Buffering Capability and Limitations in Low Dispersion
Photonic Crystal Waveguides with Elliptical Air Holes
Fang Long,* Huiping Tian, and Yuefeng Ji
Key Laboratory of Information Photonics and Optical Communications, Ministry of Education,
School of Information and Telecommunication Engineering, Beijing University of Posts &
Telecommunications, Beijing, P. R. China
Mailing dress: P.O.Box 90, BUPT, No.10, Xitucheng Road, Haidian District, Beijing,
P.R.China, 100876
Tel.: +86 10 6228 2153; fax: +86 10 6228 2153.
*
Corresponding author: longfang114@gmail.com
Abstract We propose a low dispersion photonic crystal waveguide with triangular lattice
elliptical air holes to realize compact, high-performance optical buffering application. In the
optimized structures, the constant group velocity ranging from c/41 to c/256 and the
corresponding negligible dispersion bandwidth are obtained. Due to the dispersion engineering,
the optimized structures effectively improve the buffering capability and limitations. The
maximums of buffer capacity and data rate can reach 262bit and 515Gb/s, respectively. And the
corresponding delay time is 255.4ps. These results show that the proposed structure has
considerable potential for optical buffering application.
Keywords Buffer capacity; Bit rate; Delay time; Group velocity dispersion; Photonic crystal
1
waveguides
OCIS：060.1810 Buffers, 260.2030 Dispersion, 130.5296 Photonic crystal waveguides
1. Introduction
Optical buffer is a device that temporarily stores and adjusts the timing of optical packets [1]
and a key device for future all-optical communication networks. Compared with previous
solutions [2], such as electromagnetically induced transparency (EIT) and optical fibers, optical
buffer based on photonic crystal (PhC) slow light are compatible with on-chip integration and
room-temperature operation, and can offer wide-bandwith and low group velocity propagation
[3]. Therefore PhC slow light is a promising solution for buffering and has been a research focus
in optical communications.
However, the photonic crystal waveguide (PCW) slow light is usually accompanied by
large group velocity dispersion (GVD). Dispersion-induced broadening of pulses is undesirable
as it interferes with the detection process and leads to errors if the pulse spreads outside its
allocated bit slot [4]. Clearly, GVD limits the buffer performance in PCW, such as buffer
capacity, data rate for a fixed transmission length (i.e. a specified delay time). Theocharidis’s
work [5] and Jing ’s work [6] focused on the limitation of data rate and buffer capacity in PCW,
respectively. To eliminate the undesirable dispersion, several approaches have been successfully
investigated to tailor the dispersion of slow light modes [6-12]. These PhC structures are
designed and fabricated in triangular lattice with circular holes. And some researchers also
studied the dispersion engineering in elliptical unit cell structures [1, 13]. The elliptical hole
breaks the symmetric of the circular hole. The authors of [14] presented that the deformation of
the unit cell structure to ellipse breaks the periodicity symmetry, and the wave satisfying Bragg
condition doesn’t resonant in different directions and has less power leaking. So our numerical
2
investigation is based on the PCW with elliptical air holes. We not only achieve and balance the
constant low group velocity and negligible dispersion bandwidth, but also systemically analyze
the buffering capability and limitations in our proposed structure.
In this paper, the buffer performance and limitations in low dispersion PCW with elliptical
air holes are systemically analyzed. Firstly we theoretically analyze the model of PCW. Then we
focus on optimizing the major and minor axes to achieve the low group velocity and negligible
dispersion bandwidth, and balancing the both by fine turning the structure parameters of the first
low. Finally we research the limitations of buffer performance due to the dispersion–induced
broadening, including buffer capacity, data rate and delay time. The limitation of buffer capacity
and the delay time are effectively improved in our proposed dispersion engineering PCW
compared with [6].
2. The PCW Model
The structure of the PCW consisting of a single line defect triangular lattice elliptical air
holes is illustrated in Fig. 1. The semi-major and semi-minor axes of bulk elliptical air holes is A
and B, respectively. The lattice constant is a, and the silicon refractive index is nsi=3.5. The
infinite height of the PCW in our analyses is assumed. The displacements of the first row relative
to the unmodified lattice (dashed lines) are given by d, where shifts toward the waveguide centre
are defined to be positive. And the semi-major and semi-minor axes of the first row adjacent to
the defect are A1 and B1, respectively.
A typical band diagram for transverse-electric (TE) polarized light in the line defect PCW
with elliptical air holes is obtained numerically by using the 2D plane wave expansion method
(PWE) [15, 16] with a resolution of 32 grid points, and the wave vector is divided into 500 parts
in the range of 0<k<0.5. In Fig. 2, the PCW supports even and odd modes in the photonic band
3
gap (PBG). The modes below the light line are intrinsically loss-less (i.e. not much leakage)
when the device is fabricated into the PhC slab [6]. Since the even PBG mode is seen to flatten at
large wave vectors due to the folding at the Brillouin zone-end, we just discuss the flat band of
the even mode.
3. Numerical Results and Discussions
3.1 Dispersion Engineering in PCW
The pulse of high speed optical signal is severely broadened by GVD, given by
 2  d 2 k / d  2  (1/ vg )3  d  2 / dk 2 , where ω is normalized frequency, k is wave vector and vg is
group velocity. The group velocity of the guided mode is given by the inverse of the first-order
dispersion as vg  (k /  ) 1  c / ng , where c and ng are light velocity in vacuum and group index,
respectively. We used a first-order central finite difference approximation to differentiate the
dispersion relation. However, the contribution of different frequency components to vg and β2 are
also different. Thus, assuming the Gaussian signal, we introduce a weighting factor
f (w) = 1 - (1 /Vw)2 (w - w0 )2 / 2 according to [6], Where ω0 is the central frequency. Thus, the
average group velocity and the average GVD parameter in the frequency range ∆ω become [6]
vg  
0  /2
0  /2
2  
f ( )vg ( )d /   
0  /2
0  /2
0  /2
0  /2
f ( )2 ( )d /   
(1  (1 /  )2 (  0 )2 / 2)vg ( )d / 
0  /2
0  /2
(1  (1 /  )2 (  0 )2 / 2)2 ( )d / 
(1)
(2)
We firstly study the dependence of the even PBG mode on the semi-major axis A and semiminor axis B of the elliptical air holes. Fig. 3 illustrates the movement of the even PBG mode
4
when B is fixed at 0.30a and A changes form 0.25a to 0.40a. As seen, the mode is affected
primarily at large wave vectors (i.e. the tail of the mode) when A is changed. The bandwidth and
the slope of the mode at large vectors increase in magnitude as A decreases. In addition, the
shifting tendency of the mode when B varies is shown in fig. 4. The mode moves down in
frequency when B decreases form 0.26a to 0.41a and A=0.30a is assumed. Moreover, the shape
of the mode in large wave vector remains the same. These dependent behaviors of the even mode
on A and B can be used to engineer the dispersion properties in slow light regime. We coarsetune the tail of the mode to obtain the desired group velocity with vanishing dispersion for large
wave vectors by changing A from 0.35a to 0.40a at intervals of 0.005a. Then we fine-tune the
band to the desired frequency region by adjusting B from 0.25a to 0.30a at intervals of 0.005a.
After the whole adjusting procedure, we obtain an optimal PCW with A=0.38a and B=0.27a. Fig.
5 presents the guided modes and group indices of the PCW with elliptical (A=0.38a, B=0.27a)
and circular (r=0.32a) air holes of the same filling ratio. The guided mode of the PCW with
elliptical air holes is flatter at large wave vectors than that with circular air holes. Moreover, the
negligible dispersion bandwidth with constant group index can be achieved in the PCW with
elliptical air holes. The bandwidth is single-mode range and the constant group index reaches
256. While in the PCW with circular air holes, the group index increases sharply at large wave
vectors, which can leads to large GVD.
In order to describe the influence of dispersion on the optical transmission more clearly, the
modal field distribution in PCW is analyzed by using the 2D finite difference time domain
(FDTD) simulations [17],with a resolution of 32 grid points. The temporal excitation type of the
light propagation is CW excitation at 1550nm with perfectly matched absorbing boundary layers.
The modal field distribution in the PCW with elliptical (A=0.38a, B=0.27a) and circular (r=0.32a)
5
air holes are sketched in fig. 6 (a) and (b), respectively. Compared with the PCW with circular
air holes, by the dispersion engineering, the energy is well confined inside the PCW with
elliptical air holes.
The group index ng is considered as constant within ±10% range, corresponding to [9]. Thus
the negligible dispersion bandwidth of the PCW with elliptical (A=0.38a, B=0.27a) can reach
1.23nm. In order to obtain the larger flat bandwidth over which group index is constant, we
adjust A1, B1 and d appropriately. When A1 and B1 decrease to 0.98A and 0.98B, and d equals to
0.02a, we obtain 4.4nm flat bandwidth with the constant group velocity about c/73. Then we
only decrease A1 and B1 to achieve the larger flat bandwidth. When A1 and B1 decrease to 0.94A
and 0.94B, respectively, the negligible dispersion bandwidth of the PCW can reaches 8.1nm and
the corresponding group velocity is c/41. Fig. 7 illustrates three structures of PCW with different
A1, B1 and d to highlight the evolution of the group indices, and the gray areas denote the
negligible dispersion bandwidth. The results indicate that there is a tradeoff between the
bandwidth of the data and the constant group index. The right inset depicts the GVD of the even
PBG mode in the structure Ⅱ (A1=0.98A, B1=0.98B, d=0.02a) to describe the dispersion
engineering of PCW. Considering PCW is a nanostructure, GVD is acceptable on the order of
105~106ps2/km [8, 18]. It can be seen that GVD is within the order of 106ps2/km over the
bandwidth which has a constant group index and suitable for the slow light pulse propagation in
optical buffering application.
3.2 Limitations of Buffer Performance in Dispersion Engineering PCW
In this section we focus on how the buffer capacity and the data rate are limited by
dispersion and how it can be improved by using the technique of dispersion engineering. If the
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length of delay line based on the PCW is L, then the storage time is given by Ts  L / vg and the
buffer capacity C is equal to the delay-bandwidth product TsB [19]. B=2Rb is the base-band
bandwidth of the input data, where Rb is the bit rate, assuming RZ format. Note that GVD limits
the bit rate Rb for a fixed transmission distance L [20] and one can relate GVD to Rb by using the
criterion that the broadened pulse should remain confined to its own bit slot (TB=1/Rb) [4].
Therefore, the pulse distortion is acceptable when the base-band bandwidth B  1 / (4 L | 2 |)1/2
[6]. Then the upper limitation on Rb and C can be calculated by
Rb  1 / 4( L | 2 |)1/2
C  Ts B 
L
vg
1/2


1


 4 L |  2 | 
1/2
 L 


 4 
1
vg |  2 |1/2
(4)
The structureⅡ exhibits nearly constant group index of 73 over 4.4nm bandwidth. So we
choose it to analyze the limitations of buffer performance. At the frequency ω0=0.21599, the
average GVD parameter is 7.47×104ps2/km calculated by (2), and the average group velocity is
0.0131c by (1). Then we obtain the value vg | 2 |1/2 about 3.58c·ps·km-(1/2). The upper limitations
of the data rate and buffer capacity in the structureⅡ can be deduced using equations (3) and (4).
Fig. 8 plots the maximums of data rate and buffer capacity as a function of delay length in the
structureⅡ. It can be seen that the upper limitation of the buffer capacity increases while the data
rate decreases nonlinearly as the delay length increases. Especially for the small delay length, the
maximums of data rate and buffer capacity change sharply. These results suggest that the buffer
capacity and data rate that can be supported in PCW are limited due to the dispersion–induced
broadening, especially in the fixed length where special delay time is required.
7
We now look in detail at how these limitations can be improved by using the technique of
dispersion engineering in our proposed structure. In order to describe it more clearly, we
compare the structureⅡ with the structure in [6]. The values of vg ,  2 and vg | 2 |1/2 in the
structure in [6] are 0.0206c, 7.47×104ps2/km and 3.58c·ps·km-(1/2), respectively. Fig. 9 plots the
maximum of buffer capacity and the maximum of delay time as a function of delay length in the
structureⅡ and the structure in [6]. And table 1 summarizes the values of Cmax, Ts and Rbmax in
both structures. From fig. 9 and table 1, the results indicate that the limitation of the buffer
capacity and the delay time are effectively improved in our proposed structure, due to the smaller
values of vg and vg | 2 |1/2 . To obtain the same maximum of buffer capacity 166bit, the delay
length of structureⅡ only needs 0.4mm, while that of structure in [6] needs 0.6mm. When the
delay length fixes to 1mm, the maximum of buffer capacity and the maximum of data rate are
262bit and 515Gb/s, respectively. And the corresponding delay time is 255.4ps. In addition, for
the average GVD of the structure in [6] is smaller, it can support a little higher data rate than that
of the structureⅡ, as shown in table 1. All the results show that the lower GVD and group
velocity can support larger delay time, higher buffer capacity and data rate signal transmission in
the fixed length PCW, and these limitations can be improved by using the technique of
dispersion engineering.
4. Conclusion
In short, we have not only achieved the negligible dispersion bandwidth for the requirement
of optical buffers, but also investigated the upper limitation of buffer capacity and bit rate in line
defect PCW with elliptical air holes. By adjusting the major and minor axes and the structure
parameters of the first rows, we obtained the constant group velocity ranging from c/41 to c/256
8
and the corresponding negligible dispersion bandwidth ranging from 1.23nm to 8.1nm. In
addition, the buffering capability and limitations are effectively improved in the proposed
structure by using the technique of dispersion engineering. The maximum of the buffer capacity
and the maximum of data rate can reach 262bit and 515Gb/s, respectively. And the
corresponding delay time is 255.4ps. These results show that the proposed structure has
considerable potential for realizing compact, high-performance optical buffering application in
all-optical network.
5. Acknowledgement
This research was supported in part by NSFC (No. 60707001，60932004), National 973
Program (No. 2007CB310705), National 863 Program (No.2009AA01Z214, 2009AA01Z256),
NCET (07-0110), P. R. China
Reference
[1] S. Rawal, R. Sinha, and R. M. De La Rue, "Slow light miniature devices with ultra-flattened
dispersion in silicon-on-insulator photonic crystal," Opt. Express, vol. 17, pp. 13315-13325,
2009.
[2] T. F. Krauss, "Why do we need slow light?," Nat Photon, vol. 2, pp. 448-450, 2008.
[3] T. Baba, "Slow light in photonic crystals," Nat Photon, vol. 2, pp. 465-473, 2008.
[4] G. P. Agrawal and MyiLibrary, Nonlinear fiber optics: Springer, 2001.
[5] A. Theocharidis, T. Kamalakis, A. Chipouras, and T. Sphicopoulos, "Linear and Nonlinear
Optical Pulse Propagation in Photonic Crystal Waveguides Near the Band Edge," Quantum
Electronics, IEEE Journal of, vol. 44, pp. 1020-1027, 2008.
9
[6] M. Jing and J. Chun, "Flatband Slow Light in Asymmetric Line-Defect Photonic Crystal
Waveguide Featuring Low Group Velocity and Dispersion," Quantum Electronics, IEEE
Journal of, vol. 44, pp. 763-769, 2008.
[7] A. Y. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal
waveguides," Applied Physics Letters, vol. 85, pp. 4866-4868, 2004.
[8] L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, "Photonic crystal
waveguides with semi-slow light and tailored dispersion properties," Opt. Express, vol. 14,
pp. 9444-9450, 2006.
[9] J. Li, T. P. White, L. O'Faolain, A. Gomez-Iglesias, and T. F. Krauss, "Systematic design of
flat band slow light in photonic crystal waveguides," Opt. Express, vol. 16, pp. 6227-6232,
2008.
[10] M. Jing and J. Chun, "Demonstration of Ultraslow Modes in Asymmetric Line-Defect
Photonic Crystal Waveguides," Photonics Technology Letters, IEEE, vol. 20, pp. 12371239, 2008.
[11] Y. Hamachi, S. Kubo, and T. Baba, "Slow light with low dispersion and nonlinear
enhancement in a lattice-shifted photonic crystal waveguide," Opt. Lett., vol. 34, pp. 10721074, 2009.
[12] M. Ebnali-Heidari, C. Grillet, C. Monat, and B. J. Eggleton, "Dispersion engineering of
slow light photoniccrystal waveguides using microfluidic infiltration," Opt. Express, vol. 17,
pp. 1628-1635, 2009.
[13] F. Wang, J. Ma, and C. Jiang, "Dispersionless Slow Wave in Novel 2-D Photonic Crystal
Line Defect Waveguides," J. Lightwave Technol., vol. 26, pp. 1381-1386, 2008.
10
[14] L. Jiguang and Z. R. Huang, "Low Loss Photonic Crystal Waveguide by Elliptical Unit Cell
Structure," in Lasers and Electro-Optics Society, 2006. LEOS 2006. 19th Annual Meeting of
the IEEE, 2006, pp. 827-828.
[15] S. Guo and S. Albin, "Simple plane wave implementation for photonic crystal calculations,"
Opt. Express, vol. 11, pp. 167-175, 2003.
[16] A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion
method for two-dimensional photonic crystals with arbitrary hole-shape," Physical Review
B, vol. 73, p. 075107, 2006.
[17] A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G.
Johnson, and G. W. Burr, "Improving accuracy by subpixel smoothing in the finitedifference time domain," Opt. Lett., vol. 31, pp. 2972-2974, 2006.
[18] H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. Van Hulst, T. F.
Krauss, and L. Kuipers, "Real-space observation of ultraslow light in photonic crystal
waveguides," Physical Review Letters, vol. 94, p. 073903, 2005.
[19] R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, "Slow-Light Optical Buffers: Capabilities
and Fundamental Limitations," J. Lightwave Technol., vol. 23, p. 4046, 2005.
[20] G. P. Agrawal, Fiber-optic communication systems. New York: Wiley, 2002.
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Fig. 1. Schematic of the line defect PCW with elliptical air holes.
12
Fig. 2. Typical band diagram of the line defect PCW with elliptical air holes. The odd and even
modes are in the PBG of the crystal.
13
Fig. 3. Movement of the even PBG mode when the semi-major axis A of bulk elliptical air hole
changes form 0.25a to 0.40a, where the semi-minor axis B=0.30a is assumed.
14
Fig. 4. Movement of the even PBG mode when the semi-minor axis B of bulk elliptical air hole
changes form 0.26a to 0.41a, where the semi-major axis A=0.30a is assumed.
15
Fig. 5. Guided modes and group indexes of the line defect PCW with elliptical and circular air
holes of the same filling ratio.
16
Fig. 6. Modal field distribution in line defect PCW with (a) elliptical (A=0.38a, B=0.27a) and (b)
circular (r=0.32a) air holes of the same filling ratio, for a wavelength of 1550nm.
17
Fig. 7. Group index as a function of normalized frequency with different A1, B1 and d. The
bandwidths with constant group-delay are single-mode ranges. The right inset depicts the GVD
of the even PBG mode in the structure Ⅱ. And the gray areas denote the negligible dispersion
bandwidth.
18
Fig. 8. Maximums of data rate and buffer capacity as a function of delay length in the structureⅡ.
19
Fig. 9. Maximum of buffer capacity and delay time as a function of delay length in the structure
Ⅱ and the structure in [6].
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Table 1. Maximum of buffer capacity, delay time and maximum of data rate in the structureⅡ
and the structure in [6]
Structure in [6]
StructureⅡ
L
Cmax
Ts
Rbmax
Cmax
Ts
Rbmax
(mm)
(bit)
(ps)
(Gb/s)
(bit)
(ps)
(Gb/s)
0.2
117
50.8
1153
96
32.4
1486
0.4
166
101.7
815
136
64.7
1051
0.6
203
152.6
665
166
97.1
858
0.8
235
203.5
576
192
129.4
743
1.0
262
255.4
515
215
161.8
664
21