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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 8 — Apr. 16, 2007
  • pp: 4536–4545
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Periodic dielectric waveguide beam splitter based on co-directional coupling

Pi-Gang Luan and Kao-Der Chang  »View Author Affiliations


Optics Express, Vol. 15, Issue 8, pp. 4536-4545 (2007)
http://dx.doi.org/10.1364/OE.15.004536


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Abstract

A compact beam splitter consisting of three branches of periodic dielectric waveguides (PDW) is designed and analyzed theoretically. Both the symmetrical and asymmetrical configurations of the beam splitter are studied. The band structure for the guided modes is calculated by using finite-difference time-domain (FDTD) method with Bloch-type boundary conditions applying in an appropriate supercell. The field patterns for the whole structure and the transmissions for the output ports are calculated using the multiple scattering method. By utilizing the co-directional coupling mechanism, the light injected into the input branch can be efficiently transferred into the two output branches if the phase matching conditions are satisfied. The coupling length is short and the broad-band requirement can be achieved. Bending loss is small and high transmission (above 95 %) can be preserved for arbitrarily bent PDW if the bend radius of each bend exceeds five wavelengths. This feature indicates that the periodic dielectric waveguide beam splitter (PDWBS) is a high efficiency device for power redistribution while avoiding the lattice orientation restriction of the photonic crystal waveguides (PCW).

© 2007 Optical Society of America

1. Introduction

Photonic crystals (PC) are periodic dielectric structures made for manipulating light at about wavelength scale [1

1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals-Molding the Flow of Light (Princeton University Press, 1995).

, 2

2. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).

]. The dispersion relations of the propagating modes in PC are given by the photonic band structure, which contains passbands and bandgaps. Many functional devices using photonic bandgap (PBG) effect have been proposed and are expected to play important roles in high-density optical integrated circuits. The devices based on PBG structures have the advantage of very small feature sizes comparing with their conventional counterparts. For instance, a photonic crystal waveguide (PCW) can be designed to guide light with high transmission in the sub-micron scale, which is almost an impossible task for a conventional dielectric waveguide (CDW)[3

3. A. Mekis, J. C. Chen, I. Kurland, S.-H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996). [CrossRef] [PubMed]

]. However, PCW also has several disadvantages. First of all, a sharp bend implies large interior reflection, which may reduce the transmitting power. In addition, the waveguide track of a PCW must follow the specific lattice orientation of the background PC structure, thus it cannot be bent arbitrarily. Furthermore, a PCW must be embedded in a wide enough background PC region (at least several lattice constants) in order to avoid the propagation loss, thus the PCW component occupies much space in the transverse dimension. These facts may cause some inconvenience toward fabricating real optical integrated circuits.

In order to control the flow of light completely, power division devices are also needed. The most commonly used power division devices are beam splitters (BS). In the past few years, a considerable number of photonic crystal beam splitters (PCBS) relying on different mechanisms have been proposed. The T-shape [4

4. S.-H. Fan, S. G. Johnson, J. D. Joannopoulos, G. Manoatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001). [CrossRef]

], Y-junction [5

5. S. Boscolo, M. Midrio, and T. F. Krauss, “Y junctions in photonic crystal channel waveguides: high transmission and impedance matching,” Opt. Lett. 27, 1001–1003 (2002). [CrossRef]

], and cross-type [6

6. C. C. Chen, H. T. Chien, and P. G. Luan, “Photonic crystal beam splitter,” Appl. Opt. 43, 6187, (2004). [CrossRef] [PubMed]

] PCW beam splitters (PCWBS) use one or more local defects to achieve impedance matching and power division. On the other hand, dispersion-based BS use the self-collimation effect of PC passband together with the evanescent wave tunnelling or partial reflection by line defects to realize the power division [7

7. S.-Y. Shi, A. Sharkawy, G.-H. Chen, D. M. Pustai, and D. W. Prather “Dispersion-based beam splitter in photonic crystal,” Opt. Lett. 29, 617–619 (2004). [CrossRef] [PubMed]

, 8

8. X.-F. Yu and S.-H. Fan, “Bends and splitters for self-collimated beams in photonic crystal,” Appl. Phys. Lett. 83, 3251–3253 (2003). [CrossRef]

, 9

9. S. G. Lee, S. S. Oh, J. E. Kim, H. Y. Parka, and C. S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. 87, 181106 (2005). [CrossRef]

]. More sophisticated BS devices and interferometers consisting of PBG mirrors, partial reflection line defects, and dielectric or photonic crystal waveguides have also been proposed [10

10. P. Pottier, S. Mastroiacovo, and R. M. D. L. Rue, “Power and polarization beam-splitters, mirrors, and integrated interferometers based on air-hole photonic crystals and lateral large index-contrast waveguides,” Opt. Express 14, 5617–5633 (2006). [CrossRef] [PubMed]

]. Although these PCBS devices can be made compact, they still have the same kind of disadvantages mentioned above.

Recently, PC power splitters and PC switches utilizing directional coupling mechanism were considered [11

11. T. Liu, A. R. Zakharian, M. Fallahi, J. V. Moloney, and M. Mansuripur, “Multimode Interference-Based Photonic Crystal Waveguide Power Splitter,” J. Lightwave Technol. 22, 2842–2846 (2004). [CrossRef]

, 12

12. N. Yamamoto, T. Ogawa, and K. Komori, “Photonic crystal directional coupler switch with small switching length and bandwidth,” Opt. Express 14, 1223–1229 (2006). [CrossRef] [PubMed]

]. In [11

11. T. Liu, A. R. Zakharian, M. Fallahi, J. V. Moloney, and M. Mansuripur, “Multimode Interference-Based Photonic Crystal Waveguide Power Splitter,” J. Lightwave Technol. 22, 2842–2846 (2004). [CrossRef]

], PCWBS relying on multi-mode interference were considered. A 1 × 2 waveguide splitter can be designed by appropriately choosing the length of the multi-mode section of the waveguide so that the two symmetrically distributed beam images appear correctly at the entrances of the two output singlemode waveguides. In [12

12. N. Yamamoto, T. Ogawa, and K. Komori, “Photonic crystal directional coupler switch with small switching length and bandwidth,” Opt. Express 14, 1223–1229 (2006). [CrossRef] [PubMed]

], the authors first showed that the switching length and bandwidth have a trade-off relationship in the conventional directional coupler switches. They then redesigned the dispersion curves of the modes via modifying the structure of the mode-coupling section, and a switch with compact size and wide bandwidth characteristics was successfully realized. Since the operation of these directional coupling devices are based on the phase matching conditions between coupling modes, they can be systematically designed by analyzing the dispersion curves of the modes.

2. Review of the previous work on PDW

3. Dispersion relations of the modes in the coupling section

In this paper, the dielectric constant ε = 11.56 and the radius r = 0.2a of the cylinders are chosen as the same as that used in [13

13. P. G. Luan and K. D. Chang, “Transmission characteristics of finite periodic dielectric waveguides,” Opt. Express 14, 3263–3272 (2006). [CrossRef] [PubMed]

]. The simulation methods are also the same. Two arrangements of rods shown in Fig. 1(a1) and (b1) are considered, which represent the mode-coupling section of two different designs of PDWBS. Both structures are multimode PDWs formed by three rows of rods in air. For the case of width Ly = 1.3a, the corresponding band structures (dispersion relations) are shown in Fig. 1(a2), (b2). The number of low-frequency guiding bands is equal to the number of rows. In addition, for a definite frequency, different modes correspond to different k vectors and different field patterns (See the insets of Fig. 1(a2), (b2)).

When a mode of the single-row PDW (the input waveguide) of appropriate frequency is injected into the coupling section of the PDWBS (the three rows region), mode exchange phenomenon happens. The injected mode can be expressed in this region as a superposition wave of two modes with different k, say, k 1 and k 2. Assuming that k 1 > k 2, then k 1 represents the first band mode and k 2 the second band mode. The energy of the k 1 mode is concentrated in the central row, whereas the energy of the k 2 mode is mostly distributed in the two side-rows. Along the propagation direction, the two modes interfere with each other via establishing the phase difference, thus the field pattern changes its manifestation from one to another in a propagation distance (coupling length)

Fig. 1. (Color on line) (a1) and (b1): The two kinds of multimode PDWs consisting of three rows of dielectric rods. The size of the supercells for calculating the band structures is a×14a. (a2) and (b2) are the calculated band structures of these two waveguides (Choosing Ly = 1.3a.). The mode patterns are shown in the insets.
Fig. 2. The coupling length Lc for the two cases in Fig.1 as functions of frequency. The length unit is the lattice constant a. (a) and (b) correspond to the structures in Fig. 1(a1) and (a2), respectively.
Fig. 3. (a) The structure of the PDWBS. It consists of an input waveguide, the coupling section (b), two S-shaped waveguides (c), and two output (straight) waveguides. The coupling section is denoted by the solid line rectangle in (a), and its length is Lc. The Pi and Pa,b are the input and output power evaluated at the planes indicated in the figure, respectively. The band radius is chosen as R = 19.1a and the bend angle θ is arbitrary.
Lc=πk1k2.
(1)

Since the k 1 mode is almost the same as the original single-row mode, the length of the coupling section of the PDWBS for splitting one beam to two can be approximately set as the coupling length Lc. This is the central idea for designing the PDWBS based on the directional-coupling mechanism. Here we have utilized only the simplest concept of mode interference. More thorough treatment of coupled waves can be found in [14

14. A. Yariv and P. Yeh, Optical Waves in Crystals (John Wiley & Sons, New York, 1984).

].

To design the PDWBS, we have to find the appropriate coupling length. We name the two structures in Fig. 1(a1) and (b1) as structure A and structure B, respectively. We vary the width Ly and recalculate the dispersion relations of the modes, and then derive the corresponding coupling lengths according to Eq.(1). For Ly = 1.0a, 1.2a, and 1.4a, the results are shown in Fig. 2(a) and (b). As one can see, when we reduce the width of the coupling section (the three rows region), the coupling length is also reduced. For a given width Ly the coupling length for structure A is a monotonically decreasing function of frequency. Besides, there is a cut-off frequency. For example, when Ly = 1.2a, the cut-off frequency is a/λ = 0.23. This frequency is the top frequency of the first band. In fact, it is impossible to couple two modes, one in the 1st-band and one in the 2nd-band, above this frequency. This explains why there is the cut-off frequency. On the other hand, the coupling length for structure B is almost a constant in a wide range of frequency from 0.2 to 0.24, insensitive to the frequency variation. This feature indicates that structure B is the better choice to be used as the coupling section of a wideband PDWBS. In the following simulations we choose structure B of length 8a as the coupling section of the PDWBS, and the width Ly is assumed to be 1.2a.

Fig. 4. (Color on line) The one-arm transmission spectrum for the symmetric PDWBS as function of splitting angle. The horizontal and vertical axes are the reduced frequency a/λ and the splitting angle θsp, respectively.

4. System description and numerical results

We now turn to the system description before doing numerical simulations. Fig. 3(a) illustrates the structure of the beam splitter, which consists of an input waveguide, a coupling section, two S-shaped arms, and two straight output waveguides. The input and output waveguides are single row PDWs of 20 and 10 rods, respectively. The coupling section is of the type B structure of length Lc = 8a and width Ly = 1.2a. An S-shaped arm (excluding the output straight PDW) contains 61 rods, forming the two oppositely curved bends and one straight PDW in between. Hereafter we denote the two arms as Arma and Armb. For any one of the two arms, say, Arma, the two bends (arcs) inside are identical, and can be characterized by the bending angle θa. The bending angles θa,b for the two arms take arbitrary values from 0° to 90°, and the bend radius R is chosen to be 19.1a. In a bend, one lattice spacing a contributes 3° to the bending angle. Thus for a 90° bent S-shaped arm the straight PDW between the two bends disappears. In order to calculate the transmissions Ta,b, we insert three planes of width 6a in the input and output PDWs at the positions indicated by Fig. 3(b),(c) and evaluate the energy fluxes Pi, and Pa,b. The transmissions through the two output PDWs that connected with Arma,b are then defined as Ta,b = Pa,b/Pi.

We now study the influences of the bending angles θa,b on the transmission spectra Ta,b(ω). We first study the symmetric case of θa =θb, here we define θsp =θa as the splitting angle. It is obvious that in this case we must have Ta = Tb. In order to reduce the interference effect between the two arms, we only consider the situations that θsp > 9°. The transmission spectrum Ta,b(ω) for different values of θsp is shown in Fig. 4. The simulation result reveals that on average for one arm the transmission is higher than 0.425, that is, the averaged total transmission is higher than 0.85, if 9° < θsp < 90° are considered. The total transmission is high enough so this symmetric structure can indeed be used as a beam splitter. Furthermore, high splitting ability (> 0.46) can be achieved around a/λ = 0.219. It seems that power transfer can be accomplished more efficiently in the central part of the frequency range we considered.

Fig. 5. (Color on line) The transmission spectra Ta,b(ω) for the asymmetric configuration of the PDWBS as functions of θb, taking θa to be fixed. Here “a” and “b” stand for the Arma and Armb, respectively. For θa = 0°, the calculated Ta(ω) and Tb(ω) are plotted in (a1) and (b1). For θa = 90°, the results are plotted in (a2) and (b2).

Now we explore the applicability of the asymmetric configuration of the three-branch PDW structure as a beam splitter. As mentioned before, the number of rods in the two arms are the same. We first fix the bending angle θa and study the influence of the bending angle θb on the transmission spectra Ta,b(ω). Two simplest situations are considered. In the first situation we set θa = 0° and vary θb from 9° to 90°, and the calculated Ta(ω) and Tb(ω) are shown in Fig. 5(a1) and (b1), respectively. Similarly, in the second situation we set θa = 90°, and Ta,b(ω) are shown in Fig. 5(a2) and (b2). From the results of Fig. 5(a1) and (b1), we find that Ta is in general larger than Tb, and on average we have Ta : Tb ≈ 0.63 : 0.37, implying that the internal reflection of wave in the straight arm is much smaller than in the bent arm. Moreover, the typical value of total transmission is larger than 95%, exceeding that of the symmetric structure (see Fig.4). For case of θa = 90°, the results are very different, as shown in Fig. 5(a2) and (b2). In this situation, light powers are more equally distributed between the two arms. The average splitting ratio Ta : Tb is 0.44 : 0.56. It indicates that the asymmetric configuration of three-branch structure can indeed be used as a practical beam splitter.

We now study the influence of varying both θa and θb on the transmissions Ta,b. We choose a/λ = 0.219 and plot the calculated Tb in Fig. 6. The horizontal and vertical axes are θa and θb, respectively. By symmetry consideration, the result for Ta can be easily obtained by the reflection mapping with respect to the 45°-oriented line (the line described by the equation θa = θa). According to the numerical results, the ratio θa : θb on average is 0.48 : 0.52. In addition, for the major portion of the transmission map shown in Fig. 6, Ta and Tb are almost equal. This indicates that asymmetric PDWBS can work well in most situations.

Fig. 6. (Color on line) The transmission Tb as function of θa and θb. The reduced frequency of the incident beam is a/λ = 0.219. The horizontal and vertical axes are θa and θb, respectively.

To demonstrate the splitting abilities of the symmetric and asymmetric configurations of PDWBS, we plot the corresponding field patterns at frequency a/λ = 0.219 in Fig.7 (a) and (b). In Fig. 7(a), the splitting angle for the symmetric PDWBS is taken to be θsp = 60°. In Fig. 7(b), we set θa = 75° and θb = 39° for the asymmetric PDWBS. The field patterns confirm again the beam splitting abilities of the two kinds of structures. We believe both the symmetric and asymmetric structures of PDWBS can be used as practical devices in the optical integrating circuits.

In this paper the band structure of the PDWBS is calculated using the FDTD method, whereas the transmission data in Fig. 4–6 and the field pattern in Fig. 7 are obtained by implementing the multiple scattering method. One can also calculate the transmission data using FDTD method, however, it takes a lot of time. The multiple scattering method is the faster method in dealing with such kind of calculation when compared with the FDTD method. However, the multiple scattering method also has its weakness. For instance, the method is hard to implement directly if the cross section of the dielectric cylinder is not circular. Besides these differences, the FDTD method is as good as the multiple scattering method when be used to calculate the field distribution. We re-calculate the field distribution for the symmetric structure of Fig. 7(a) using FDTD method by launching the light from the end of single row waveguide. A snapshot of the field pattern (the real part) around the coupling section of the PDWBS is shown in Fig. 8. The result confirms that the light energy can be successfully switched to the two output rows of rods.

Recently, Zhang and Li proposed to use line-defect bending waveguides in photonic crystal for optical interconnections [16

16. Y. Zhang and B.-J. Li, “Photonic crystal-based bending waveguides for optical interconnections,” Opt. Express 14, 5723–5732 (2006). [CrossRef] [PubMed]

]. According to their numerical simulations, high transmission over 98.5% can be achieved if the various bending waveguides are operated at frequencies within the photonic band gap. It seemed possible to design compact beam splitter based on this kind of waveguide for power division while avoiding the lattice orientation restriction.

Fig. 7. (Color on line) The field patterns for the symmetric (a) and asymmetric (b) PDWBS.
Fig. 8. (Color on line) The field distribution (real part) around the coupling section of the symmetric PDWBS. This result is obtained by using the FDTD method. The structure and simulation parameters are the same as that used in Fig. 7(a).

5. Conclusion

In conclusion, we have studied the transmission properties of PDWBS numerically. Effects of symmetric and asymmetric bending of the arms on the transmissions are considered. By employing a simple analysis on the mode coupling condition, the coupling length can be determined. Numerical results reveal that high transmissions can be preserved for both the symmetric and asymmetric structures. The advantage that the geometric restriction of PCW can be avoided might be helpful in making compact optical circuits.

Acknowledgments

The authors gratefully acknowledge financial support from National Science Council (Grant No. NSC 95-2221-E-008-114-MY3) and Ministry of Economic Affairs (Grant No. 95-EC-17-A-08-S1-0006) of the Republic of China, Taiwan.

References and links

1.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals-Molding the Flow of Light (Princeton University Press, 1995).

2.

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).

3.

A. Mekis, J. C. Chen, I. Kurland, S.-H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996). [CrossRef] [PubMed]

4.

S.-H. Fan, S. G. Johnson, J. D. Joannopoulos, G. Manoatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001). [CrossRef]

5.

S. Boscolo, M. Midrio, and T. F. Krauss, “Y junctions in photonic crystal channel waveguides: high transmission and impedance matching,” Opt. Lett. 27, 1001–1003 (2002). [CrossRef]

6.

C. C. Chen, H. T. Chien, and P. G. Luan, “Photonic crystal beam splitter,” Appl. Opt. 43, 6187, (2004). [CrossRef] [PubMed]

7.

S.-Y. Shi, A. Sharkawy, G.-H. Chen, D. M. Pustai, and D. W. Prather “Dispersion-based beam splitter in photonic crystal,” Opt. Lett. 29, 617–619 (2004). [CrossRef] [PubMed]

8.

X.-F. Yu and S.-H. Fan, “Bends and splitters for self-collimated beams in photonic crystal,” Appl. Phys. Lett. 83, 3251–3253 (2003). [CrossRef]

9.

S. G. Lee, S. S. Oh, J. E. Kim, H. Y. Parka, and C. S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. 87, 181106 (2005). [CrossRef]

10.

P. Pottier, S. Mastroiacovo, and R. M. D. L. Rue, “Power and polarization beam-splitters, mirrors, and integrated interferometers based on air-hole photonic crystals and lateral large index-contrast waveguides,” Opt. Express 14, 5617–5633 (2006). [CrossRef] [PubMed]

11.

T. Liu, A. R. Zakharian, M. Fallahi, J. V. Moloney, and M. Mansuripur, “Multimode Interference-Based Photonic Crystal Waveguide Power Splitter,” J. Lightwave Technol. 22, 2842–2846 (2004). [CrossRef]

12.

N. Yamamoto, T. Ogawa, and K. Komori, “Photonic crystal directional coupler switch with small switching length and bandwidth,” Opt. Express 14, 1223–1229 (2006). [CrossRef] [PubMed]

13.

P. G. Luan and K. D. Chang, “Transmission characteristics of finite periodic dielectric waveguides,” Opt. Express 14, 3263–3272 (2006). [CrossRef] [PubMed]

14.

A. Yariv and P. Yeh, Optical Waves in Crystals (John Wiley & Sons, New York, 1984).

15.

Y. Y. Chen and Z. Ye, “Acoustic Attenuation by Two-Dimensional Arrays of Rigid Cylinders,” Phys. Rev. Lett. 87, 1843011–4 (2001). [CrossRef]

16.

Y. Zhang and B.-J. Li, “Photonic crystal-based bending waveguides for optical interconnections,” Opt. Express 14, 5723–5732 (2006). [CrossRef] [PubMed]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(250.5300) Optoelectronics : Photonic integrated circuits

ToC Category:
Photonic Crystals

History
Original Manuscript: February 8, 2007
Revised Manuscript: March 22, 2007
Manuscript Accepted: March 22, 2007
Published: April 3, 2007

Citation
Pi-Gang Luan and Kar-Der Chang, "Periodic dielectric waveguide beam splitter based on co-directional coupling," Opt. Express 15, 4536-4545 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-4536


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References

  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals-Molding the Flow of Light (Princeton University Press, 1995).
  2. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).
  3. A. Mekis, J. C. Chen, I. Kurland, S.-H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996). [CrossRef] [PubMed]
  4. S.-H. Fan, S. G. Johnson, J. D. Joannopoulos, G. Manoatou, and H. A. Haus, "Waveguide branches in photonic crystals," J. Opt. Soc. Am. B 18, 162-165 (2001). [CrossRef]
  5. S. Boscolo, M. Midrio, and T. F. Krauss, "Y junctions in photonic crystal channel waveguides: high transmission and impedance matching," Opt. Lett. 27, 1001-1003 (2002). [CrossRef]
  6. C. C. Chen, H. T. Chien, and P. G. Luan, "Photonic crystal beam splitter," Appl. Opt. 43, 6187, (2004). [CrossRef] [PubMed]
  7. S.-Y. Shi, A. Sharkawy, G.-H. Chen, D.M. Pustai, and D.W. Prather "Dispersion-based beam splitter in photonic crystal," Opt. Lett. 29, 617-619 (2004). [CrossRef] [PubMed]
  8. X.-F. Yu, and S.-H. Fan, "Bends and splitters for self-collimated beams in photonic crystal," Appl. Phys. Lett. 83, 3251-3253 (2003). [CrossRef]
  9. S. G. Lee, S. S. Oh, J. E. Kim, H. Y. Parka, and C. S. Kee, "Line-defect-induced bending and splitting of selfcollimated beams in two-dimensional photonic crystals," Appl. Phys. Lett. 87, 181106 (2005). [CrossRef]
  10. P. Pottier, S. Mastroiacovo, and R. M. D. L. Rue, "Power and polarization beam-splitters, mirrors, and integrated interferometers based on air-hole photonic crystals and lateral large index-contrast waveguides," Opt. Express 14, 5617-5633 (2006). [CrossRef] [PubMed]
  11. T. Liu, A. R. Zakharian,M. Fallahi, J. V. Moloney, andM.Mansuripur, "Multimode Interference-Based Photonic Crystal Waveguide Power Splitter," J. Lightwave Technol. 22, 2842-2846 (2004). [CrossRef]
  12. N. Yamamoto, T. Ogawa, and K. Komori, "Photonic crystal directional coupler switch with small switching length and bandwidth," Opt. Express 14, 1223-1229 (2006). [CrossRef] [PubMed]
  13. P. G. Luan, and K. D. Chang, "Transmission characteristics of finite periodic dielectric waveguides," Opt. Express 14, 3263-3272 (2006). [CrossRef] [PubMed]
  14. A. Yariv, P. Yeh, Optical Waves in Crystals (John Wiley & Sons, New York, 1984).
  15. Y. Y. Chen and Z. Ye, "Acoustic Attenuation by Two-Dimensional Arrays of Rigid Cylinders," Phys. Rev. Lett. 87, 1843011-4 (2001). [CrossRef]
  16. Y. Zhang and B.-J. Li, "Photonic crystal-based bending waveguides for optical interconnections," Opt. Express 14, 5723-5732 (2006). [CrossRef] [PubMed]

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