Díaz, Luis Javier Martínez,
Pablo Aitor Postigo(S)
Instituto de Microelectrónica de Madrid, Centro Nacional de Microelectrónica, Consejo Superior de
Investigaciones Científicas, Isaac Newton 8, PTM Tres Cantos, 28760 Madrid, Spain.
(*) Dirección actual: Instituto de Ciencia y Tecnología de Polímeros, Consejo Superior de Investigaciones
Científicas, Juan de la Cierva 3, 28006 Madrid, Spain.
* Email: gonzalo@ictp.csic.es S: miembro de SEDOPTICA / SEDOPTICA member
Recibido / Received: 5 – Mar – 2007; Aceptado / Accepted: 22 – Mar – 2007
REFERENCES AND LINKS
[1] J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton
University Press, Princeton, NJ (1995).
[2] E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics”, Phys. Rev. Lett.
58, 2059-2062 (1987).
[3] E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor
and acceptor modes in photonic band structure”, Phys. Rev. Lett. 67, 3380-3383 (1991).
[4] S. G. Johnson, J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice, Kluwer
Academic Publishers, Massachusetts (2002).
RESUMEN:
En este trabajo se investiga la formación de bandas prohibidas para fotones (PBG) en varios
diseños de láminas de cristales fotónicos bidimensionales. Los cristales están diseñados desde el
enfoque de la reducción de simetría de la red y están basados en la red triangular de agujeros
triangulares. El análisis se ha realizado computando los autoestados de las ecuaciones de Maxwell
para estructuras dieléctricas periódicas con algoritmos de iteración por bloques con condiciones
previas y una base de ondas planas. Encontramos un ensanchamiento de las bandas prohibidas
para fotones en comparación con la red de agujeros triangulares simples al tiempo que aumenta la
separación entre motivos lo que facilita la fabricación con las posibilidades experimentales
actuales.
Palabras clave: Nanofotónica, Cristales Fotónicos, Diseño de Nanoestructuras.
ABSTRACT:
We investigate the complete photonic band gap (PBG) formation in several designs of two-
dimensional photonic crystal (2D-PC) slabs using the symmetry reduction approach. We have
based our designs in the triangular lattice of triangular holes. The analysis has been performed by
computing eigenstates of Maxwell’s equations for periodic dielectric structures with pre-
conditioned block-iterative algorithms and a plane-wave basis. We have found an enlargement of
the complete PBG with respect to that for plain triangles at the same time that the dielectric walls
are enlarged what makes easier the fabrication with the present experimental capabilities.
Keywords: Nanophotonics, Phtonic crystals, Nanostructures design.
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Opt. Pura Apl. 40 (3) 243-248 (2007) - 244 - © Sociedad Española de Óptica
[5] S. Takayama, H. Kitagawa, Y. Tanaka, T. Asano, S. Noda, “Experimental demonstration of complete
photonic band gap in two-dimensional photonic crystal slabs”, Appl. Phys. Lett. 87, 061107 (2005).
[6] T. Trifonov, L. F. Marsal, A. Rodríguez, J. Pallarès, R. Alcubilla, “Effects of symmetry reduction in two-
dimensional square and triangular lattices”, Phys. Rev. B 69, 235112 (2004).
[7] L. J. Martínez, A. García-Martín P. A. Postigo, “Photonic band gaps in a two-dimensional hybid triangular-
graphite lattice”, Opt. Express 12, 5684-5689 (2004).
[8] S. G. Johnson, J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in
planewave basis”, Opt. Express 8, 173-190 (2001).
[9] S. G. Johnson, J. D. Joannopoulos, The MIT Photonic-Bands Package home page: http://ab-
initio.mit.edu/mpb/.
[10] M. Qiu, “Effective index method for heterostucture-slab-waveguide-based two-dimensional photonic
crystals”, Appl. Phys. Lett. 81, 1163-1165 (2002).
1. Introduction
Photonic crystals (PCs) are periodic dielectric
structures in one or more spatial directions [1].
Many unusual optical properties of these PC, such as
the suppression of spontaneous emission [2] and the
possibility of creating localized defect modes [3] lie
in the existence of a photonic band gap (PBG), i.e., a
frequency range for which light propagation is
forbidden inside the structure. For their possibilities
in practical applications, three-dimensional PCs
exhibiting at least a complete band gap within the
range of telecommunications frequencies have been
intensely looked for but the fabrication of useful
three-dimensional PCs in the near IR spectra is a
difficult task because of technological limitations at
the submicronic length scales. An alternative system
to achieve a band gap in the photonic spectrum is a
two-dimensional photonic crystal (2D-PC) slab, a
thin dielectric structure with a 2D-PC pattern that
uses index guiding to confine light in the third
dimension. In this structure, as in two dimensions,
one is able to decompose the guided modes into two
non-interacting classes. However, the lack of
translational symmetry in the vertical direction
means that the states are not purely TM or TE
polarized (electric and magnetic fields are normal to
the plane of periodicity respectively). Instead, the
guided bands can be classified as even (TE-like) or
odd (TM-like) with respect to reflections through the
plane bisecting the slab4. The physical properties of
PCs slabs can be significantly different from those
of the corresponding 2D systems. For instance, it is
well known that the triangular lattice of circular air
holes in a purely 2D photonic crystal exhibit a
complete band gap at sufficiently large air fractions,
nevertheless, the same pattern in a 2D-PC slab does
not longer show a complete PBG due to the
appearance of degeneracy between the first and
second band at highly symmetric J point of the
Brillouin zone [5].
Recently it was suggested and experimentally
demonstrated by Takayama et al [5] that a 2D-PC
slab consisting of a triangular lattice of triangular air
holes (TLTH) gives rise to a complete PBG because
the reduction of the symmetry of the unit-cell
structure solves the degeneracy at the J point. The
optimization and improvement of the experimental
viability of this structure as well as the design of
similar structures is of great technological interest.
Using the symmetry reduction approach [6,7] and
basing our designs on the TLTH we have
investigated the existence of a complete PBG in 2D-
PC slabs with several patterns as well as their
experimental feasibility as a function of the pattern
parameters from numerical simulations of the
photonic bands.
2. Structures description and simulations
The patterns of 2D lattices under consideration are
depicted in Fig. 1. The basic structure from which
we have designed the rest of the patterns of the 2D-
PC slabs is the TLTH (Fig.1a). The modified
structures are the triangular lattice of triangular air
holes with interstitial circular holes (TLTH-CH), the
triangular lattice of triangular air holes with
interstitial triangular holes (TLTH-TH) and the
triangular lattice of rounded triangular holes
(TLRTH) (Fig. 1b, 1c and 1d respectively). In Fig.
1a, 1b and 1c the parameter L denotes the length of
the side of the triangular holes. The parameters r and
l denote the radius of the included circular holes and
the length of the side of the included triangular holes
in Fig. 1b and 1c respectively. In the case of Fig. 1d
theparameter r denotes the radius of the circles that
conform the rounded triangles while l denotes the
distance between the centres of such circles. The
angle of rotation θ of the motifs is defined as the
angle between the bottom side of the triangle and the
lattice vector a1. Fig.1e represents the first Brillouin
zone and the path Γ-X-J-Γ along the calculations
were performed.
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Fig.1. Patterns of the 2D-PC slabs under consideration: a) the basic structure is the triangular lattice of triangular air holes (TLTH) and the
modifications in the designs of the 2D-PC slabs are b) triangular lattice of triangular air holes with interstitial circular holes (TLTH-CH), c)
triangular lattice of triangular air holes with interstitial triangular holes (TLTH-TH) and d) triangular lattice of rounded triangular air holes
(TLRTH). We have also represented the lattice vectors and the Wiegner-Seitz cell. e) The first Brilloin zone for all the cases.
In order to obtain the dispersion relations we have
used a plane-wave expansion method consisting of
the fully vectorial solution of Maxwell’s equations
with periodic boundary conditions computed by
preconditioned conjugate-gradient minimization of
the Rayleigh quotient [8,9]. Moreover, to take into
account the finite thickness of the slab we have used
the effective refractive index [10], i.e., the mode
refractive index of the guided waves without PC in
the slab. To compare our results with that on Ref. [5]
we have calculated the effective index assuming a
slab of Si of 320 nm thick suspended in air
obtaining a values neven=2.92 for the TE-like mode
and nodd=2.51 for the TM-like mode.
3. Photonic band gaps
In this section we will present the results of the
simulations we have performed on the structures
described above. We have limited our study to the
first and second band for the even and odd modes in
order to ensure that the PBG frequency lies well
below the light cone so the PBG occurs between
guided modes in the slab. To analyze the formation
of the PBG we have calculated the gap-maps of the
2D-PC slabs as a function of the size of the motifs
and in some cases as a function of the angle of
rotation of them.
3.a. Triangular lattice of triangular air holes
(TLTH)
We begin our discussion with the basic TLTH (Fig.
1a). Contrary to the triangular lattice of circular
holes, that have a C∞ symmetry, the utilization of
triangular-shaped holes (C3ν symmetry) solves the
degeneracy at the J point of the Brillouin zone for
the odd modes. This leads to a maximum normalized
width of the complete PGB of about Δω/ωg=6.56%
centred at ωg=0.294a/λ for L=0.92a and θ=0º. Here,
the parameters Δω and ωg denote the frequency
width of the gap and the frequency at the middle of
the gap respectively. This values, for a gap centred
at wavelength of 1550 nm, correspond to an
experimental lattice parameter of a=456 nm and a
critical distance, i.e., the minimum distance between
adjacent holes, of Δ=36 nm.
In Fig. 2a are drawn the frequencies of gap
boundaries between first and second bands as a
function of the side of the triangles for an angle of
rotation θ=0º until adjacent triangles begin to
overlap (L/a=1). The gap map shows that the
maximum complete PBG appears at L/a=0.92 that
corresponds to the size of the triangle when the even
mode PBG and the odd mode PBG begin to non
completely overlap.
a2
a1
θ
L
a)
a2
a1
r
L
b)
a2
a1
l
L
c)
X J
Γ
e)
d)
r l
a2
a1
θ
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Fig. 2. Photonic gap-maps between first and second bands
of the triangular lattice of triangular air holes as a function
of a) the side of the triangle for a fixed angle of 0º and as
b) the angle of rotation for a fixed side of the triangle of
L/a=0.92.
The study of the evolution of the PBG with the
angle of rotation indicates that the maximum
absolute gap occurs for θ=0º, or the analogous
situation θ=60º, and disappears completely for
θ=30º, as shown in Fig. 2b for triangular holes of
side L/a=0.92. This behaviour reveals the fact that to
solve the degeneracy at the J point of the Brillouin
zone for the odd modes the vertices of the triangular-
shaped holes must point in the real space at the J
points of the Wiegner-Seitz cell while pointing at the
X points (θ=30º) the degeneracy is recovered.
Moreover the band gaps exhibit a symmetry with
respect to a rotation angle θ=30º due to the inverse
symmetry of the structure. We would also like to
remark the opposite behaviour of the even mode
PBG and the odd mode PBG, that is, when the even
mode gap increases the odd mode gap decreases and
vice versa, but since the even mode PBG is
considerably greater than that for the odd mode is
the latter who limits the complete PBG.
3.b. Triangular lattices of triangular air holes
with interstitial figures (TLTH-CH and
TLTH-TH)
The next structures we have investigated are the
basic TLTH with the inclusion of holes with
different shapes in the interstitial positions in order
to increase the air filling fraction (FF) as a way to
increase the gap size. The first structure that we
consider is the TLTH-CH (Fig. 1b). For this
structure, instead of the expected behaviour, the gap
for the even mode decreases as the radius of the
interstitial circle is increased while the gap for the
odd mode decreases for small radii, closes
completely (the degeneracy is recovered) and
reopens for large radii (the degeneracy is again
solved) reaching a width three times larger than that
for plain triangles but now both gaps don’t overlap
(Fig. 3). Although for the odd modes the gap is
considerably increased for large radii, the even mode
PBG rapidly decreases and the first and interstitial
circle is increased, as is shown in Fig. 4. For this
reasons this structure doesn’t improve the results
obtained with plain triangles only.
Fig. 3. Photonic gap-map between first and second band
for the triangular lattice of triangular air holes with
interstitial a) circles as a function of the circle radius and b)
triangles as a function of the size of the included triangles
for a fixed size of the main triangles of L/a=0.92. Both
gap-maps are calculated until adjacent figures begin to
overlap.
0,4 0,5 0,6 0,7 0,8 0,9 1,0
0,22
0,24
0,26
0,28
0,30
0,32
0,34
0,36
Frequency (ωa/2πc)
L/a
even
odd
a)
0 102030405060
0,22
0,24
0,26
0,28
0,30
0,32
0,34
0,36
Frequency (ωa/2πc)
θ (deg)
even
odd
b
)
0,0 0,1 0,2 0,3 0,4 0,5
0,28
0,30
0,32
0,34
0,36
0,38
0,40
0,42
0,44
0,46
Frequency (ωa/2πc)
l/a
even
odd
0,00 0,05 0,10 0,15 0,20 0,25 0,30
0,28
0,30
0,32
0,34
0,36
0,38
0,40
0,42
0,44
0,46
Frequency (ωa/2πc)
r/a
even
odd
b)
a
)
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Another way to increase the air FF trying to avoid
the tendency of the even modes to appear degenerate
at the J point is to introduce triangles as interstitial
figures (Fig. 1c). Even though it is true that the
tendency of the even modes to degenerate is reduced
with respect to the previous situation this design
doesn’t lead to an enlargement of the complete PBG.
Contrary to that, the behaviour of the gaps for both
modes is the same as in the case of interstitial
circles, the PBG for the odd modes decreases as the
included triangle increases for small sizes closing
completely to reopens at larger lengths of the side of
the interstitial triangle without overlapping the gap
for the even modes (Fig. 3b). It is also remarkable
that the included triangles don’t lead to an
enlargement of the odd mode PBG as strong as the
included circles. This is probably due to the fact that
the air-FF is considerably greater when the circle is
about to overlap the adjacent triangles than when the
included triangle is near the close-packed condition.
So, even though the utilization of large air-FF is in
general an effective method to increase the gap size,
in this case the inclusion of interstitial figures
doesn’t improve the results obtained with plain
triangles. This is mainly because of the tendency of
the even mode to degenerate at the J point as the size
of the included figure increases and because of the
lack of overlap between the gaps for both modes at
large air-FF.
0,20
0,25
0,30
0,35
0,40
0,45
0,50
Γ
J
X
Γ
Frequency (ωa/2πc)
even
odd
0,20
0,25
0,30
0,35
0,40
0,45
0,50
ΓJ
X
Γ
Frequency (ωa/2πc)
even
odd
a) b)
c) d)
0,20
0,25
0,30
0,35
0,40
0,45
0,50
Γ
JXΓ
Frequency (ωa/2πc)
even
odd
0,25
0,30
0,35
0,40
0,45
0,50
ΓJ
X
Γ
Frequency (ωa/2πc)
even
odd
Fig. 4 Dispersion relations of the triangular lattice of
triangles with interstitial circular holes for a fixed side
of the triangle L/a=0.92 and a radius a) r/a=0.10, b)
r/a=0.15, c) r/a=0.20 and d) with r/a=0.25. The black
line is the light cone
012345678
0
2
4
6
8
10
12
Δω/ω
g
(%)
β=l/r
r/a=0.10
r/a=0.15
r/a=0.20
r/a=0.25
r/a=0.30
r/a=0.35
0 20 40 60 80 100 120 140 160 180
0
2
4
6
8
10
12
Δω/ω
g
(%)
Δ (nm)
r/a=0.10
r/a=0.15
r/a=0.20
r/a=0.25
r/a=0.30
r/a=0.35
a)
b)
Fig. 5 Width Δω/ωg of the complete gap for the triangular
lattice of rounded triangles as a function of a) the
parameter β=l/r and as a function of b) the critical distance
for a gap centred at λ=1550 nm. For each radius we vary
the distance between the centres of the circles.
3.c. Triangular lattice of rounded triangular air
holes (TLRTR)
For the TLRTH we define a parameter β=l/r with the
intention to investigate if there’s an optimal relation
between the radii of the circles that conform the
rounded triangles and the distance between its
centres finding that this relation is not general (Fig.
5a). For a particular radius there’s an optimal
distance l leading to an optimal β value different
than that for other radii, but at the optimal β value
for each radius the width of the complete PBG
doesn’t differ a lot. It varies from a minimum value
Δω/ωg=9.17% for r=0.350a and l=0.225a (β=0.643)
centred at ωg=0.377a/λ and a maximum value
Δω/ωg=11.47% for r=0.250a and l=0.400a (β=1.600)
centred at ωg=0.347a/λ. The best result enlarges the
complete PBG for plain triangles by a factor 1.75.
The values of the parameters that lead to the
maximum PBG correspond, for a gap centred at
wavelength of 1550 nm, to an experimental lattice
parameter a=537 nm and a critical distance Δ=54nm,
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so not only the width of the PBG is improved but
also the dielectric walls are increased by a factor 1.5.
The fact that with rounded vertices we’re able to
increase the air FF, increasing the critical distance at
the same time, gives a clue of the improvement of
the width of the gap. The inspection of the air FF at
the optimous parameteres for plain triangles
(FF=42.3%) and for rounded triangles (FF=65.3%)
highlights this fact.
To study the experimental feasibility of this
structure we represent the normalized width of the
complete gap as a function of the critical distance for
a wavelength of 1550 nm for different radii (Fig.
5b). It seems that, albeit there’s not a general
relation between the radii of the circles and the
distance between its centres, there’s an optimal
value of the critical distance at about 60 nm and
there’s also gaps with similar widths as in the case
of the plain triangles with a critical distance at about
100 nm. This means that this structure is easier to
fabricate not only because its rounded vertices but
also because the dielectric walls are bigger for the
same width of the gap.
Finally, in Fig. 6 we have represented the gap-map
of this structure for β=1 as a function of the radii of
the circles (and as a function of l since for the case
β=1 the distance between the centres of the circles is
equal to the radius) and as a function of the angle of
rotation for a rounded triangle with l=r=0.300a in
order to compare this structure with the TLTH. The
main difference between the two structures is that
making round the vertices of the triangles rises
slightly the photonic bands and enlarge both the
even mode and the odd mode gaps. The behaviour of
the gaps with respect to the angle of rotation of the
rounded triangles is, as expected, the same as for
plain triangles, i.e., the gap is maximum for θ=0º
and closes at θ=30º.
4. Summary
We have investigated the photonic band gaps of 2D-
PC slabs with several patterns basing our designs on
the basic triangular lattice of triangular air holes
finding that making round the vertices of the
triangles produces an enlargement of the complete
PBG by a factor 1.75 and increases the dielectric
walls by a factor 1.5 what makes this structure easier
to fabricate. The fact that with rounded vertices
we’re able to increase the air FF, increasing the
critical distance at the same time, gives a clue of the
improvement of the width of the gap.
Fig. 6 Photonic gap-maps between first and second bands
of the triangular lattice of rounded triangles for β=1 as a
function of a) the size of the triangle for a fixed angle of 0º
and as b) the angle of rotation for l/a=r/a=0.300.
Acknowledgements
L. J. Martínez and I. Prieto thank an I3P fellowship
from the CSIC and projects IST-2-511616-NOE
(PHOREMOST) and NMP4-CT-2004-500101
(SANDIE), Spain-Italy Integrated Action HI2004-
0367/IT2304, projects NAN2004-08843-C05-04,
TEC-2005-05781-C03-01, NAN2004-09109-C04-
01, CONSOLIDER-Ingenio 2010 (CSD2006-
00019).
0,10 0,15 0,20 0,25 0,30
0,20
0,25
0,30
0,35
0,40
0,45
0,50
0,55
Frequency (
ω
a/2
π
c)
l/a=r/a
even
odd
0 102030405060
0,20
0,25
0,30
0,35
0,40
0,45
0,50
0,55
Frequency (
ω
a/2
π
c)
θ
(deg)
even
odd
b)
a
)
