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In this paper, we investigate bi-periodic photonic crystal structures, including photonic crystal waveguides, bends and splitters. In these structures, periodicity of the two rows next to guiding region altered. It is shown that bi-periodic

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This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institutionand sharing with colleagues.Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third partywebsites are prohibited.In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further informationregarding Elsevier’s archiving and manuscript policies areencouraged to visit:http://www.elsevier.com/copyright
Author's personal copy
Bi-periodic photonic crystal Y-splitter
A. Ghaffari
, M. Djavid, M.S. Abrishamian
Department of Electrical Engineering, K.N. Toosi University of Technology, Tehran, Iran
a r t i c l e i n f o
Article history:
Received 15 July 2008Received in revised form2 February 2009Accepted 17 April 2009Available online 3 May 2009
PACS:
42.70.Qs42.82.
m
Keywords:
Bi-periodicBent waveguidesY-splitterPhotonic crystal
a b s t r a c t
In this paper, we investigate bi-periodic photonic crystal structures, including photonic crystalwaveguides, bends and splitters. In these structures, periodicity of the two rows next to guiding regionaltered. It is shown that bi-periodic structures can show much better performances compared to theirconventional counterparts. Then a new bi-periodic Y-splitter is presented. The transmission windowcorresponding to the conventional Y-splitter has been widened and increased in amplitude in the newbi-periodic Y-splitter.
&
2009 Elsevier B.V. All rights reserved.
1. Introduction
Photonic crystals (PCs) are artiﬁcially engineered materialswith periodic arrangement of dielectric constants. They prohibitpropagation of light for frequencies within the photonic bandgaps.The interest in photonic crystals and bandgap structures is stillincreasing and many optical devices based on PCs have beenproposed. Several planar photonic crystal components topologyoptimized for TE-polarized light, including 60
1
bends andY-splitters have been investigated theoretically [1] and experi-mentally [2]. A key optical component is the splitter which iswidely used in interferometers and (de)multiplexers [3–5].Optical power splitters or dividers are essential components inintegrated photonic devices used in ﬁber optic networks. Ideally,the splitters should divide an input power equally into outputports without signiﬁcant reﬂection or radiation losses, and shouldbe small in size [1]. The most straightforward Y-splitter designconsists of three single-defect (W
1
) waveguides joined together at120
1
, which leads to strong reﬂections and narrow-bandwidthoperation [2]. To have the output channels of the Y-splitter beparallel to the input channel, the two output channels have a 60
1
bend. Both the Y-junction and the 60
1
bend represent severediscontinuities in the PC waveguides (PCWs) and are potentiallyregions in which the single-mode operation might suffer fromlarge transmission losses. Therefore the discontinuities in theseregions must carefully designed.Most theoretical studies conducted so far have investigatedarrays of dielectric rods in air. The advantage of this model systemis that waveguides created by removing a single line of rods aresingle moded. Getting light to travel around sharp bendswith high transmission is then relatively straightforward, andT-junctions and Y-junctions have already been proposed [6].Unfortunately, the ‘rods in air’ approach does not providesufﬁcient vertical conﬁnement and is difﬁcult to implement formost practically useful device implementations in the opticalregime. Therefore the PCs that are suitable for planar integratedoptics are in the form of a lattice of circular holes in a dielectricmaterial such as Si or GaAs. Intensive investigations have beencarried out recently on these structures. For a two-dimensional(2D) triangular lattice of air holes in a dielectric medium, the 2Dphotonic bandgap (PBG) is larger than their corresponding squarelattice counterparts due to their more circular Brillouin zone.In the present paper we consider triangular lattice of air holes.The problem encountered for the ‘holes in dielectric’ approach isthat the single-defect PC waveguide becomes multi-moded, whichmakes it difﬁcult to get light to ﬂow efﬁciently around the circuitbecause higher-order modes are easily excited at discontinuities[7–9]. Single-mode PCWs were demonstrated by increasing thesize of air holes in two rows of the structure next tothe line defect[10], such single-mode structures are necessary to avoid un-desired signal distortion. Boscolo et al. showed that poortransmission can srcinate from modal mismatch at the junction
ARTICLE IN PRESS
Contents lists available at ScienceDirectjournal homepage: www.elsevier.com/locate/physe
Physica E
1386-9477/$-see front matter
&
2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physe.2009.04.025
Corresponding author.
E-mail address:
afshin.ghafari@ee.kntu.ac.ir (A. Ghaffari).Physica E 41 (2009) 1495–1499
Author's personal copy
of waveguides. Placing additional holes at the junction thenclearly increases the bandwidth and power transmission [1].The outline of the papers is as follows. In Section 2 we discussthe computational method used in our simulations. In Section 3we investigate transmission characteristics of bi-periodic straightwaveguides. In Section 4 we give numerical results of thetransmission spectrum for a bi-periodic bent waveguide. Finally,considering results of Sections 3 and 4, we propose a new bi-periodic Y-splitter and compare with the conventional ones.
2. Numerical analysis
The PC under study is a triangular lattice of air holes etched ina dielectric substrate with an effective dielectric constant of 7.9which corresponds to the effective permittivity of the silicon oninsulator (SOI) conﬁgurations commonly used in the realization of planar photonic crystal structures. The structure is assumed to betwo dimensional, i.e., the air holes are inﬁnitely long with a radiusof
r
¼
0.3
a
(
a
is the lattice constant). The structure is excited withTE polarization. Using a ﬁnite difference time domain (FDTD) codethe spectrum of the power transmission is obtained. The outputpower is calculated ateach port byintegrating the Poynting vectorover the cells of the output ports. The spectrum of the powertransmission is calculated in our FDTD code during 50,000 timestep. The FDTD mesh size and time step are
D
x
¼
D
y
¼
a
/24 and
D
t
¼
D
x
/2*
c
(
c
is speed of light in free space).Since we consider a ﬁnite structure here, the whole computa-tional domain is surrounded by perfectly matched layers (PMLs)to absorb the outgoing wave. By using plane wave expansion code,the PBG of the structure is found to be between the normalizedfrequencies 0.253
o
a
/
l
o
0.32 (Fig.1). In the upper half of the PBG,radiation and leaky modes coexist. Therefore, we mainlyconcentrate on the behavior of the structures in the lower partof the PBG.The PCW under study is connected to an input slab waveguidewith an embedded Huygens source, which radiates energy only inone direction. The length of the slabwaveguide is twice the latticeconstant, while the length of the PBG waveguide shown inFig. 2(a) is 15
a
. Coupling of a dielectric slab waveguide to a PBGwaveguide have been discussed previously. A dielectric slabwaveguide can be coupled efﬁciently in a wide frequency rangeto a PBG waveguide and vice versa [11].To reduce computational effort the effective index approxima-tion for the vertical direction and the 2D FDTD method for thein-plane propagation is applied, which is a good approximation.For a suitable choice of effective refractive index, 2D simulationsgive the same quantitative characteristics as 3D simulations. Thelattice constant and radius of holes are set to be 444nm and
r
¼
133.2nm, respectively.
3. Transmission characteristics of bi-periodic straight waveguides
A guided mode with linear dispersion is desired for integratedoptics applications. While within the transmission window of theconventional PCW, the fundamental TE mode suffers fromthe nonlinear dispersion and this nonlinear dispersion increasesthe PCW modal loss. The ﬂattening frequency is inversely relatedto the periodicity of the structure in the guiding direction. We canchange the ﬂattening frequency by altering the period of holes of the PCW in only two rows next to the guiding region.It has previously shown that by perturbing the period of thetwo rows next to the guiding region and by considering a newperiodicity (
a
0
) the nonlinear dispersion and modegap of PCWscan be overcome [12]. The frequency range inside the PBG forwhich no guided mode exists is referred to as the modegap [13]. Abi-periodic photonic crystal waveguide coupled to the slabwaveguide is shown in Fig. 2(a).Now we investigate the effect of changing the period (
a
0
) andradius(
r
0
) of the holes next to the guiding region. Fig. 2(b) and (c)shows the power transmission spectra for PCWs with ﬁvedifferent values of
a
0
while the radii remains unchanged (
r
0
¼
r
).It is seen that decreasing (increasing) the periodicity shift theedge of modegap toward higher (lower) frequencies. By reducing
a
0
we can increase the mode ﬂattening frequency and eventuallymove it out of the PBG. It can be shown theoretically that thefundamental mode of this structure with (
a
0
¼
0.7
a
) has lineardispersion over the entire PBG. This structure can have very smallloss coefﬁcient. Two reasons for obtaining such a low loss are thelinear dispersion of the guided mode and the bi-periodic nature of the structurewhich limits the couplingof the guided modes totheradiation modes of the structure [12].Also effects of changing the radius (
r
0
) of the holes next to theguiding region while
a
0
¼
a
is depicted in Fig. 3. It shows thepower transmission spectra for ﬁve different values of
r
0
whilethe lattice constant remains unchanged. It is clear that increase(decrease) in the radius of holes shifts the edge of modegaptoward higher (lower) frequencies.
ARTICLE IN PRESS
Fig. 1.
Band diagram of hexagonal PC lattice (
r
/
a
¼
0.3).
A. Ghaffari et al. / Physica E 41 (2009) 1495–1499
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4. Transmission characteristics of bi-periodic bent waveguides
We generalize the application of bi-periodic structures andshow other bi-periodic structures such as bends and splitters canalso be implemented. Based on previous section results, byselecting
a
0
/
a
¼
0.7 and
r
0
/
r
¼
1 (optimized values for two rowsnext to the guiding region) it is possible to have guiding with hightransmission coefﬁcient and linear dispersion diagram over thewhole photonic bandgap. First we simulate conventional andbi-periodic PC bends. Normalized transmission spectra forbi-periodic and conventional bends are depicted in Fig. 4(b). It isseen that bi-periodic photonic crystal bends can show betterperformances. The transmission window has been widened, andincreased in amplitude. The validity of our computation has beenveriﬁed by comparing our result with Ref. [14].
5. Bi-periodic Y-splitter
Now, based on previous results, we propose a new bi-periodicY-splitter. We show that this Y-splitter can also have much betterperformances compared to their conventional counterparts. TheY-splitters are symmetric with a smaller hole with radius of 0.7
r
positioned at the centre of the junction (Fig. 5), giving rise touniform splitting and high transmission efﬁciency.Transmission window corresponding to the conventionalY-splitter has been widened and increased in amplitude in thecorresponding bi-periodic Y-splitter as shown in Fig. 6. Besides,conventional Y-splitter has zero transmission with no guidedmode over a large frequency range (modegap of the conventionalY-splitter) in the useful part of the PBG. But there is a new hightransmission window in the spectrum of the bi-periodic Y-splitterat this lower part of the PBG (important part of the PBG).
6. Conclusions
In this paper, ﬁrst we investigated transmission properties of the guided mode in a bi-periodic PCW and we showed that bychanging the periodicity and radius of the two rows of air holesthat are adjacent to guiding region, transmission characteristics
ARTICLE IN PRESS
0.26 0.27 0.28 0.29 0.3 0.3100.20.40.60.81
T r a n s m i t i o n C o e f f i c i e n t
a'/a = 0.7a'/a = 0.9a'/a = 1
0.26 0.27 0.28 0.29 0.3 0.3100.20.40.60.81
T r a n s m i t i o n C o e f f i c i e n t
a'/a = 1.2a'/a = 1.1a'/a = 1
Normalized Frequency a/
λ
Normalized Frequency a/
λ
aa'2r'2r
Fig. 2.
(a) Coupling of a slab waveguide to a bi-periodic photonic crystal waveguide.
r
0
and
a
0
are radius and period of the air holes next to the guiding region, (b) theevolution of normalized transmission as
a
’ changes, increased periods (
a
0
/
a
¼
1.1, 1.2) and (c) decreased periods (
a
0
/
a
¼
0.7, 0.9).
0.26 0.27 0.28 0.29 0.3 0.3100.20.40.60.81Normalized Frequency a/
λ
T r a n s m i t i o n C o e f f i c i e n t
r'/r = 1r'/r = 1.1r'/r = 1.2r'/r = 1r'/r = 0.9r'/r = 0.8
0.26 0.27 0.28 0.29 0.3 0.3100.20.40.60.81
T r a n s m i t i o n C o e f f i c i e n t
Normalized Frequency a/
λ
Fig. 3.
Normalized transmission as the radius of the air holes in the two rows nextto the guiding region changes (a) increased radius (
r
0
/
r
¼
1.1,1.2) and (b) decreasedradius (
r
0
/
r
¼
0.8,0.9).
A. Ghaffari et al. / Physica E 41 (2009) 1495–1499
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Author's personal copy
can be improved. Based on these results we chose optimized valuefor
r
0
,
a
0
to have highest guiding bandwidth, transmissionefﬁciency and linear dispersion within the PBG. Then we general-ized application of bi-periodic structures to bent waveguides andY-splitters. We have presented a new bi-periodic Y-splitter.Transmission window of this new Y-splitter has been widened
ARTICLE IN PRESS
Fig. 4.
(a) Bi-periodic bends and (b) the transmission coefﬁcient of the bi-periodic and conventional bends.
Fig. 5.
(a) Conventional Y-splitter and (b) bi-periodic Y-splitter.
Fig. 6.
Normalized transmission spectra of the bi-periodic and conventional Y-splitter (summation of the two outputs).
A. Ghaffari et al. / Physica E 41 (2009) 1495–1499
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