systematic analysis of hole size, hole-type and rows shifting on slowight characteristics of photonic crystal waveguides with ring-shapedoles
ulya Bagci1, Baris Akaoglu ∗
nkara University, Faculty of Engineering, Department of Engineering Physics, 06100 Besevler, Ankara, Turkey
r t i c l e i n f o
rticle history:eceived 7 June 2013ccepted 5 November 2013vailable online xxx
a b s t r a c t
A photonic crystal waveguide embedded in silica is proposed and the effects of number of defect rowsadjacent to the line-defect, number of rows shifted in transverse direction to the light propagation and thetypes of holes in these rows on slow light properties are investigated without changing the line-width byplane-wave expansion method. We observe that the structure with one row of ring-shaped holes exhibits
eywords:hotonic crystal waveguidelow lightroup indexroup velocity dispersionropagation loss
better slow light properties than the structure with two and three innermost rows of ring-shaped holeswhen outer and inner radius of the holes are considered as free parameters. Shifting the second innermostrows of holes is found to be preferable than shifting the second innermost rows of rings. Besides, shiftingthe second and third innermost rows together does not make considerable enhancement on the slowlight properties as shifting only the second innermost rows, no matter the shifting holes are ring-shapedholes or only holes.
. Introduction
Slow light has become an intense research area due to its greatotential for applications in optical buffering, in nonlinear optics asell as in many applications requiring various types of time domainrocessing of optical signals and for the miniaturization of opticalevices [1,2]. Slow light can be practically realized in photonic crys-al waveguides (PCW) with many advantages as room-temperatureperation, easy on-chip integration, tailorable wide bandwidthnd dispersion-free propagation [1]. Line-defect photonic crystalaveguides (PCW) have been the most straightforward structuralesigns to achieve tailorable slow light over a wide wavelengthange via engineering structural dispersion properties of these sys-ems.
The dispersion band of PCW structures consists of index-guidednd gap-guided modes, where anticrossing occurs at the intersect-ng points of these modes [3]. The anticrossing point is greatlynfluenced by the structure adjacent to the waveguide [4,5]. By tail-
Please cite this article in press as: F. Bagci, B. Akaoglu, A systeslow light characteristics of photonic crystal waveguides with rhttp://dx.doi.org/10.1016/j.ijleo.2013.11.032
ring structure properties, the anticrossing point can be shifted andhe dispersion can be controlled. Hence, flat band slow light regions
can be positioned away from the highly dispersive Brillouin zoneedges.
All approaches for tailoring the dispersion curves aim to havea high and constant group-index (nG) in a wide bandwidth withlow group velocity dispersion (GVD) properties. However, group-index and bandwidth properties are inversely proportional andthe bandwidth can only be extended at the cost of group index.Therefore, a figure-of-merit called normalized delay-bandwidthproduct (NDBP) is defined, which is proportional to the groupindex-bandwidth product per unit length and the NDBP is aimedto be maximized for many of slow light applications. Various dis-persion engineering methods have been developed to achieve thispurpose with very low GVD [2–26]. These methods can be sub-stantially grouped into two categories. The first one manipulatesthe speed of light using chirped waveguide structures in direc-tional coupler geometries, engineering dispersion-compensatedslow light [6,7]. The second one is based on obtaining nearly zerodispersion by changing the waveguide properties and it is investi-gated by altering the waveguide width [3,4,8], changing the size ofholes [1,5,9–11], merging coupled-cavities [12], introducing ring-shaped holes [13–17], infiltrating microfluidic into air holes [18,19],shifting the rows of holes adjacent to the line-defect in lateral[20–22], transverse [23–25] or both directions [26] and adopting
matic analysis of hole size, hole-type and rows shifting oning-shaped holes, Optik - Int. J. Light Electron Opt. (2014),
oblique lattice structures [17,21,24]. Among these methods, chang-ing the radius and position of the rows of holes adjacent to theline-defect has been the most applied and general procedures forgenerating slow light [1,9–11,13–17,20–26].
In this study, we propose a photonic crystal waveguide withing-shaped holes and focus on the systematic identification ofhe effects of innermost rows on slow light properties, rather thanolely optimizing the PCW structure. In a systematic study of flatand slow light in PCWs with different radii of innermost holes, itas been shown that for the bands that are close to the dielectricontinuum, the field distribution reaches up to the fourth row [12].n this study, by utilizing ring-shaped holes, the guided bands arebtained to be down-facing and close to the dielectric continuum.herefore, we investigate the effect of the number of innermostows on slow light properties by introducing ring-shaped holesp to the third row. In addition, we shift the innermost rows inhe direction transverse to light propagation. Hereby, we explorehe effects of the number of shifting rows and the type of holes inhese rows on slow light properties. It should be noted that PCWsith enlarged line-widths may lead to multimode operation [27]hereas the scattering losses increase as the waveguide width is
educed [28]. In order to avoid such problems and unambiguouslyustify the results, we keep the first rows constant and shift theecond and both the second and third rows adjacent to the line-efect, without changing the line-width. To best of our knowledge,he effects of shifting of innermost rows in transverse directiono the light propagation on slow light properties independent ofine-width have not been reported.
. Waveguide design and modeling
The PCW structure proposed in this study is illustrated in Fig. 1.t consists of a triangular lattice of silica (n = 1.45) holes in a high-�i dielectric material embedded in silica. We preferred an oxide-lad structure in this study because oxide-clad structures offer aymmetric configuration as membrane structures. However, theyre more robust, stable, unaffected from environmental contami-ation and can be easily integrated with electronic components on
single chip [29]. In addition, they are CMOS-compatible and areery suitable for Mach-Zehnder optical modulator (MZM) designs30]. The line-defect waveguide is created by removing a row ofoles in the center along the � –K direction. Ring-shaped holes are
ormed in the place of the holes adjacent to the line-defect. Ringsre introduced up to the third rows adjacent to the line-defect.or each PCW structure and various outer radii of the rings andackground holes, a scan of the inner radii is performed. To furthernalyze the effect of the number of innermost rows and the typesf holes in these rows, for the selected outer and inner radii com-ination of ring-shaped holes, second and both second and thirdows adjacent to the line-defect are shifted toward the waveguideenter. The shifts of second and, both the second and third rows
Please cite this article in press as: F. Bagci, B. Akaoglu, A systeslow light characteristics of photonic crystal waveguides with rhttp://dx.doi.org/10.1016/j.ijleo.2013.11.032
re denoted with “s2” and “s2, s3” symbols, respectively. The shiftsf rows toward the waveguide center are remarked with a ‘+’ signnd those from the waveguide center are remarked with a ‘−’ sign.
ig. 1. Schematic picture of the photonic crystal waveguide structure. Second rowhifts are denoted as s2 and third row shifts are denoted as s3. Shifts are in theemarked direction transverse to the light propagation. The inner radius of the ring-haped holes is denoted as “r” and the outer radius of the ring-shaped holes and theadius of the background holes are denoted as “rout”.
PRESS xxx (2014) xxx–xxx
2.1. Figures of merit
Normalized delay-bandwidth product is one of the main figuresof merit describing the slow light performance. NDBP is defined as:
NDBP = 〈nG〉 × �ω
ω0(1)
where 〈nG〉 is the average group index for the frequency bandwidth�ω [1] and ω0 denotes the central normalized frequency. The aver-age group indices and bandwidths are determined within the 10%range of the minimum group index.
Large group velocity dispersion (GVD) effects lead to large signaldistortion during pulse propagation [31]. Therefore a high NDBPwith a sufficiently high nG value should be accompanied with alow GVD parameter. The GVD parameter, D� can be given as:
D� = −2�c
�2× ∂2k
∂ω2(2)
where c is the speed of light in vacuum, � is the wavelength and kis the wave number of light.
The proposed PCW structure supports one even and one oddmode in the photonic band gap below the silica light line. Thepolarization is assumed to be TE-even in all the simulations sinceefficient coupling can only be realized for evenly lateral modes [4].The slow light regions below the silica light line are investigated.In our calculations, we have utilized MIT photonic-bands package(MPB) that uses plane-wave expansion (PWE) method with peri-odic boundary conditions [32]. The group indices are obtained byusing the built-in function of MPB. To approximately realize 3Dcalculations in 2D, an effective index medium with a neff of 2.98 isemployed for the background medium [33].
3. Results and discussions
3.1. Effects of radius change
First we investigated the effects of modification of the innerradius of holes adjacent to the line-defect on slow light properties.Dispersion properties of the PCW with only one row of ring-shapedholes adjacent to the line-defect is calculated for rout = 0.30a asshown in Fig. 2. As the inner radius (r) is increased, the bandsshift to lower frequencies (Fig. 2(a)). Both index-guided and gap-guided modes are observed to be influenced but the shift of theindex-guided modes is slightly weaker. If it is reminded that theinteraction of index-guided modes with surroundings of the line-defect is relatively weaker, the slightly stronger shift in frequencyof the gap-guided modes is reasonable [3]. As a result, the anti-crossing point shifts to smaller wave-numbers. Furthermore, as theinner radius is increased, the nG–frequency curves change from ashoulder to a step shape. At r = 0.19a, 〈nG〉 = 18.76 with a 15.96 nmbandwidth and NDBP = 0.1935. When r is increased (r = 0.21a), 〈nG〉decreases together with a relatively enhanced increase in band-width, which consequently causes the NDBP to be relatively largeas 0.259.
Fig. 3 demonstrates the nG characteristics for the one, two andthree rows of ring-shaped holes adjacent to the line-defect for opti-mized r values at various outer radii of the ring-shaped holes from0.30a to 0.38a. In this analysis, step-like nG curves are sought due totheir high nG and low GVD character. As the number of introducedrows with ring-shaped holes is increased, the bands approach todielectric-continuum. Group indices for the three structures exhibitsimilar behavior: first, as the outer radius is increased, the opti-
matic analysis of hole size, hole-type and rows shifting oning-shaped holes, Optik - Int. J. Light Electron Opt. (2014),
mized r values become smaller. Second, the nG values increase andbandwidths decrease as the outer radius of the rings is increased.These findings are consistent with the studies of slow light employ-ing up to two innermost rows of ring-shaped holes [15–17].
ARTICLE IN PRESSG ModelIJLEO-54136; No. of Pages 6
F. Bagci, B. Akaoglu / Optik xxx (2014) xxx–xxx 3
Fig. 2. (a) Dispersion and (b) group index characteristics of the PCW with only one row of ring-shaped holes adjacent to the line-defect for various inner radii. rout is set to0.30a.
F hapeda
idtrbiSeot
Fitti
Fv
ig. 3. Group index characteristics of the PCW structure with (a) one row of ring-sdjacent to the line-defect for various outer radii and optimized inner radii.
When Fig. 3(a)–(c) are compared, introduction of additionalnnermost rows of ring-shaped holes is found to be leading aecrease in nG. The PCW with one row of ring-shaped holes exhibitshe best slow light properties in terms of group-index and also fab-ication concerns. For this structure, the average nG values changeetween 17.63 and 33.93 over 22.72–10.57 nm bandwidth center-
ng at 1550 nm and NDBPs change between 0.2022 and 0.2592.ince the PCW structure with three rows of ring-shaped holesxhibits smaller group index values than those with two rows orne row of ring-shaped holes, we do not further analyze this struc-ure in the next sections.
Fig. 4 depicts the band shape structure of the guided bands ofig. 3(a) as the outer radius is increased from 0.30a to 0.38a. As rout
ncreases, both the bandgap bottom edge and guided modes shift
Please cite this article in press as: F. Bagci, B. Akaoglu, A systeslow light characteristics of photonic crystal waveguides with rhttp://dx.doi.org/10.1016/j.ijleo.2013.11.032
o higher frequencies. Since all the radii of the lattice holes change,he shift of the bandgap bottom edge mode is reasonable. However,f it is closely looked, it is noticed that the shift in the frequency
ig. 4. Dispersion diagram of the guided and the bandgap bottom edge modes forarious rout values. The r values correspond to the PCW structures given in Fig. 3(a).
holes, (b) two rows of ring-shaped holes and (c) three rows of ring-shaped holes
of the guided bands are smaller in comparision to the bandgapbottom edge bands. Therefore the frequency interval between theedge and the guided modes decreases. As the band gets close to itscorresponding edge mode, the interaction between them becomesstronger, leading to a flatter band [13]. This observation explainsthe enhanced nG behavior in Fig. 3 for rout increase. The anticross-ing point shifts to smaller wave-numbers as rout increases, thatremoves the slow-light region more from the Brillouin-zone edge.
3.2. Effects of row shift
3.2.1. Group-index analysisShifting of innermost rows has been reported to be more prefer-
able than changing the size of holes in terms of fabrication control[20–26]. We also analyze the effect of shifting the rows of holesadjacent to the line-defect in the transverse direction withoutchanging the width of the line-defect, that is, without shifting thefirst innermost rows of ring-shaped holes. We apply these shiftsfor the two PCW structures given in Fig. 3. In these analyses, rout isselected to be 0.36a with r = 0.15a.
In the PCW structure with one row of ring-shaped holes(Fig. 3(a)), the second innermost rows and both the second andthird innermost rows are approached to the line-defect and the nGcharacteristics are determined for various shifts as shown in Fig. 5.It is clearly observed that the increase in 〈nG〉 values is weak whenboth the second and the third rows are shifted toward the line-defect as shown in Fig. 5(b). Even for s2 and s3 values, greater than0.04a, there is no considerable change in 〈nG〉 and �ω (Table 1(c)).However, shifting only the second innermost rows lead to consid-erably large increase in 〈nG〉 values (Table 1(a)). As the magnitudesof shifting increase, the group index-frequency curves change froma step shape to U-shape in both Fig. 5(a) and (b).
matic analysis of hole size, hole-type and rows shifting oning-shaped holes, Optik - Int. J. Light Electron Opt. (2014),
Fig. 6 shows the nG characteristics for the PCW structurewith two innermost rows of ring-shaped holes when the secondinnermost rows and, both the second and third innermost rowsare approached to the line-defect. When only second rows are
ARTICLE IN PRESSG ModelIJLEO-54136; No. of Pages 6
4 F. Bagci, B. Akaoglu / Optik xxx (2014) xxx–xxx
F ed ho(
antie(
ssiswaor〈viωwsdsNa
taibmcls
TAcr
ig. 5. Group index characteristics of the PCW structure with one row of ring-shapb) both second and third innermost rows toward the line-defect.
pproached, there is almost a linear increase in the minimum ofG (nGmin
) from 19.80 (s2 = 0) to 36.05 (s2 = 0.08a). However, ashe third rows are also approached to the line-defect, the increasen nGmin
is small. Similar to Fig. 5(b), the 〈nG〉 and �ω appar-ntly remain nearly the same from s2 = s3 = 0.04a to s2 = s3 = 0.08aTable 1(d)).
The average group index, bandwidth and NDBPs of the PCWtructures with shifted second innermost rows of holes/ring-haped holes that are depicted in Figs. 5(a) and 6(a) are presentedn Table 1(a and b). For these PCW structures, as the magnitude of2 shift increases, the average group index increases and the band-idth decreases. However, due to different rates of change in 〈nG〉
nd �ω in Table 1(a), the NDBPs do not show totally increasingr decreasing behavior. For the PCW structure with one innermostow of ring-shaped holes (Table 1(a)), as s2 approaches 0.08a, thenG〉 reaches of 143.2 whereas the bandwidth decreases down toalues of 1.25 nm. As s2 increases from 0.06a to 0.08a, although 〈nG〉s considerably increased, the decrease in �ω and the increase in
0 in large rates lead to a decrease in NDBP. For the PCW structureith two innermost rows of rings (Table 1(b)), in contrast to the
tructure given in Table 1(a), the NDBP values exhibit continuouslyecreasing behavior as s2 increases from 0 to 0.08a. Consequently,hifting of second innermost row of rings has a negative effect onDBP values. However, this structure might be preferred in somepplications where bandwidth properties are more essential.
In a study of PCW with one innermost row of ring-shaped holes,he average group index was 82 in a 0.93 nm bandwidth centeredt 1550 nm for an outer radius of 0.365a [16]. In this study, at a sim-lar outer radius, slab equivalent index and nG values, the slow lightandwidth is considerably increased by shifting the second inner-
Please cite this article in press as: F. Bagci, B. Akaoglu, A systeslow light characteristics of photonic crystal waveguides with rhttp://dx.doi.org/10.1016/j.ijleo.2013.11.032
ost rows of holes in transverse direction (Table 1(a)). The latticeonstant is required to be 399.4 nm in order that 1550 nm wave-ength is centered to the flat band region for the PCW structure with2 = 0.08a (Table 1(a)). Thereby, the width of the ring-shaped hole
able 1verage group indices (〈nG〉), bandwidths around 1550 nm (�ω) and normalized delay-bontaining (a) one row of rings, (b) two rows of rings, and for the PCW structure with vaings.
(a) s2 (Fig. 5(a)) 〈nG〉 �ω (nm) NDBP
0 29.49 9.55 0.1817
0.02a 39.71 7.38 0.2464
0.04a 56.58 5.54 0.2022
0.06a 86.81 4.30 0.2369
0.08a 143.19 1.25 0.1189
(b) s2 (Fig. 6(a)) 〈nG〉 �ω (nm) NDBP
0 20.10 23.04 0.3083
0.02a 24.28 19.55 0.3077
0.04a 28.31 11.00 0.2013
0.06a 33.51 7.78 0.1682
0.08a 37.17 6.17 0.1478
les adjacent to the line-defect for various shifts of (a) second innermost rows and
is calculated as 83.9 nm. Silica filling of this region can be realizedexperimentally by spin-on-glass or silica chemical vapor depositionprocesses [34,35].
The average group index, bandwidth and NDBPs of the PCWstructures, where both the second and third innermost rows ofholes/ring-shaped holes are shifted, are presented in Table 1(c andd). These two PCW structures also show common behaviors: as themagnitude of s2 and s3 shifts increase, in general, the average groupindices increase and the bandwidths decrease. However, when thes2 and s3 shifts increase from 0.06a to 0.08a, a distinct behaviorstands out, that is, the 〈nG〉 values decrease. This behavior will beinvestigated in more detail in the next section.
When the left column of Table 1 is compared to right column, itis observed that the shift of the only second innermost rows (only s2shift) improves slow light properties stronger than the shift of theboth second and third innermost rows (s2 and s3 shifts). Shiftingof third rows has rarely been used in the literature for slow lightgeneration in PCWs [10,36]. A constant group index of ∼40 hasbeen measured for a PC slab on SOI substrate by shifting the thirdrows in lateral direction to light propagation by an amount of 0.2a(90 nm) [10]. We achieve nearly the same group index from 2D PWEcalculations by shifting the second and third rows with a ten timessmaller amount (0.02a, Table 1(c)) in transverse direction to lightpropagation.
3.2.2. Dispersion characteristics of the guided bandsWe considered the dispersion characteristics of the PCW struc-
tures in Figs. 5(a), 6(a) and 5(b) in Fig. 7(a)–(c), respectively to betterexplain the slow light characteristics given in Table 1. In all the casesof Fig. 7, as the rows shift more to the line-defect, the frequencies
matic analysis of hole size, hole-type and rows shifting oning-shaped holes, Optik - Int. J. Light Electron Opt. (2014),
of the guided modes increase because the Hz field concentratesmore in the air region and the average index probed by the fielddecreases. However, the frequencies of index-guided modes andband gap bottom edge modes remain almost the same. Since the
andwidth products (NDBPs) for the PCW structure with various second rows shiftrious second and third rows shift containing (c) one row of rings, (d) two rows of
ARTICLE IN PRESSG ModelIJLEO-54136; No. of Pages 6
F. Bagci, B. Akaoglu / Optik xxx (2014) xxx–xxx 5
Fig. 6. Group index characteristics of the PCW structure with two rows of ring-shaped holes adjacent to the line-defect for various shifts of (a) second innermost rows and(b) both second and third innermost rows toward the line-defect.
F the Po us shiv
pfb
iasblsnkTt
ettitalttTfatracianb
structure and 16 ps/(mm-nm) for s2 = 0.06a structure, respectively.It should be noted that D� parameter can be further decreased to2 ps/(mm-nm) at the expense of bandwidth. If 10,000(a/2�c2) is
ig. 7. Dispersion diagram of the guided and the bandgap bottom edge modes forf second innermost rows, (b) two innermost rows of ring-shaped holes and varioarious shifts of both second and third innermost rows.
osition of the first rows adjacent to the line-defect remains fixedor the shifts, and the background lattice holes do not change, thisehavior appears reasonable.
First, the effect of the shape of the innermost holes is takennto account, when the second innermost rows are holes (Fig. 7(a))nd, ring-shaped holes (Fig. 7(b)). When the second rows are ring-haped holes, the cut-off frequency of the bands is lower, as cane expected from the higher average refractive index. Since the
ow k part of the dispersion curves has the same character, thelope of the band curvature is higher for Fig. 7(b), which makes theG values smaller. In addition, in Fig. 7(b) the band curve around
= 0.39(2�/a) starts to face a little upward as s2 becomes 0.04a.his behavior is more prominent for larger s2 values which altershe nG curves from a step shape to U-shape in Fig. 6(a).
Second, the effect of the number of shifting innermost rows isxamined, when the second innermost rows (Fig. 7(a)) and, bothhe second and third innermost rows (Fig. 7(c)) are approachedo the line-defect. Since at the Brillouin-zone boundary, the modes very little affected from the changes in the third rows adjacento the line-defect [13], the cut-off frequency values of Fig. 7(a)nd (c) are observed to be almost the same (k = 0.5(2�/a)). Simi-ar to shifts of s2 = 0.04a, s2 = 0.06a and s2 = 0.08a given in Fig. 7(b),he band region around k = 0.39(2�/a) is found to be facing a lit-le upward in Fig. 7(c) as the s2 and s3 shifts increase from 0.04a.his kind of modifications in the bands has also been observedor the third innermost rows shift in the lateral direction [36]nd the second innermost rows shift in the transverse direc-ion [25]. The frequency of the band at the Brillouin zone edgeegion (∼k = 0.5(2�/a)) does not increase as much as the regionround k = 0.39(2�/a) and therefore the slope of the dispersionurve increases. The upward facing of the band leads to chang-
Please cite this article in press as: F. Bagci, B. Akaoglu, A systeslow light characteristics of photonic crystal waveguides with rhttp://dx.doi.org/10.1016/j.ijleo.2013.11.032
ng of the nG–frequency curves from a step shape to a U-shapes can be seen in Fig. 5(b). This upward facing is more promi-ent for the s2 and s3 shifts of 0.08a and the 〈nG〉 is found toe decreasing from s2 = s3 = 0.06a to s2 = s3 = 0.08a in Table 1(c).
CW structure with (a) one innermost row of ring-shaped holes and various shiftsfts of second innermost rows and (c) one innermost row of ring-shaped holes and
This observation indicates that as the shifting amount toward thewaveguide center increases, the group index curves turn from step-like to U-shape, which would decrease the available bandwidth ofGVD.
3.2.3. Group velocity dispersion characteristicsThe group velocity dispersion characteristics are examined for
various shifts of second rows in the PCW with one row of ring-shaped holes, since these structures given in Table 1(a) are morepromising in terms of slowness of light. The GVD curves withs2 = 0.02a, s2 = 0.04a and s2 = 0.06a are shown in Fig. 8. In the slowlight bandwidth (±10% nGmin
), the maximum GVD parameters, D�
are determined to be 2 ps/(mm-nm) for s2 = 0 structure, 4 ps/(mm-nm) for s2 = 0.02a structure and 9.5 ps/(mm-nm) for s2 = 0.04a
matic analysis of hole size, hole-type and rows shifting oning-shaped holes, Optik - Int. J. Light Electron Opt. (2014),
Fig. 8. Group velocity dispersion characteristics of the PCW structure with oneinnermost row of ring-shaped holes and various shifts of the second innermost rows.
egarded as ultralow GVD [16], according to Eq. (2) and for a lat-ice constant of 399.4 nm and a central wavelength of 1.55 �m, thisorresponds to a GVD parameter of 55.4 ps/(mm-nm). Inspectingig. 8, all the structures show ultralow GVD in a bandwidth of morehan 10 nm spectral range.
As the amount of shift increases, the D� increases with thencrease in nG. It is revealed both theoretically [37] and exper-mentally [38] that the scattering loss scales inversely with thequared group velocity. The PCW structure with 0.08a shiftedecond innermost rows gives a maximum GVD parameter valuef 40 ps/(mm-nm) in the slow light bandwidth of 1.25 nm. Dueo its comparatively high GVD character, it is not shown inig. 8.
. Conclusions
In summary, the effects of the number of defect rows adja-ent to the line-defect and the effects of hole-types on slow lightroperties are discussed in detail both for non-shifted row andhifted-row PCW configurations. The structure with one innermostow of ring-shaped holes is preferable to two innermost rows andhree innermost rows of ring-shaped holes in terms of slow lightroperties. Smaller amount of shifting of the second innermostows of the PCW structure with one innermost row of rings, inomparison to shifting of the second innermost row of the PCWtructure with two innermost rows of rings, is more effective. Whilehifting the second row increases the group index substantially forhe PCW structure with one innermost row of rings, for the PCWtructure with two innermost rows of rings, shifting the secondnnermost row leads an almost linear increase in the group index.esides, shifting the second and third innermost rows together doesot enhance the slow light properties for the shifts greater than.06a due to the upward facing of the guided band below the silica
ight line.
cknowledgments
We gratefully acknowledge Ankara University for the finan-ial support (BAP project 12B4343011). Part of the numericalalculations reported in this paper is performed at TUBITAKLAKBIM, High Performance and Grid Computing Center (TRGrid-Infrastructure).
eferences
[1] T. Baba, Slow light in photonic crystals, Nat. Photon. 2 (2008) 465–473.[2] T.F. Krauss, Why do we need slow light, Nat. Photon. 2 (2008) 448–450.[3] N. Natomi, K. Yamada, A. Shinya, T. Takahashi, C. Takahashi, I. Yokohama,
Extremely large group-velocity dispersion of line-defect waveguides in pho-tonic crystal slabs, Phys. Rev. Lett. 87 (2001) 253902-1–253902-4.
[4] A.Y. Petrov, M. Eich, Zero dispersion at small group velocities in photonic crystalwaveguides, Appl. Phys. Lett. 85 (2004) 4866–4868.
[6] D. Mori, T. Baba, Wideband and low dispersion slow light by chirped photoniccrystal coupled waveguide, Opt. Express 13 (2005) 9398–9408.
[7] T. Baba, T. Kawasaki, H. Sasaki, J. Adachi, D. Moki, Large delay bandwidth prod-uct and tuning of slow light pulse in photonic crystal coupled waveguide, Opt.Express 16 (2008) 9245–9253.
Please cite this article in press as: F. Bagci, B. Akaoglu, A systeslow light characteristics of photonic crystal waveguides with rhttp://dx.doi.org/10.1016/j.ijleo.2013.11.032
[8] M.D. Settle, R.J.P. Engelen, M. Salib, A. Michaeli, L. Kuipers, T.F. Krauss, Flat-band slow light in photonic crystals featuring spatial pulse compression andterahertz bandwidth, Opt. Express 15 (2007) 219–226.
[9] F.H. Wang, J. Ma, C. Jiang, Dispersionless slow wave in novel 2-D photoniccrystal line-defect waveguides, J. Lightwave Technol. 26 (2008) 1381–1386.
[
[
PRESS xxx (2014) xxx–xxx
10] S. Kubo, D. Mori, T. Baba, Low-group-velocity and low-dispersion slow light inphotonic crystal waveguides, Opt. Lett. 32 (2007) 2981–2983.
11] H. Kurt, K. Üstün, L. Ayas, Study of different spectral regions and delay band-width relation in slow light photonic crystal waveguides, Opt. Express 18 (2010)26965–26977.
12] K. Üstün, H. Kurt, Ultra slow light achievement in photonic crystals by mergingcoupled cavities with waveguides, Opt. Express 18 (2010) 21155–21161.
13] A. Saynatjoki, M. Mulot, J. Ahopelto, H. Lipsanen, Dispersion engineering ofphotonic crystal waveguides with ring-shaped holes, Opt. Express 15 (2007)8323–8328.
14] A. Saynatjoki, M. Mulot, K. Vynck, D. Cassagne, J. Ahopelto, H. Lipsanen, Prop-erties, applications and fabrication of photonic crystals with ring-shaped holesin silicon-on-insulator, Photon. Nanostruct. Fundam. Appl. 6 (2008) 42–46.
15] L. Dai, C. Jiang, Ultrawideband low dispersion slow light waveguides, J. Light-wave Technol. 27 (2009) 2862–2868.
16] J. Hou, D. Gao, H. Wu, R. Hao, Z. Zhou, Flat band slow light in symmetricline defect photonic crystal waveguides, IEEE Photon. Technol. Lett. 21 (2009)1571–1573.
17] Y. Zhai, H. Tian, Y. Ji, Slow light property improvement and optical buffer capa-bility in ring-shape-hole photonic crystal waveguide, J. Lightwave Technol. 29(2011) 3083–3090.
18] M.E. Heidari, C. Grillet, C. Monat, B.J. Eggleton, Dispersion engineering of slowlight photonic crystal waveguides using microfluidic infiltration, Opt. Express17 (2009) 1628–1635.
19] S.Y. Lü, J.L. Zhao, D. Zhang, Flat band slow light in asymmetric photonic crystalwaveguide based on microfluidic infiltration, Appl. Opt. 49 (2010) 3930–3934.
20] R. Hao, E. Cassan, H. Kurt, J. Hou, X. Le Roux, D. Marris-Morini, L. Vivien, D.Gao, Z. Zhou, X. Zhang, Novel kind of semislow light photonic crystal wavegui-des with large delay-bandwidth product, IEEE Photon. Technol. Lett. 22 (2010)1041–1135.
21] R. Hao, E. Cassan, X.L. Roux, D. Gao, V.D. Khanh, L. Vivien, D. Marris-Morrini,Z. Zhang, Improvement of delay-bandwidth product in photonic crystal slow-light waveguides, Opt. Express 18 (2010) 16309–16319.
22] Y. Hamachi, S. Kubo, T. Baba, Slow light with low dispersion and nonlinearenhancement in a lattice-shifted photonic crystal waveguide, Opt. Lett. 34(2009) 1072–1074.
23] J.T. Li, T.P. White, L. O’Faolain, A. Gomez-Iglesias, T.F. Krauss, Systematic designof flat band slow light in photonic crystal waveguides, Opt. Express 16 (2008)6227–6232.
24] F.C. Leng, W.Y. Liang, B. Liu, T.B. Wang, H.Z. Wang, Wideband slow light anddispersion control in oblique lattice photonic crystal waveguides, Opt. Express18 (2010) 5707–5712.
25] J. Wu, Y.P. Li, C. Peng, Z.Y. Wang, Wideband and low dispersion slow light inslotted photonic crystal waveguide, Opt. Commun. 283 (2010) 2815–2819.
26] J. Liang, L.-Y. Ren, M.-J. Yun, X. Han, X.-J. Wang, Wideband ultraflat slow lightwith large group index in a W1 photonic crystal waveguide, J. Appl. Phys. 110(2011) 063103-1–063103-6.
27] M. Galli, D. Bajoni, M. Patrini, G. Guizzetti, D. Gerace, L.C. Andreani, M. Belotti,Y. Chen, Single-mode versus multimode behavior in silicon photonic crystalwaveguides measured by attenuated total reflectance, Phys. Rev. B 72 (2005)125322-1–125322-10.
28] S.A. Schulz, L. O’Faolain, D.M. Beggs, T.P. White, A. Melloni, T.F. Krauss, Disper-sion engineered slow light in photonic crystals: a comparison, J. Opt. 12 (2010)104004-1–104004-10.
29] T.P. White, L. O’Faolain, J. Li, L.C. Andreani, T.F. Krauss, Silica-embedded siliconphotonic crystal waveguides, Opt. Express 16 (2008) 17076–17081.
30] H.C. Nguyen, S. Hashimoto, M. Shinkawa, T. Baba, Compact and fast photoniccrystal silicon optical modulators, Opt. Express 20 (2012) 22465–22474.
31] R.J.P. Engelen, Y. Sugimoto, Y. Watanabe, J.P. Korterik, N. Ikeda, N.F. Van Hulst,K. Asakawa, L. Kuipers, The effect of higher-order dispersion on slow light prop-agation in photonic crystal waveguides, Opt. Express 14 (2006) 1658–1672.
32] S.G. Johnson, J.D. Joannopoulos, Block-iterative frequency-domain methods forMaxwell’s equations in a planewave basis, Opt. Express 8 (2001) 173–190.
33] M. Qui, Effective index method for heterostructure-slab-waveguide based two-dimensional photonic crystals, Appl. Phys. Lett. 81 (2002) 1163–1165.
34] L. O’Faolain, D.M. Beggs, T.P. White, T. Kampfrath, K. Kuipers, T.F. Krauss, Com-pact optical switches and modulators based on dispersion engineered photoniccrystals, IEEE Photon. J. 2 (2010) 404–414.
35] J.H. Moon, S. Yang, W. Dong, J.W. Perry, A. Adibi, S.-M. Yang, Core–shelldiamond-like silicon photonic crystals from 3D polymer templates created byholographic lithography, Opt. Express 14 (2006) 6297–6302.
36] A. Hosseini, D. Kwong, H. Subbaraman, R.T. Chen, Low dispersion slow light insilicon-on-insulator photonic crystal waveguide, Proc. SPIE 7609 (2010), pp.76091A-1–76091A-7.
matic analysis of hole size, hole-type and rows shifting oning-shaped holes, Optik - Int. J. Light Electron Opt. (2014),
37] S. Hughes, L. Ramunno, J.F. Young, J.E. Sipe, Extrinsic optical scattering loss inphotonic crystal waveguides: role of fabrication disorder and photon groupvelocity, Phys. Rev. Lett. 94 (2005) 033903-1–033903-4.
38] J.P. Hugonin, P. Lalanne, T.P. White, T.F. Krauss, Coupling into slow light pho-tonic crystal waveguides, Opt. Lett. 32 (2007) 2638–2640.
