Open Access
Add to BibSonomyAbstract
We have investigated the transmission properties of a photonic crystal waveguide (PCW) formed by holographic lithography for the first time with a two-dimensional (2D) triangular holographic photonic crystal (PhC) including a line defect with two 60° bends. Calculations have shown that for this PCW high transmission (>90%) through sharp corners can be obtained in a wide frequency range from 0.298 to 0.310 (ωa/2πc) with the relative band gap of 4% when the dielectric contrast is 7.6:1. As far as we know, this result should be the widest frequency range with high transmission (>90%) in the waveguide of similar 2D triangular PhCs ever reported. We have also found that the specific holographic designs of PhC have strong influence on the resonance between the two waveguide bends, and thus this fact can be used as an effective means to improve the transmission property of 2D holographic PCW. In addition to the simplicity and low cost of holographic fabrication of PhCs, these features may reveal the possibly better guiding ability of holographic PCW than the conventional waveguide and the promising potential of the former in the application of photonic integrated circuits.
©2008 Optical Society of America
1. Introduction
Photonic crystals (PhCs) have the ability to control the propagation of light within a frequency band (i.e., a stop band) for their periodic arrangements of dielectric [1, 2]. Utilizing the band gap effect one may introduce a line defect in a photonic crystal to guide the electromagnetic wave of the frequency in the stop gap [3]. Such a line defect is called a photonic crystal waveguide (PCW). The PCW is very compact (its typical width is around λ/2) and allows sharp bends without losses [4]. In theory, this full control of light can be achieved only by using three-dimensional (3D) PhCs [5, 6], but the introduction of linear defects in such 3D structures is not easy. A real alternative system is a dielectric slab of a two-dimensional (2D) PhC, also called PhC slab [7] consisting of a 2D PhC of finite height, where light is confined in the vertical direction by total internal reflection (TIR). A waveguide can be created in the PhC slab by introducing a linear defect in the in-plane 2D periodic structure. Nevertheless, waveguides in simple 2D PhCs (vertical confinement is not considered) and fundamental mode confined in these structures will be used throughout this paper. The effect studied here is based on the fact that the finite thickness of the slab can be approximated by using effective refractive index of the zeroth-order guided mode of the dielectric slab, therefore the field patterns of eigenmodes of the 2D PhC and 2D PhC slab are almost identical [8].
Since the ability to guide light waves around sharp corners with high efficiency is crucial for photonic integrated circuits, many studies have been carried out concerning waveguide bends through sharp bends in 2D PhC slabs [9-13] and various methods are used to improve the transmission property of waveguide bends, such as adding an appropriate defect [14] or fabricating a new taper structure [15]. However, in all these papers the authors limit their studies to the structures formed by air rods with regular circular cross sections. As far as we know, none of these works dealt with the PhCs formed by holographic lithography (HL) which usually have irregular “atoms” or columns. It is well known that the photonic band gap (PBG) property of resultant structure varies with the shape of “atoms” or columns, thus the PBGs for PhCs made by HL will be different from those of regular structures, so will the propagation properties. Besides, interference lithography is easier and quicker for patterning microstructures in large scale than many other methods [16-18]. In this paper, we investigate the transmission property of the waveguide with two 60° bends in PhCs made by HL with the 2D triangular structures reported in our previous work [19] as an example. The result of analyses demonstrates that a holographic PCW can efficiently guide light in a wide range of frequency around sharp corners, and the widest frequency range of 0.012 (ωa/2πc) from 0.298 to 0.310 (ωa/2πc) with high transmission (>90%) can be obtained, which improves the bandwidth more than twice compared with the previous report [9]. This study also reveals that the resonance between the two bends has a close relation with the configuration of PhC and can be used as an important factor to improve the transmission property of 2D PCW. In the following we will first explain the 2D PhC structure and the line defect considered in our work, and then discuss their transmission properties, where the band structures of holographic PCWs are calculated by the 2D plane-wave expansion method with supercell technique [20, 21], and a finite-difference time-domain (FDTD) method [22] is used to calculate the transmission property of the guided mode.
2. Structures and waveguide design
Fig. 1. Different cross section shapes of the considered inverse structures corresponding to (a) It=3.0, f=61.2%; (b) It=2.5, f=48.7%; (c) It=2.1, f=32.8%; where the black parts denote the dielectric areas and the white parts air.
Fig. 2. Schematic of a 2D PhC waveguide with two 60° bends. Two arrows indicate the Γ-J and Γ-X direction of the crystal respectively.
The 2D periodic structure considered in this work is a triangular Bravais lattice formed by the interference technique of three noncoplanar beams [19]. The intensity of interference field for the formation of this lattice can be expressed as
A certain lattice can be formed if we wash away the region of I< It, where It is a chosen intensity threshold. By filling this structure with a material of high refractive index and then removing the template, an inverse structure can be obtained [23]. When the filling ratio f of the dielectric material is chosen from 61.2% to 32.8%, corresponding to It changing from 3.0 to 2.1, the shape of air holes of this inverse structure changes gradually from a circle to a hexagon approximately as shown in Fig. 1.
It has been known that some peaks of transmission may result from the resonance between two bends in the waveguide of ordinary PhCs with circular air holes [9]. It will be interesting to see whether or not the resonance between two bends has the effect on the transmission property of waveguide in the PhC formed by HL. To investigate this problem, we introduce a waveguide with two 60° bends by filling a row of air holes in Γ-J direction (0°) in the 2D PhC mentioned above and then turning this line defect by 30° to Γ-X direction and finally straightening it again to the Γ-J direction as shown in Fig. 2. The distance between the two bends is 9a, where a is the lattice constant in Eq. (1). If the PCW is not long enough, the input pulse and reflected pulse at the bend will overlap which will affect the precision of calculation, so the length of PCW for calculation is selected to be 200a with dielectric contrast of 7.6:1.
3. Numerical results and analysis
Fig. 3. Band diagrams for TE-like modes of (a) the triangular lattice configuration with f=48.7% and (b) the selected single-line-defect waveguide in this configuration. The gray areas show the PhC modes that can propagate inside the PhC configuration; the solid color lines show waveguide modes.
For the reason that the triangular lattice with irregular air holes embeded in a dielectric material possesses a very large gap for TE-like waves, in this paper we consider only the TE modes as in other literatures [8, 9]. We simulate the propagation of light inside this waveguide by 2D FDTD and use the Gaussian pulse to calculate the transmission and reflection spectra. The calculation result shows that when It is close to 2.5, corresponding to f=48.7%, the normalized frequency range for high transmission (>90%) reaches the widest of 0.012 (ωa/2πc) spanning from 0.298 to 0.310 (ωa/2πc), which is much wider than the result of 0.005 obtained in Ref. 9. 2D plane-wave expansion method is used to calculate the band diagram of the PhC configuration with It=2.5 and f=48.7% for the TE-like mode. As shown in Fig. 3(a), for TE mode, a wide photonic band gap exists from frequency 0.290 to 0.418 (ωa/2πc). Next, we create a waveguide by filling a row of air holes in the Γ-J direction and use a supercell with the waveguide (as inset in Fig. 3(b)) to calculate the band diagram of this 2D PCW. The supercells containing 13 unit cells and about 1000 plane waves are used. The band diagram of the waveguide is shown in Fig. 3 (b), where the gray areas show the PhC modes that can propagate inside the PhC structure and two guided modes appear inside the PBG. For these frequencies, the holographic PCW is strictly single-mode and the fundamental mode is the only one that will be excited inside the structure while the higher order modes remain unexcited. Therefore we have worked around the lower edge of the fundamental guided mode, in the range around 0.290 to 0.310 (ωa/2πc).
Figure 4(a) indicates the transmission and reflection spectra of the waveguide in the PhC structure of It=2.5 and f=48.7% with ε=7.6. The red curve here corresponds to the transmission spectra and the blue one to the reflection spectra. High transmission of more than 90% can be obtained in a wide frequency range from 0.298 to 0.310 (ωa/2πc) with the relative band gap of 4%. To the best of our knowledge, this result should be the widest frequency range with high transmission (>90%) in the waveguide of similar 2D triangular PhCs in ever reported papers. In order to see more clearly the influence of resonance between the two bends on the transmission property of waveguide in the PhC formed by HL, the transmission spectrum of the similar PCW with only one 60° bend is given in Fig. 4(b) for comparison. Obviously the frequency range with high transmission (>90%) in Fig. 4(a) is much wider than that in Fig. 4(b). This difference convincingly proves that the existence of resonance between two bends has strong effect on the transmission spectra of a PCW, and this fact can be used to optimize the transmission property of a PCW effectively. In addition, it can be seen that the result here calculated by using 2D FDTD accords well with the band structure in Fig. 3(b).
Fig. 4. Transmission (red curve) and reflection spectra (blue curve) of the PhC waveguide (a) with two 60° bends and (b) with one 60° bend when It=2.5, f=48.7% and ε=7.6.
Fig. 5. Variation of transmission spectrum with different filling ratios for the triangular configuration through the waveguide with two 60° bends.
Variation of transmission spectrum in the holographic PCWs with different filling ratios is shown in Fig. 5 with various color curves. It is clear that the high transmission of more than 90% can be achieved easily through two 60° waveguide bends in these holographic PhCs. The output detector locates at a point of 15a away form the second bend. The main output pulse is first achieved and then, for the two 60° waveguide bends have the similar configuration, the resonance between two bends leads to the appearance of several output pulses which give rise to the second transmission peak in the vicinity of primary transmission peak. It is worth noting that the value of the resonated transmission peak and the distance between the resonated peak and the primary peak have a close relation with the configuration of PhCs of different filling ratios. For example, when the filling ratio f=61.2%, the shape of air holes embedded in dielectric material is approximately circular, the resonated transmission peak is far from the primary peak with a low value of transmission. With the decreasing of filling ratio, the resonated transmission peak approaches the primary one gradually and finally connects with it. When It is close to 2.5, corresponding to f=48.7%, the normalized frequency range for high transmission (>90%) reaches the widest of 0.012 (ωa/2πc) spanning from 0.298 to 0.310 (ωa/2πc). When the filling ratio keeps on decreasing further, the shape of air holes becomes approximately hexagonal and the resonated peak trends to go away from the primary peak again. In addition to the wide frequency range with high transmission, the span of filling ratio to obtain a high transmission (>90%) is very wide in this case. These results have shown that the resonance between two bends has a close relation with the configuration of PhC and is an important factor to improve the transmission property of 2D PCW. If the configuration of PhC is properly designed, the high transmission in a wide frequency range can be obtained for the existence of resonance between two bends.
Fig. 6. (Media 1) Snapshot of the TE polarization wave propagation at the frequency of 0.305 (ωa/2πc) in the inverse structure of It=2.5 or f=48.7 % with dielectric contrast of 7.6 :1. The linked movie shows how the pulse with frequency spanning from 0.298 to 0.308 (ωa/2πc) travels through the waveguide as time increases.
To demonstrate the transmission properties of the holographic PCW with two 60° bends, in Fig. 6 we plot the amplitude distribution of magnetic field in the waveguide for the case when frequency is 0.305 (ωa/2πc). The blue and red spots correspond to the negative and positive values of the amplitude respectively as indicated at the bottom of this figure. The linked movie shows how the pulse with the frequency spanning from 0.298 to 0.308 (ωa/2πc) travels through the waveguide with time increasing. Obviously both the light of 0.305 (ωa/2πc) and the pulse are well confined to the waveguide with little loss or leakage into the photonic crystal. These pictures have convincingly verified that high transmission in a wide frequency range truly exists in the waveguide of holographic PhC with f=48.7% when dielectric contrast is 7.6:1.
4. Conclusions
In summary, we have investigated the transmission properties of a holographic PCW for the first time and demonstrated that a holographic PCW can efficiently guide light in a wide range of frequencies around sharp corners. Specifically, the widest frequency span of 0.012(ωa/2πc) with high transmission (>90%) is obtained through two 60° bends in the triangular holographic PhC with irregular air holes when It=2.5, corresponding to f=48.7%, for ε=7.6. Analyses have revealed that the resonance between two 60° bends has a close relation with the configuration of PhC, and thus the transmission property of a 2D holographic PCW can be effectively improved by the proper design of the PhC. This work makes holographic PhCs promising for application in the range of photonic integrated circuits and provides a guideline for optimizing the transmission property of a 2D holographic PCW. This idea and the method of analysis here may open a new freedom for PCW engineering.
Acknowledgment
This work is sponsored by the National Natural Science Foundation (grant 60777008 and 10804063), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and the Natural Science Foundation of Shandong Province (grant Y2004G01 and Y2006A09), China.
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. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000). [CrossRef]
4. A. Mekis, J. C. Chen, I. Kurland, S. 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]
5. A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S.W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405, 437–440 (2000). [CrossRef] [PubMed]
6. S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998). [CrossRef]
7. M. Loncar, T. Doll, J. Vuckovic, and A. Scherer, “Design and fabrication of silicon photonic crystal optical waveguides,” J. Lightwave Technol. 18, 1402–1411 (2000). [CrossRef]
8. J. García, P. Sanchis, and J. Martí, “Detailed analysis of the influence of structure length on pulse propagation through finite-size photonic crystal waveguides,” Opt. Express 14, 6879–6893 (2006). [CrossRef] [PubMed]
9. A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488–4492 (2000). [CrossRef]
10. A. Talneau, L. L. Gouezigou, N. Bouadma, M. Kafesaki, and C. M. Soukoulis, “Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55µm,” Appl. Phys. Lett. 80, 547–549 (2002). [CrossRef]
11. J. Smajic, C. Hafner, and D. Erni, “Design and optimization of an achromatic photonic crystal bend,” Opt. Express 11, 1378–1384 (2003). [CrossRef] [PubMed]
12. X. D. Cui, C. Hafner, R. Vahldieck, and F. Robin, “Sharp trench waveguide bends in dual mode operation with ultra-small photonic crystals for suppressing radiation,” Opt. Express 14, 4351–4356 (2006). [CrossRef]
13. J. García, P. Sanchis, and J. Martí, “Detailed analysis of the influence of structure length on pulse propagation through finite-size photonic crystal waveguides,” Opt. Express 14, 6879–6893 (2006). [CrossRef] [PubMed]
14. A. Chutinan, M. Okano, and S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 80, 1698–1700 (2002). [CrossRef]
15. A. Talneau, P. Lalanne, M. Agio, and C.M. Soukoulis, “Low-reflection photonic-crystal taper for efficient coupling between guide sections of arbitrary widths,” Opt. Lett. 27, 1522–1524 (2002). [CrossRef]
16. X. X. Shen, X. Q. Yu, X. L. Yang, L. Z. Cai, Y. R. Wang, G. Y. Dong, X. F. Meng, F. X., and Xu, “Fabrication of periodic microstructures by holographic photopolymerization with a low-power continuous-wave laser of 532 nm,” J. Opt. A: Pure Appl. Opt. 8, 672–676 (2006). [CrossRef]
17. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: The face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991). [CrossRef] [PubMed]
18. S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full three-dimensional photonic bandgap crystals at near-infrared wavelengths,” Science 289, 604–606 (2000). [CrossRef] [PubMed]
19. X. L. Yang, L. Z. Cai, and Q. Liu, “Theoretical bandgap modeling of two-dimensional triangular photonic crystals formed by interference technique of three-noncoplanar beams,” Opt. Express 11, 1050–1055 (2003). [CrossRef] [PubMed]
20. F. Ramos-Mendieta and P. Halevi, “Electromagnetic surface modes of a dielectric superlattice: the supercell method,” J. Opt. Soc. Am. B 14, 370–381 (1997). [CrossRef]
21. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures” Phys. Rev. Lett. 65, 3152–3155 (1990). [CrossRef] [PubMed]
22. M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000). [CrossRef]
23. T. Y. M. Chan, O. Toader, and S. John, “Photonic bandgap templating using optical interference lithography,” Phys. Rev. E 71, 046605 (2005). [CrossRef]
