Wideband slab photonic crystal waveguides for slow light using differential optofluidic infiltration Amir Khodamohammadi,1,2 Habib Khoshsima,1,2,* Vahid Fallahi,3,4 and Mostafa Sahrai2 1 2 3
Photonics-Physics Group, Aras International Campus, University of Tabriz, 51665-163 Tabriz, Iran
Research Institute for Applied Physics and Astronomy, University of Tabriz, 51665-163 Tabriz, Iran
Department of Physics and Energy Engineering, Amirkabir University of Technology, Hafez Ave., 1591634311 Tehran, Iran 4
e-mail: [email protected]
*Corresponding author: [email protected] Received 28 October 2014; revised 27 December 2014; accepted 4 January 2015; posted 6 January 2015 (Doc. ID 225788); published 3 February 2015
A new type of wideband slow light with a large delay bandwidth product in a slab photonic crystal waveguide with a triangular lattice of circular air holes in a silicon-on-insulator substrate based on optofluidic infiltration is demonstrated. It is shown that dispersion engineering through infiltrating optical fluids— with different refractive indices n1f and n2f —in the first two rows of the air holes innermost to the waveguide results in an improved normalized delay bandwidth product ranging from 0.187 to 0.377 with large bandwidth 12 nm < Δλ < 32 nm and group index 14.20 < ng < 24.62 around 1550 nm. The nearly zero group velocity dispersion on the order of 10−20 s2 ∕m is achieved in all of the structures. These results are obtained by numerical simulation based on a three-dimensional-plane-wave expansion method. © 2015 Optical Society of America OCIS codes: (260.2030) Dispersion; (130.5296) Photonic crystal waveguides; (130.2790) Guided waves; (230.7400) Waveguides, slab. http://dx.doi.org/10.1364/AO.54.001002
1. Introduction
The slow-light effect, which mainly refers to the reduction of the group velocity vg, has opened new possibilities for signal processing systems, optical buffers, and optical logic gates in next-generation communication [1–4]. Correspondingly, the novel properties of slow light can be exploited in a wide range of potential application areas such as optical integrated circuits and controlled enhanced light– matter interaction [5,6]. Furthermore, the devices based on the slow-light effect can be utilized in different applications including modulators, optical switches, biosensors, phase shifters, and data storage devices [7–11]. There are typically two approaches for slowing down the group velocity of the light: material slow 1559-128X/15/051002-08$15.00/0 © 2015 Optical Society of America 1002
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light and structural slow light [12]. The first one is based on dispersion properties associated with the resonance structure of a material medium, such as those that occur in electromagnetically induced transparency (EIT) [13] and coherent population oscillations (CPO) [14]. The second one is based on structural resonances such as those that occur in photonic crystal waveguides (PCWs) [15,16] and fiber Bragg gratings (FBGs) [17]. Recently, slow light in PCWs has attracted great deal of attention due to its capacity to work at room temperature and its ability to provide larger bandwidth compared with EIT and CPO technologies [1]. A W1 slab waveguide, which is formed by a missing central row of air holes from a perfect twodimensional photonic crystal lattice, has been considered more than other types of PCWs [18,19]. However, slow-light propagation is inevitably accompanied by high group velocity dispersion (GVD) and extremely narrow bandwidth of slow light, the two
major shortcoming with almost all W1 waveguides. Such disadvantages are due to the flatband edge, which usually distorts the optical signals and limits the applications of slow-light devices. Therefore, it seems necessary to decrease the GVD and increase the bandwidth in the W1-type PCWs. Researchers have recently made numerous efforts to optimize these parameters using various techniques. One of these techniques based on dispersion compensation can be exploited in the chirped photonic crystalcoupled waveguides [20,21]. However, manufacturing of such intricate devices is difficult and can limit their practical applications in integrated photonics devices. The other techniques are based on nearly zero dispersion, in which the PCW geometry is changed. The most appropriate parameters to be changed are the width of the waveguide [22], the hole size [23], the position of holes in the first two rows (usually with different displacement in the first row instead of the second one) [24–27], and usage of the annular holes [28]. Nevertheless, fabrication of such structures requires nanometer-scale precision that can limit the applications of PCWs. Optofluidic infiltration of PCW air holes has been introduced recently as an alternative postfabrication technology. Optofluidics is a new branch of photonics that combines microfluidics and photonics [29–32]. Due to the intrinsic holey nature of slab PCWs, fluids with different optical properties can be injected into some PCW holes. Transferring a precisely controlled amount of liquid in holes can be realized by a standard microscope equipped with a confocal laser scanning microscope and a microinfiltration system. The microinfiltration equipment consists of a hollow submicron-size pipette that can be moved by a micromanipulator in nanometer scale on the sample. The infiltration process is dominated by physical capillary force due to submicron dimensions of photonic crystal holes and the pipette. The holes are completely filled up by contact with the meniscus of the optical liquid inside the pipette [33,34]. The optofluidic infiltration has been achieved by Smith et al. [35] using a tapered glass microtip with an apex diameter of 220 nm. The microtip was controlled by a three-axis piezo-electric stage and was initially inserted within a meniscus of the filling fluid. Bedoya et al. [36] also exploited a micropipette with a 500 nm outer diameter tip that was computer controlled by a three-axis Eppendorf TransferMan NK 2 micromanipulator with a resolution of approximately 40 nm per actuator step. Optofluidic infiltration technology has many advantages such as the ability to be tuned, flexible, and reconfigured. In particular, it is a key technology for applications such as nonlinear enhancement in photonic devices [37], optical switches [38], and channel drop filters [39]. Dispersion engineering can be accomplished by manipulating the refractive indices of the fluids injected into the first two rows of air holes adjacent to the waveguide. High-index ionic liquid 1-ethyl3-methylimidazolium heptaiodide with refractive
indices ranging from 1.40 to 2.10 in intervals of 0.01 can be used as optical immersion fluid for selective optofluidic infiltration [40]. This liquid as well as other ionic liquids have melting points below room temperature that have recently attracted considerable attention for optofluidic applications. The interest is mainly due to their peculiar properties such as lack of vapor pressure, good thermal and chemical stability, and very good dissolution properties of both organic and inorganic compounds. Cargille optical liquids with high refractive indices from 1.30 to 2.31 at specific intervals of 0.005 and 0.002 refractive index units are promising alternatives as optical fluids for infiltration [41]. Specifically, these liquids find application in nonlinear Kerr effects in slow-light PCWs [38]. It has been demonstrated both theoretically and experimentally that optimizing slow-light performance can be achieved by choosing one [30,42] and two [43] kinds of liquid infiltration. In this paper, we theoretically develop a simple and flexible optofluidic-based technique to achieve optimized low GVD, wide bandwidth, and large normalized delay bandwidth product (NDBP). We infiltrate the first and second rows adjacent to the W1 PCW with optical fluids of refractive indices n1f and n2f , respectively. By choosing the appropriate infiltrated liquids with high-refractive-index differences (Δnf ), there is a possibility of greatly increasing the bandwidth and NDBP. The values of Δnf for optofluids have been chosen to be 0.2, 0.1, 0.05, and 0.025 in this analysis. Simulation results show that the value of Δnf can be used as a control parameter for increasing or decreasing the group index of the waveguide without loss in the NDBP. 2. Design and Computational Modeling
The proposed PCW geometry is presented in Fig. 1, which is a silicon-on-insulator (SOI) structure. The silicon slab layer has a thickness of 210 nm and is
Fig. 1. Schematic view of the proposed SOI-based W1 slab PCW. The first row of holes (blue) adjacent to the waveguide is filled with an optical fluid of refractive index n1f , and row 2 (red holes) is filled with another optical fluid of refractive index n2f . 10 February 2015 / Vol. 54, No. 5 / APPLIED OPTICS
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perforated by a triangular lattice of circular air holes. This layer is supported by a 1210-nm-thick silica layer on the bottom, and the whole is placed in the air. The radius of the holes is r 0.3a, in which a stands for the slab PC lattice constant. The parameters r and a are 126 and 420 nm, respectively, in our structure. The refractive indices for silicon, silica, and air are 3.4, 1.5, and 1.0, respectively. Strong confinement of light in the vertical direction as a result of a total internal reflection mechanism is provided by high-refractive index contrast attainable in SOI structures, while the photonic bandgap (PBG) effect of the PC provides confinement of the light in the horizontal direction. Upon the results of the previous studies, rows 1 and 2 are the most effective parts on GVD, bandwidth, and group velocity control. Thus, in our design, row 1 is filled with an optical fluid of refractive index n1f and row 2 is filled by another optical fluid of refractive index n2f . The refractive indices of the optical fluids are arranged from 1.7 to 2.3. It is noteworthy that such a simple and distinctive PC waveguide geometry design does not cause any problems in manufacturing orientation in comparison, for example, with other solutions using manipulation of the row geometry. In this study, we have calculated the band diagrams of the proposed structure with the MIT Photonic Bands software package, which implements a 3D-plane-wavelength expansion (PWE) method based on the supercell approach [44]. We used a supercell with four air holes on either side of the line defect of length a, which is the same as the periodicity along the Γ − K direction. An additional supercell of eight periods is imposed in the direction perpendicular to the waveguide for implementation of the periodic boundary condition. It is also assumed that a periodic sequence of slabs separated by a thick layer of silica as the bottom cladding and a sufficient amount of an air layer acting as the top cladding are used. Bloch boundary conditions are implemented for each side of the supercell. The PWE method [45–48] solves the following Maxwell’s equation in the frequency domain: ⃗ × ∇
1 ⃗ ⃗ ∇×H ϵr ⃗r
ω2 ⃗ H; c2
(1)
where ϵr ⃗r is the relative dielectric function of PCW, and c is the speed of light. Since ϵr ⃗r is periodic, its inverse can be expanded in the form of a Fourier series as η⃗r
X 1 ⃗ ηG⃗ ejG·⃗r ; ϵr ⃗r ⃗
(2)
G
where ηG⃗ is the Fourier coefficients given by 1 ηG⃗ V 1004
Z
1 −jG·⃗ ⃗ e r dv; ⃗ r ϵ V r
(3)
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⃗ and V is the volume of the supercell. The vector G m1 g⃗ 1 m2 g⃗ 2 m3 g⃗ 3 is the reciprocal lattice vector and g⃗ i (i 1; 2; 3) are the primitive reciprocal lattice vectors. According to Bloch’s theorem, the magnetic field can be expanded on a plane-wave basis ⃗ r H⃗
2 XX ⃗ λ1 G
⃗
⃗
ˆ λ ejkG·⃗r ; hG;λ ⃗ ε
(4)
where k⃗ is a wave vector in the Brillouin zone of the cell, and εˆ 1 , εˆ 2 are two polarization vectors ⃗ because of the transverse perpendicular to k⃗ G nature of the magnetic field. Substituting Eqs. (2) and (4) into the wave equation, Eq. (1), one can obtain the following matrix equations: 2 XX ⃗ 0 λ1 G
0
H λ;λ ⃗ 0 ;λ ⃗ ⃗ 0 hG G;G
2 0 ω2 X X Cλ;λ ⃗ 0 ;λ ; ⃗ G ⃗ 0 hG c2 ⃗ 0 λ1 G;
(5)
G
0
0
λ;λ where the matrices H λ;λ ⃗ ⃗ 0 and C ⃗ ⃗ 0 are given by G;G
⃗ ⃗ ⃗ ⃗ H λ;λ ⃗ G ⃗ 0 jk Gjjk G j ⃗ ⃗ 0 ηG− 0
0
G;G
G;G
εˆ 2 · εˆ 02 −ˆε1 · εˆ 02
−ˆε2 · εˆ 01 ; (6) εˆ 1 · εˆ 01
and 0 Cλ;λ ⃗ G ⃗ 0 G;
εˆ 1 · εˆ 01 εˆ 2 · εˆ 01
εˆ 1 · εˆ 02 : εˆ 2 · εˆ 02
(7)
By choosing the polarization vectors as εˆ 1 yˆ and ⃗ the electromagnetic fields can be εˆ 2 yˆ × k⃗ G, decoupled into two transversely polarized modes, namely the TM-like (ˆε1 ) and the TE-like (ˆε2 ) modes. The band diagram calculations show that the bandgap of a hexagonal lattice of air holes in silicon is relatively small for the TM-like modes and specifically disappears for r∕a < 0.385 [49,50]. Therefore, we only focus on the more robust TE-like gap for waveguiding purposes. Since the guided modes above the light line are essentially lossy, the slow guided-light mode is located within the PBG region of bulk PC ranging from 0.22a∕λ to 0.31a∕λ and below the silica light line. 3. Theoretical Background of Slow Light A. Group Velocity and Group Index Characteristics
In the previous section, the W1 slab PCW with a triangular lattice of circular air holes, which shows a flatband in the dispersion curve below the silica light line, has been developed. To define key parameters, we start from group velocity (vg ). It is the slope of the dispersion diagram for the guided modes within the photonic bandgap region. The group velocity of the guided modes is calculated as the derivate of the dispersion relation, and is expressed by [1]
vg
Z
dω ; dk
(8)
where ω is the light frequency, and k is the wavevector along the waveguide. The group index ng is inversely proportional to vg and shows the slowing of the propagating pulse in the medium, as given by ng
c dk ; vg dω
(9)
where c is the velocity of light in vacuum. The concept of the group index is mostly used to determine the performance of slow light. Accordingly, low vg denotes high ng and therefore it can be used as a factor to determine whether vg is reduced or not. In most cases, the region in which the variations of ng are less than 10% is considered as the slow-light bandwidth. There is often an inter-relationship between the bandwidth and ng such that as one increases, the other decreases. B.
ω0 Δω 2
ω0 −Δω 2
ng ω ·
dω : Δω
(13)
The main purpose of this study is to achieve a slowlight PCW with high ng , low vg, and small GVD in a wide bandwidth that corresponds to a high NDBP. The guided modes in PCWs in terms of the field distribution are classified into the gap-guided modes (GGMs) and the index-guided modes (IGMs). The GGMs with respect to the periodic structure are restricted by distributed Bragg reflection, while IGMs are confined by the total internal reflection because of the index contrast between the waveguide and the surrounding medium [32]. The holes near to the core of the slab waveguide are the coupling points of IGM and GGM. Altering the two rows can have twofold effects: it can change the intrinsic interaction between GGMs and IGMs and can regulate the dispersion, which in turn can broaden the flatband region for slow light or increase bandwidth. Indeed, the formation of the guided mode can be affected by modifying the parameters of the two rows adjacent to the waveguide.
Group Velocity Dispersion
GVD is another significant parameter in a slow-light device that causes some disadvantages due to substantial distortions in optical signals and broadening of the optical pulses [1]. The GVD, denoted by β, is defined as the second order derivative of the dispersion relation or the derivative of the inverse of vg with respect to frequency d2 k d 1 1 d2 ω − : β dω2 dω vg v3g dk2 C.
n~ g
(10)
Delay-Bandwidth Product
The DBP is used for comparing the performance of the slow-light devices with different frequencies and lengths [1]. The DBP defines the number of bits and buffering capacity that can be stored in a slow-light device and is defined as [51] DBP T d · B;
(11)
where T d is the propagation time of the pulse in the waveguide, and B is the bandwidth of the slow light. Comparing the capacity of the slow-light PCWs with different lengths necessitates the modification of the above expression for DBP to a useful form, i.e., the NDBP which is expressed as NDBP n~ g ·
Δω ; ω0
(12)
where Δω∕ω0 is the normalized bandwidth of the slow-light region, ω0 the normalized central frequency, and n~ g is the average group index in the frequency range Δω and is given by
4. Results and Discussion
The optofluidic-based technique in W1 slab PCW can produce slow-light performance. To demonstrate the phenomena, we studied the influence of infiltrating optical fluids with different refractive indices n1f and n2f into the two rows adjacent to the line defect in a proposed PCW on the waveguide dispersion and group index curves. The dispersion diagrams are calculated using the 3D-PWE method. The analysis was done for three different situations. In the first situation, the first and second rows of holes are filled with different optical fluids. In the second situation, the refractive index of the first row was left unchanged (with the best value of the NDBP obtained in the first situation) and the second row was injected with various values of n2f . Last, for the third analysis, optical fluids with different refractive indices were injected only into the first row. The refractive indices of the optical fluids were considered as the standard series of Cargille Liquids [41]. In the following subsections, these analyses are elaborated in detail. A. Changing the Refractive Indices of the First and Second Rows
The effect of infiltrating two different optical fluids into the two rows adjacent to the line defect on slow-light properties was studied. In our calculations, Δnf was assumed to be 0.2, 0.1, 0.05, and 0.025. The dispersion diagrams and the group index curves were calculated for a related guided mode. The band diagram variations for n1f ranging from 1.9 to 2.3 and n2f ranging from 1.7 to 2.1 with the fixed index difference of 0.2 (based on our calculations, this index difference produces the best results) are shown in Fig. 2(a). The figure shows that by increasing the refractive indices of the optical fluids, 10 February 2015 / Vol. 54, No. 5 / APPLIED OPTICS
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the band structure shifts to the lower frequencies. The corresponding group indices are shown in Fig. 2(b). The main guided mode for the flatband region is located between 0.39 < k < 0.46. Infiltrating optofluids into the two rows leads to a negative slope for the dispersion curve and forms a chair-shaped curve for ng. As it is obvious, when n1f and n2f increase, the ng in the flat region decreases, and the corresponding bandwidth increases. To make a comparison, the results obtained here and the results of some previously published papers have been presented in Table 1. It can be seen that by increasing the values of n1f and n2f , the group index and the bandwidth centered at the wavelength of 1550 nm change from 24.62 to 14.20 and from 2 to 32 nm, respectively, and
Table 1. Bandwidth (Δλ), Group Index (n g ), and NDBP for Different Refractive Indices n 1f and n 2f with a Fixed Δn f of 0.2 Compared with Different References
The present work
Ref. Ref. Ref. Ref.
[24] [7] [27] [43]
n1f
n2f
Δλ (nm)
ng
NDBP
1.9 2.0 2.1 2.2 2.3
1.7 1.8 1.9 2.0 2.1
1.38
1.72
12 21 28 32 30 11 ∼18 6.7 6.9
24.62 21.32 18.67 16.62 14.20 44 23 60 50
0.187 0.258 0.329 0.377 0.331 0.312 ∼0.3 0.257 0.2072
the NDBP parameter changes from 0.187 to 0.377. The maximum value of NDBP (0.377) is obtained for n1f 2.2 and n2f 2.0. The corresponding GVD, denoted by β, is indicated in Fig. 2(c). It is clear that GVD in all cases is near to zero. If we can ignore the β values on the order of 10−20 s2 ∕m, the proposed structure has a very low GVD. B. Changing the Refractive Index of the Second Row (n 2f )
In the next step, the refractive index of the first row was left unchanged, which was set on the value of 2.2 (produces the best value for NDBP according to the results of the previous subsection), and Δnf ranges from 0.025 to 0.2. Figure 3(a) shows the band diagram curves of the guided modes. The group indices corresponding to the guided modes are shown in Fig. 3(b), in which ng has been considered as a constant if its variations are within 10%. Similar to Fig. 2(a), as the value of n2f increases, the dispersion diagrams shift to the lower frequencies. Some calculated parameters of the slow modes have been given in Table 2. Increasing the value of n2f clearly results in decreasing the group index from 16.62 to 14.5 and increasing the bandwidth from 32 to 43 nm. Moreover, as observed in Table 2, NDBP in all cases has the same 0.377 value. Additionally, the proposed waveguide allows appropriate control of the group index and the bandwidth, opening opportunities for tunable slow-light devices. The GVD curves for different refractive indices n2f with different Δnf 0.2, 0.1, 0.05, and 0.025 are shown in Fig. 3(c). The GVD for all cases has the same value and is near zero (0.5 × 10−20 s2 ∕m). C. Changing the Refractive Index of Only the First Row (n 1f )
Fig. 2. (a) Dispersion, (b) group index ng , and (c) GVD curves for different refractive indices n1f and n2f with a fixed Δnf of 0.2. 1006
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The influence of injecting an optical fluid with different refractive indices (n1f ) only into the first innermost row around the waveguide has been studied in the third step of this work. The value of n1f was gradually changed from 2.0 to 2.3 with a pace of 0.1, while the refractive index of the second and the other rows was left unchanged (n1f 1). The dispersion diagrams of the guided modes are shown in Fig. 4(a). With an increasing n1f value, the guided modes shifted to the lower frequencies. The
Fig. 3. (a) Dispersion, (b) group index ng , and (c) GVD curves for n1f 2.2 and n2f with different Δnf ranges from 0.2 to 0.025.
Fig. 4. (a) Dispersion, (b) group index ng , and (c) GVD curves for fixed n2f 1.0 and a different refractive index n2f with a step of 0.1.
corresponding ng curves versus normalized frequency are plotted in Fig. 4(b). Table 3 shows the significant factors of the slow-light performance in this analysis. Based on the results of Table 3, by increasing the value of n1f , the bandwidth of the slow-light
performance increases while the group index value ng decreases. In this case, the NDBP ranges from 0.103 to 0.212, and the bandwidth changes from 3 to 7 nm, which are the minimum amounts achieved in the three performed analyses. On the other hand, in this case,
Table 2.
Δnf 0.2 0.1 0.05 0.025
Bandwidth (Δλ), Group Index (n g ) and NDBP for n 1f 2.2 and n 2f with Different Δn f Ranges from 0.2 to 0.025.
Table 3. Bandwidth (Δλ), Group Index (n g ), GVD, and NDBP for Fixed n 2f 1.0 and a Different Refractive Index n 1f with a Step of 0.1
n1f
n2f
Δλ (nm)
ng
NDBP
n1f
n2f
Δλ (nm)
ng
GVD ×10−20 s2 ∕m
NDBP
2.200 2.200 2.200 2.200
2.000 2.100 2.150 2.175
32 39 42 43
16.62 15.08 14.38 14.05
0.3773 0.3772 0.3774 0.3775
2.0 2.1 2.2 2.3
1.0 1.0 1.0 1.0
3 4 6 7
66.11 60.38 54.13 47.26
3.088 2.474 1.790 1.725
0.103 0.140 0.188 0.212
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ng ranges from 66.11 to 47.26, which is the maximum value achieved in the analyses. The corresponding GVD values, shown in Fig. 4(c), have increased almost six-fold relative to the two aforementioned cases. Nevertheless, their maximum values are still very small at 1.725 × 10−20 s2 ∕m < β < 3.088 × 10−20 s2 ∕m. From the technological point of view, since only the first innermost row is filled with an optical fluid, such a structure is straightforward and easy to fabricate. However, according to the obtained results in Section 4.A, it is better to consider the second kind of liquid infiltration in the second row adjacent to the waveguide in order to improve the bandwidth and NDBP of the waveguide.
10.
11. 12. 13. 14. 15.
5. Conclusion
A new slow-light PCW based on optofluidic infiltration with large bandwidth, high group index, and nearly zero dispersion on the order of 10−20 s2 ∕m has been theoretically investigated. It has been shown that by carefully changing the refractive indices of the first two rows innermost to the line defect waveguide with different values of n1f and n2f , a versatile control of light with a group index ranging from 14.20 to 24.62 and bandwidth ranging from 12 to 32 nm would be achieved around the wavelength of 1550 nm. The results of this study are in agreement with the previous works. However, the proposed structure has the technological advantage over previous works in that the hole positions in the first two rows have been changed. Due to the wide bandwidth of the proposed PCW, it has potential for practical applications in photonic devices such as optical buffers and optical modulators, and for nonlinear effects.
16.
17. 18. 19. 20. 21. 22. 23.
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24. 25.
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adjustment and optofluidic infiltration,” Appl. Opt. 52, 8184– 8189 (2013). F. Intonti, S. Vignolini, V. Türck, M. Colocci, P. Bettotti, L. Pavesi, S. L. Schweizer, R. Wehrspohn, and D. Wiersma, “Rewritable photonic circuits,” Appl. Phys. Lett. 89, 211117 (2006). S. Vignolini, Sub-Wavelength Probing and Modification of Complex Photonic Structures, Premio tesi di dottorato (Firenze University, 2010). C. L. Smith, U. Bog, S. Tomljenovic-Hanic, M. W. Lee, D. K. Wu, L. O’Faolain, C. Monat, C. Grillet, T. F. Krauss, C. Karnutsch, R. C. McPhedran, and B. J. Eggleton, “Reconfigurable microfluidic photonic crystal slab cavities,” Opt. Express 16, 15887–15896 (2008). A. C. Bedoya, P. Domachuk, C. Grillet, C. Monat, E. Mägi, E. Li, and B. J. Eggleton, “Reconfigurable photonic crystal waveguides created by selective liquid infiltration,” Opt. Express 20, 11046–11056 (2012). M. Ebnali-Heidari, C. Monat, C. Grillet, and M. MoravvejFarshi, “A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration,” Opt. Express 17, 18340–18353 (2009). S. Bakhshi, M. K. Moravvej-Farshi, and M. Ebnali-Heidari, “Design of an ultracompact low-power all-optical modulator by means of dispersion engineered slow light regime in a photonic crystal Mach–Zehnder interferometer,” Appl. Opt. 51, 2687–2692 (2012). M. Bitarafan, M. Moravvej-Farshi, and M. Ebnali-Heidari, “Proposal for postfabrication fine-tuning of three-port photonic crystal channel drop filters by means of optofluidic infiltration,” Appl. Opt. 50, 2622–2627 (2011). M. Freemantle, T. Welton, and R. D. Rogers, An Introduction to Ionic Liquids (Royal Society of Chemistry, 2009). Introduction to Optical Liquids, a Cargille Labs report, 2011, available at http://www.cargille.com/opticalintro.shtml.
42. A. Casas-Bedoya, C. Husko, C. Monat, C. Grillet, N. Gutman, P. Domachuk, and B. J. Eggleton, “Slow-light dispersion engineering of photonic crystal waveguides using selective microfluidic infiltration,” Opt. Lett. 37, 4215–4217 (2012). 43. Y. Zhao, Y. Zhang, and Q. Wang, “Optimization of slow light in slotted photonic crystal waveguide with liquid infiltration,” J. Lightwave Technol. 31, 2448–2454 (2013). 44. S. Johnson and J. Joannopoulos, “Block-iterative frequencydomain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001). 45. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999). 46. S. S. Xiao, L. F. Shen, and S. L. He, “A plane-wave expansion method based on the effective medium theory for calculating the band structure of a two-dimensional photonic crystal,” Phys. Lett. A 313, 132–138 (2003). 47. S. Shi, C. Chen, and D. W. Prather, “Plane-wave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers,” J. Opt. Soc. Am. A 21, 1769–1775 (2004). 48. J. O. Vasseur, P. A. Deymier, B. Djafari-Rouhani, Y. Pennec, and A.-C. Hladky-Hennion, “Absolute forbidden bands and waveguiding in two-dimensional phononic crystal plates,” Phys. Rev. B 77, 085415 (2008). 49. C. Jamois, R. B. Wehrspohn, L. C. Andreani, C. Hermann, O. Hess, and U. Gösele, “Silicon-based two-dimensional photonic crystal waveguides,” Photon. Nanostruct. Fundam. Appl. 1, 1–13 (2003). 50. O. Painter, J. Vučkovič, and A. Scherer, “Defect modes of a twodimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B 16, 275–285 (1999). 51. P. Janes, J. Tidstrom, and L. Thylen, “Limits on optical pulse compression and delay bandwidth product in electromagnetically induced transparency media,” J. Lightwave Technol. 23, 3893–3899 (2005).
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Photonics-Physics Group, Aras International Campus, University of Tabriz, 51665-163 Tabriz, Iran
Research Institute for Applied Physics and Astronomy, University of Tabriz, 51665-163 Tabriz, Iran
Department of Physics and Energy Engineering, Amirkabir University of Technology, Hafez Ave., 1591634311 Tehran, Iran 4
e-mail: [email protected]
*Corresponding author: [email protected] Received 28 October 2014; revised 27 December 2014; accepted 4 January 2015; posted 6 January 2015 (Doc. ID 225788); published 3 February 2015
A new type of wideband slow light with a large delay bandwidth product in a slab photonic crystal waveguide with a triangular lattice of circular air holes in a silicon-on-insulator substrate based on optofluidic infiltration is demonstrated. It is shown that dispersion engineering through infiltrating optical fluids— with different refractive indices n1f and n2f —in the first two rows of the air holes innermost to the waveguide results in an improved normalized delay bandwidth product ranging from 0.187 to 0.377 with large bandwidth 12 nm < Δλ < 32 nm and group index 14.20 < ng < 24.62 around 1550 nm. The nearly zero group velocity dispersion on the order of 10−20 s2 ∕m is achieved in all of the structures. These results are obtained by numerical simulation based on a three-dimensional-plane-wave expansion method. © 2015 Optical Society of America OCIS codes: (260.2030) Dispersion; (130.5296) Photonic crystal waveguides; (130.2790) Guided waves; (230.7400) Waveguides, slab. http://dx.doi.org/10.1364/AO.54.001002
1. Introduction
The slow-light effect, which mainly refers to the reduction of the group velocity vg, has opened new possibilities for signal processing systems, optical buffers, and optical logic gates in next-generation communication [1–4]. Correspondingly, the novel properties of slow light can be exploited in a wide range of potential application areas such as optical integrated circuits and controlled enhanced light– matter interaction [5,6]. Furthermore, the devices based on the slow-light effect can be utilized in different applications including modulators, optical switches, biosensors, phase shifters, and data storage devices [7–11]. There are typically two approaches for slowing down the group velocity of the light: material slow 1559-128X/15/051002-08$15.00/0 © 2015 Optical Society of America 1002
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light and structural slow light [12]. The first one is based on dispersion properties associated with the resonance structure of a material medium, such as those that occur in electromagnetically induced transparency (EIT) [13] and coherent population oscillations (CPO) [14]. The second one is based on structural resonances such as those that occur in photonic crystal waveguides (PCWs) [15,16] and fiber Bragg gratings (FBGs) [17]. Recently, slow light in PCWs has attracted great deal of attention due to its capacity to work at room temperature and its ability to provide larger bandwidth compared with EIT and CPO technologies [1]. A W1 slab waveguide, which is formed by a missing central row of air holes from a perfect twodimensional photonic crystal lattice, has been considered more than other types of PCWs [18,19]. However, slow-light propagation is inevitably accompanied by high group velocity dispersion (GVD) and extremely narrow bandwidth of slow light, the two
major shortcoming with almost all W1 waveguides. Such disadvantages are due to the flatband edge, which usually distorts the optical signals and limits the applications of slow-light devices. Therefore, it seems necessary to decrease the GVD and increase the bandwidth in the W1-type PCWs. Researchers have recently made numerous efforts to optimize these parameters using various techniques. One of these techniques based on dispersion compensation can be exploited in the chirped photonic crystalcoupled waveguides [20,21]. However, manufacturing of such intricate devices is difficult and can limit their practical applications in integrated photonics devices. The other techniques are based on nearly zero dispersion, in which the PCW geometry is changed. The most appropriate parameters to be changed are the width of the waveguide [22], the hole size [23], the position of holes in the first two rows (usually with different displacement in the first row instead of the second one) [24–27], and usage of the annular holes [28]. Nevertheless, fabrication of such structures requires nanometer-scale precision that can limit the applications of PCWs. Optofluidic infiltration of PCW air holes has been introduced recently as an alternative postfabrication technology. Optofluidics is a new branch of photonics that combines microfluidics and photonics [29–32]. Due to the intrinsic holey nature of slab PCWs, fluids with different optical properties can be injected into some PCW holes. Transferring a precisely controlled amount of liquid in holes can be realized by a standard microscope equipped with a confocal laser scanning microscope and a microinfiltration system. The microinfiltration equipment consists of a hollow submicron-size pipette that can be moved by a micromanipulator in nanometer scale on the sample. The infiltration process is dominated by physical capillary force due to submicron dimensions of photonic crystal holes and the pipette. The holes are completely filled up by contact with the meniscus of the optical liquid inside the pipette [33,34]. The optofluidic infiltration has been achieved by Smith et al. [35] using a tapered glass microtip with an apex diameter of 220 nm. The microtip was controlled by a three-axis piezo-electric stage and was initially inserted within a meniscus of the filling fluid. Bedoya et al. [36] also exploited a micropipette with a 500 nm outer diameter tip that was computer controlled by a three-axis Eppendorf TransferMan NK 2 micromanipulator with a resolution of approximately 40 nm per actuator step. Optofluidic infiltration technology has many advantages such as the ability to be tuned, flexible, and reconfigured. In particular, it is a key technology for applications such as nonlinear enhancement in photonic devices [37], optical switches [38], and channel drop filters [39]. Dispersion engineering can be accomplished by manipulating the refractive indices of the fluids injected into the first two rows of air holes adjacent to the waveguide. High-index ionic liquid 1-ethyl3-methylimidazolium heptaiodide with refractive
indices ranging from 1.40 to 2.10 in intervals of 0.01 can be used as optical immersion fluid for selective optofluidic infiltration [40]. This liquid as well as other ionic liquids have melting points below room temperature that have recently attracted considerable attention for optofluidic applications. The interest is mainly due to their peculiar properties such as lack of vapor pressure, good thermal and chemical stability, and very good dissolution properties of both organic and inorganic compounds. Cargille optical liquids with high refractive indices from 1.30 to 2.31 at specific intervals of 0.005 and 0.002 refractive index units are promising alternatives as optical fluids for infiltration [41]. Specifically, these liquids find application in nonlinear Kerr effects in slow-light PCWs [38]. It has been demonstrated both theoretically and experimentally that optimizing slow-light performance can be achieved by choosing one [30,42] and two [43] kinds of liquid infiltration. In this paper, we theoretically develop a simple and flexible optofluidic-based technique to achieve optimized low GVD, wide bandwidth, and large normalized delay bandwidth product (NDBP). We infiltrate the first and second rows adjacent to the W1 PCW with optical fluids of refractive indices n1f and n2f , respectively. By choosing the appropriate infiltrated liquids with high-refractive-index differences (Δnf ), there is a possibility of greatly increasing the bandwidth and NDBP. The values of Δnf for optofluids have been chosen to be 0.2, 0.1, 0.05, and 0.025 in this analysis. Simulation results show that the value of Δnf can be used as a control parameter for increasing or decreasing the group index of the waveguide without loss in the NDBP. 2. Design and Computational Modeling
The proposed PCW geometry is presented in Fig. 1, which is a silicon-on-insulator (SOI) structure. The silicon slab layer has a thickness of 210 nm and is
Fig. 1. Schematic view of the proposed SOI-based W1 slab PCW. The first row of holes (blue) adjacent to the waveguide is filled with an optical fluid of refractive index n1f , and row 2 (red holes) is filled with another optical fluid of refractive index n2f . 10 February 2015 / Vol. 54, No. 5 / APPLIED OPTICS
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perforated by a triangular lattice of circular air holes. This layer is supported by a 1210-nm-thick silica layer on the bottom, and the whole is placed in the air. The radius of the holes is r 0.3a, in which a stands for the slab PC lattice constant. The parameters r and a are 126 and 420 nm, respectively, in our structure. The refractive indices for silicon, silica, and air are 3.4, 1.5, and 1.0, respectively. Strong confinement of light in the vertical direction as a result of a total internal reflection mechanism is provided by high-refractive index contrast attainable in SOI structures, while the photonic bandgap (PBG) effect of the PC provides confinement of the light in the horizontal direction. Upon the results of the previous studies, rows 1 and 2 are the most effective parts on GVD, bandwidth, and group velocity control. Thus, in our design, row 1 is filled with an optical fluid of refractive index n1f and row 2 is filled by another optical fluid of refractive index n2f . The refractive indices of the optical fluids are arranged from 1.7 to 2.3. It is noteworthy that such a simple and distinctive PC waveguide geometry design does not cause any problems in manufacturing orientation in comparison, for example, with other solutions using manipulation of the row geometry. In this study, we have calculated the band diagrams of the proposed structure with the MIT Photonic Bands software package, which implements a 3D-plane-wavelength expansion (PWE) method based on the supercell approach [44]. We used a supercell with four air holes on either side of the line defect of length a, which is the same as the periodicity along the Γ − K direction. An additional supercell of eight periods is imposed in the direction perpendicular to the waveguide for implementation of the periodic boundary condition. It is also assumed that a periodic sequence of slabs separated by a thick layer of silica as the bottom cladding and a sufficient amount of an air layer acting as the top cladding are used. Bloch boundary conditions are implemented for each side of the supercell. The PWE method [45–48] solves the following Maxwell’s equation in the frequency domain: ⃗ × ∇
1 ⃗ ⃗ ∇×H ϵr ⃗r
ω2 ⃗ H; c2
(1)
where ϵr ⃗r is the relative dielectric function of PCW, and c is the speed of light. Since ϵr ⃗r is periodic, its inverse can be expanded in the form of a Fourier series as η⃗r
X 1 ⃗ ηG⃗ ejG·⃗r ; ϵr ⃗r ⃗
(2)
G
where ηG⃗ is the Fourier coefficients given by 1 ηG⃗ V 1004
Z
1 −jG·⃗ ⃗ e r dv; ⃗ r ϵ V r
(3)
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⃗ and V is the volume of the supercell. The vector G m1 g⃗ 1 m2 g⃗ 2 m3 g⃗ 3 is the reciprocal lattice vector and g⃗ i (i 1; 2; 3) are the primitive reciprocal lattice vectors. According to Bloch’s theorem, the magnetic field can be expanded on a plane-wave basis ⃗ r H⃗
2 XX ⃗ λ1 G
⃗
⃗
ˆ λ ejkG·⃗r ; hG;λ ⃗ ε
(4)
where k⃗ is a wave vector in the Brillouin zone of the cell, and εˆ 1 , εˆ 2 are two polarization vectors ⃗ because of the transverse perpendicular to k⃗ G nature of the magnetic field. Substituting Eqs. (2) and (4) into the wave equation, Eq. (1), one can obtain the following matrix equations: 2 XX ⃗ 0 λ1 G
0
H λ;λ ⃗ 0 ;λ ⃗ ⃗ 0 hG G;G
2 0 ω2 X X Cλ;λ ⃗ 0 ;λ ; ⃗ G ⃗ 0 hG c2 ⃗ 0 λ1 G;
(5)
G
0
0
λ;λ where the matrices H λ;λ ⃗ ⃗ 0 and C ⃗ ⃗ 0 are given by G;G
⃗ ⃗ ⃗ ⃗ H λ;λ ⃗ G ⃗ 0 jk Gjjk G j ⃗ ⃗ 0 ηG− 0
0
G;G
G;G
εˆ 2 · εˆ 02 −ˆε1 · εˆ 02
−ˆε2 · εˆ 01 ; (6) εˆ 1 · εˆ 01
and 0 Cλ;λ ⃗ G ⃗ 0 G;
εˆ 1 · εˆ 01 εˆ 2 · εˆ 01
εˆ 1 · εˆ 02 : εˆ 2 · εˆ 02
(7)
By choosing the polarization vectors as εˆ 1 yˆ and ⃗ the electromagnetic fields can be εˆ 2 yˆ × k⃗ G, decoupled into two transversely polarized modes, namely the TM-like (ˆε1 ) and the TE-like (ˆε2 ) modes. The band diagram calculations show that the bandgap of a hexagonal lattice of air holes in silicon is relatively small for the TM-like modes and specifically disappears for r∕a < 0.385 [49,50]. Therefore, we only focus on the more robust TE-like gap for waveguiding purposes. Since the guided modes above the light line are essentially lossy, the slow guided-light mode is located within the PBG region of bulk PC ranging from 0.22a∕λ to 0.31a∕λ and below the silica light line. 3. Theoretical Background of Slow Light A. Group Velocity and Group Index Characteristics
In the previous section, the W1 slab PCW with a triangular lattice of circular air holes, which shows a flatband in the dispersion curve below the silica light line, has been developed. To define key parameters, we start from group velocity (vg ). It is the slope of the dispersion diagram for the guided modes within the photonic bandgap region. The group velocity of the guided modes is calculated as the derivate of the dispersion relation, and is expressed by [1]
vg
Z
dω ; dk
(8)
where ω is the light frequency, and k is the wavevector along the waveguide. The group index ng is inversely proportional to vg and shows the slowing of the propagating pulse in the medium, as given by ng
c dk ; vg dω
(9)
where c is the velocity of light in vacuum. The concept of the group index is mostly used to determine the performance of slow light. Accordingly, low vg denotes high ng and therefore it can be used as a factor to determine whether vg is reduced or not. In most cases, the region in which the variations of ng are less than 10% is considered as the slow-light bandwidth. There is often an inter-relationship between the bandwidth and ng such that as one increases, the other decreases. B.
ω0 Δω 2
ω0 −Δω 2
ng ω ·
dω : Δω
(13)
The main purpose of this study is to achieve a slowlight PCW with high ng , low vg, and small GVD in a wide bandwidth that corresponds to a high NDBP. The guided modes in PCWs in terms of the field distribution are classified into the gap-guided modes (GGMs) and the index-guided modes (IGMs). The GGMs with respect to the periodic structure are restricted by distributed Bragg reflection, while IGMs are confined by the total internal reflection because of the index contrast between the waveguide and the surrounding medium [32]. The holes near to the core of the slab waveguide are the coupling points of IGM and GGM. Altering the two rows can have twofold effects: it can change the intrinsic interaction between GGMs and IGMs and can regulate the dispersion, which in turn can broaden the flatband region for slow light or increase bandwidth. Indeed, the formation of the guided mode can be affected by modifying the parameters of the two rows adjacent to the waveguide.
Group Velocity Dispersion
GVD is another significant parameter in a slow-light device that causes some disadvantages due to substantial distortions in optical signals and broadening of the optical pulses [1]. The GVD, denoted by β, is defined as the second order derivative of the dispersion relation or the derivative of the inverse of vg with respect to frequency d2 k d 1 1 d2 ω − : β dω2 dω vg v3g dk2 C.
n~ g
(10)
Delay-Bandwidth Product
The DBP is used for comparing the performance of the slow-light devices with different frequencies and lengths [1]. The DBP defines the number of bits and buffering capacity that can be stored in a slow-light device and is defined as [51] DBP T d · B;
(11)
where T d is the propagation time of the pulse in the waveguide, and B is the bandwidth of the slow light. Comparing the capacity of the slow-light PCWs with different lengths necessitates the modification of the above expression for DBP to a useful form, i.e., the NDBP which is expressed as NDBP n~ g ·
Δω ; ω0
(12)
where Δω∕ω0 is the normalized bandwidth of the slow-light region, ω0 the normalized central frequency, and n~ g is the average group index in the frequency range Δω and is given by
4. Results and Discussion
The optofluidic-based technique in W1 slab PCW can produce slow-light performance. To demonstrate the phenomena, we studied the influence of infiltrating optical fluids with different refractive indices n1f and n2f into the two rows adjacent to the line defect in a proposed PCW on the waveguide dispersion and group index curves. The dispersion diagrams are calculated using the 3D-PWE method. The analysis was done for three different situations. In the first situation, the first and second rows of holes are filled with different optical fluids. In the second situation, the refractive index of the first row was left unchanged (with the best value of the NDBP obtained in the first situation) and the second row was injected with various values of n2f . Last, for the third analysis, optical fluids with different refractive indices were injected only into the first row. The refractive indices of the optical fluids were considered as the standard series of Cargille Liquids [41]. In the following subsections, these analyses are elaborated in detail. A. Changing the Refractive Indices of the First and Second Rows
The effect of infiltrating two different optical fluids into the two rows adjacent to the line defect on slow-light properties was studied. In our calculations, Δnf was assumed to be 0.2, 0.1, 0.05, and 0.025. The dispersion diagrams and the group index curves were calculated for a related guided mode. The band diagram variations for n1f ranging from 1.9 to 2.3 and n2f ranging from 1.7 to 2.1 with the fixed index difference of 0.2 (based on our calculations, this index difference produces the best results) are shown in Fig. 2(a). The figure shows that by increasing the refractive indices of the optical fluids, 10 February 2015 / Vol. 54, No. 5 / APPLIED OPTICS
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the band structure shifts to the lower frequencies. The corresponding group indices are shown in Fig. 2(b). The main guided mode for the flatband region is located between 0.39 < k < 0.46. Infiltrating optofluids into the two rows leads to a negative slope for the dispersion curve and forms a chair-shaped curve for ng. As it is obvious, when n1f and n2f increase, the ng in the flat region decreases, and the corresponding bandwidth increases. To make a comparison, the results obtained here and the results of some previously published papers have been presented in Table 1. It can be seen that by increasing the values of n1f and n2f , the group index and the bandwidth centered at the wavelength of 1550 nm change from 24.62 to 14.20 and from 2 to 32 nm, respectively, and
Table 1. Bandwidth (Δλ), Group Index (n g ), and NDBP for Different Refractive Indices n 1f and n 2f with a Fixed Δn f of 0.2 Compared with Different References
The present work
Ref. Ref. Ref. Ref.
[24] [7] [27] [43]
n1f
n2f
Δλ (nm)
ng
NDBP
1.9 2.0 2.1 2.2 2.3
1.7 1.8 1.9 2.0 2.1
1.38
1.72
12 21 28 32 30 11 ∼18 6.7 6.9
24.62 21.32 18.67 16.62 14.20 44 23 60 50
0.187 0.258 0.329 0.377 0.331 0.312 ∼0.3 0.257 0.2072
the NDBP parameter changes from 0.187 to 0.377. The maximum value of NDBP (0.377) is obtained for n1f 2.2 and n2f 2.0. The corresponding GVD, denoted by β, is indicated in Fig. 2(c). It is clear that GVD in all cases is near to zero. If we can ignore the β values on the order of 10−20 s2 ∕m, the proposed structure has a very low GVD. B. Changing the Refractive Index of the Second Row (n 2f )
In the next step, the refractive index of the first row was left unchanged, which was set on the value of 2.2 (produces the best value for NDBP according to the results of the previous subsection), and Δnf ranges from 0.025 to 0.2. Figure 3(a) shows the band diagram curves of the guided modes. The group indices corresponding to the guided modes are shown in Fig. 3(b), in which ng has been considered as a constant if its variations are within 10%. Similar to Fig. 2(a), as the value of n2f increases, the dispersion diagrams shift to the lower frequencies. Some calculated parameters of the slow modes have been given in Table 2. Increasing the value of n2f clearly results in decreasing the group index from 16.62 to 14.5 and increasing the bandwidth from 32 to 43 nm. Moreover, as observed in Table 2, NDBP in all cases has the same 0.377 value. Additionally, the proposed waveguide allows appropriate control of the group index and the bandwidth, opening opportunities for tunable slow-light devices. The GVD curves for different refractive indices n2f with different Δnf 0.2, 0.1, 0.05, and 0.025 are shown in Fig. 3(c). The GVD for all cases has the same value and is near zero (0.5 × 10−20 s2 ∕m). C. Changing the Refractive Index of Only the First Row (n 1f )
Fig. 2. (a) Dispersion, (b) group index ng , and (c) GVD curves for different refractive indices n1f and n2f with a fixed Δnf of 0.2. 1006
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The influence of injecting an optical fluid with different refractive indices (n1f ) only into the first innermost row around the waveguide has been studied in the third step of this work. The value of n1f was gradually changed from 2.0 to 2.3 with a pace of 0.1, while the refractive index of the second and the other rows was left unchanged (n1f 1). The dispersion diagrams of the guided modes are shown in Fig. 4(a). With an increasing n1f value, the guided modes shifted to the lower frequencies. The
Fig. 3. (a) Dispersion, (b) group index ng , and (c) GVD curves for n1f 2.2 and n2f with different Δnf ranges from 0.2 to 0.025.
Fig. 4. (a) Dispersion, (b) group index ng , and (c) GVD curves for fixed n2f 1.0 and a different refractive index n2f with a step of 0.1.
corresponding ng curves versus normalized frequency are plotted in Fig. 4(b). Table 3 shows the significant factors of the slow-light performance in this analysis. Based on the results of Table 3, by increasing the value of n1f , the bandwidth of the slow-light
performance increases while the group index value ng decreases. In this case, the NDBP ranges from 0.103 to 0.212, and the bandwidth changes from 3 to 7 nm, which are the minimum amounts achieved in the three performed analyses. On the other hand, in this case,
Table 2.
Δnf 0.2 0.1 0.05 0.025
Bandwidth (Δλ), Group Index (n g ) and NDBP for n 1f 2.2 and n 2f with Different Δn f Ranges from 0.2 to 0.025.
Table 3. Bandwidth (Δλ), Group Index (n g ), GVD, and NDBP for Fixed n 2f 1.0 and a Different Refractive Index n 1f with a Step of 0.1
n1f
n2f
Δλ (nm)
ng
NDBP
n1f
n2f
Δλ (nm)
ng
GVD ×10−20 s2 ∕m
NDBP
2.200 2.200 2.200 2.200
2.000 2.100 2.150 2.175
32 39 42 43
16.62 15.08 14.38 14.05
0.3773 0.3772 0.3774 0.3775
2.0 2.1 2.2 2.3
1.0 1.0 1.0 1.0
3 4 6 7
66.11 60.38 54.13 47.26
3.088 2.474 1.790 1.725
0.103 0.140 0.188 0.212
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ng ranges from 66.11 to 47.26, which is the maximum value achieved in the analyses. The corresponding GVD values, shown in Fig. 4(c), have increased almost six-fold relative to the two aforementioned cases. Nevertheless, their maximum values are still very small at 1.725 × 10−20 s2 ∕m < β < 3.088 × 10−20 s2 ∕m. From the technological point of view, since only the first innermost row is filled with an optical fluid, such a structure is straightforward and easy to fabricate. However, according to the obtained results in Section 4.A, it is better to consider the second kind of liquid infiltration in the second row adjacent to the waveguide in order to improve the bandwidth and NDBP of the waveguide.
10.
11. 12. 13. 14. 15.
5. Conclusion
A new slow-light PCW based on optofluidic infiltration with large bandwidth, high group index, and nearly zero dispersion on the order of 10−20 s2 ∕m has been theoretically investigated. It has been shown that by carefully changing the refractive indices of the first two rows innermost to the line defect waveguide with different values of n1f and n2f , a versatile control of light with a group index ranging from 14.20 to 24.62 and bandwidth ranging from 12 to 32 nm would be achieved around the wavelength of 1550 nm. The results of this study are in agreement with the previous works. However, the proposed structure has the technological advantage over previous works in that the hole positions in the first two rows have been changed. Due to the wide bandwidth of the proposed PCW, it has potential for practical applications in photonic devices such as optical buffers and optical modulators, and for nonlinear effects.
16.
17. 18. 19. 20. 21. 22. 23.
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