In this paper, a bandpass transmission filter realized in phase-shifted waveguide gratings based on photonic crystals (PCs) is proposed. Phase-shift regions each composed of one period of photonic crystal (PC) waveguide are incorporated into PC waveguide gratings. The magnitudes of the phase-shifts are modified by involving small changes in the size of the border rods in the phase-shift regions. Using standard coupled-mode theory and finite-difference time-domain (FDTD) method, we show that by properly choosing the magnitudes of phase-shifts and the lengths of waveguide gratings, a flat-top and sharp roll-off response with a narrow bandwidth is theoretically and numerically achieved by the designed filter. A further analysis shows that the center frequency of the transmission band can be changed by altering the magnitude of the phase-shift and the response performance exhibits relaxed sensitivity to the phase-shift variation. As a specific application, we theoretically demonstrate a third-order Chebyshev bandpass filter based on compound phase-shifted PC waveguide gratings. The filter performance is suitable for dense wavelength-division-multiplexed (DWDM) optical communication systems with a channel spacing of 100-GHz.
©2007 Optical Society of America
Photonic crystals (PCs) [1–3] are periodically patterned materials with a strong dielectric contrast. The periodic structures forbid electromagnetic waves propagation within a frequency range, which is known as photonic bandgap (PBG). Because of these unique features, in recent years optical devices based on photonic crystals have attracted great attention. Various kinds of photonic crystal (PC) devices such as PC waveguides  and filters [5–7], have been studied so far due to their compactness and potential applications in photonic integrated circuits and all-optical communication network.
Among various devices based on PCs, optical filters are key components receiving great consideration because they can act as a demultipexler to select a particular channel or multiple channels in dense wavelength-division-multiplexed (DWDM) optical communication systems. In order to be used in DWDM optical communication systems, the filters must have flap-top response and sharp roll-off from the passband to the stopband. One kind of bandpass filters based on PC have been proposed by R. Costa using PC-based Fabry-Perot (F-P) cavities in , and the partial reflecting mirrors of the F-P cavities are implemented by inserting appropriate defects into a two-dimensional (2-D) PC waveguide. In this kind of filters, reflectivity, phase and group optical lengths of the PC-based F-P cavities should be carefully designed to determine the resonance frequency and the bandwidth. Furthermore, the relative location of the inserted F-P cavities with respect to the lattice of the background 2-D PC should also be carefully designed to improve the filter performance, making an effective design more difficult to achieve. In reference , one third-order Chebyshev bandpass filter based on coupled defect resonators embedded in PC waveguide is designed in the range of the optical communication frequency band. In this device, appropriate phase retardation between neighboring resonators has to be designed to avoid deep ripples in the passband. Therefore, the parameters of the defects of the resonators should be accurately designed to enhance the filter performance.
In this paper, a bandpass transmission filter realized in phase-shifted waveguide gratings [8–9] based on PCs is proposed. Phase-shift regions each composed of one period of PC waveguide are inserted into PC waveguide gratings [10–12]. In the phase-shift regions, electromagnetic waves at the Bragg frequency of PC waveguide gratings propagate with a π/2 phase-shift. The configuration of our optical system is similar to that of optical filters composed of distributed feedback (DFB) structure  or ring resonators . The transmission characteristics of the phase-shifted PC waveguide gratings are theoretically analyzed by the transfer-matrix formalism based on standard coupled-mode theory [15, 16]. The coupling coefficients in the formalism can be calculated by the numerical approximation method mentioned in . The detailed derivations of the equations for the transmission calculations are introduced in . The numerical simulations are done by using the 2-D finite-difference time-domain (FDTD) method to verify the theoretical results. According to our analysis, a third-order Chebyshev bandpass filter based on 2-D PCs with high dielectric rods in air is designed within the optical communication frequency band.
2. Design of filter structure
First, we consider single phase-shifted PC waveguide gratings as depicted in Fig. 1, which is based on the 2-D PC made from a square lattice of dielectric rods in air. The rod has a dielectric constant ε=11.56 and radius r=0.2a, where a is the lattice constant. In this 2-D PC, PBG exists only for transverse magnetic (TM) modes which extends from ω=0.289 to 0.42 (in units of (2πc/a)) , where ω is the normalized frequency. The PC single mode waveguide is formed by eliminating a row of rods. The rods adjacent to both sides of the waveguide are alternately replaced by rods with different radius r 1=0.16a to construct PC waveguide gratings acting as a bandpass reflection filter. The reflection band is centered at the Bragg frequency ω 0=0.37426 (2πc/a) defined as the frequency at which the reflection efficiency is the maximum . The phase-shift region realized by one period of PC waveguide, which has the same lattice constants as the background PC, is placed in accordance with the background PC lattice and located in the center of the PC waveguide gratings. The lengths of the two uniform grating sections are both L 1 in units of Λ. ϕ 1 is the phase-shift occurring in the phase-shift region.
In the phase-shift region, the dispersion relation of the fundamental mode of the PC waveguide can be modified by altering the radius of the two border rods (denoted by rods with black color in Fig. 1) . Figure 2(a) shows the dispersion curves of the PC waveguide for different radius of the border rods r 0=0.16a, 0.17a, 0.18a, 0.19a and 0.20a, respectively. As the radius of the border rods is increased, the dispersion curves shift to lower frequencies. The values of the frequency (defined as ω 1) at the propagation constant β(ω)=0.25 (2π/a) for different radius r 0 are calculated and shown in Fig. 2(b). From this one can see that the frequency ω 1 for the radius r 0=0.18a is located at 0.3743 (2πc/a), which is equal to the Bragg frequency ω 0 of PC waveguide gratings with r 1=0.16a. The phase-shift ϕ 1=βa occurring in the phase-shift region is equal to π/2 for the electromagnetic waves propagation at the Bragg frequency ω 0 of the PC waveguide gratings with r 1=0.16a.
The theoretically calculated transmission spectra of single phase-shifted PC waveguide gratings with the radius r0=0.18a for different waveguide grating length L 1 are shown in Fig. 3. Transmission peaks with a Lorentzian line shape are centered at the Bragg frequency ω 0. As the length of the waveguide gratings is increased, the transmission band exhibits a reducing line width (measured by full width at half maximum (FWHM)).
Compound phase-shifted PC waveguide gratings consist of cascaded M single phase-shifted PC waveguide gratings with the waveguide grating lengths L 1,L 2,…,L M and M phase-shift regions, providing the phase shifts ϕ 1=ϕ 2=…=ϕ M. Here only three phaseshifted PC waveguide gratings are considered as depicted in Fig. 4, because the waveguide gratings with three phase-shift regions are the optimum practical choice for the filter design . The device is designed symmetrically with the outer waveguide grating lengths L 1=L 3=L out and inner waveguide grating length L 2=L in.
The theoretically calculated transmission spectrum of three phase-shifted PC waveguide gratings with L in=L out=12Λ and r 0=0.18a is shown in Fig. 5(a) denoted by the solid line.
We can see that the transmission band, which is the same as the spectrum of a third-order Chebyshev bandpass filter, exhibits a symmetrical rectangular line shape with ripples about 0.35 dB in the passband. By properly adjusting the lengths of the outer and inner waveguide gratings, realizing higher reflection in the middle of the structure, the ripples in the passband can be reduced [9, 14]. Here the ripples are reduced to about 0.1dB by reducing the outer waveguide grating length L out from 12Λ to 11Λ. The optimized transmission spectrum is shown in Fig. 5(a) denoted by the dashed line. However, the factor s, which is defined as ratio between the -1dB and the -10dB bandwidth to characterize the selectivity of the transmission band, is reduced from 0.756 to 0.712. Therefore, by slightly modifying the outer waveguide grating length, the ripples in the passband are effectively reduced and the selectivity of the transmission band has been little influenced.
Figure 5(b) shows the transmission spectra of the designed filter calculated by the 2-D FDTD method. The FDTD calculated results are in good agreement with the theoretical results, demonstrating the effectiveness of the bandpass filter design presented above. The in-band frequency response distortion can be attributed to the FDTD computational errors . The center frequency of the transmission band is located at normalized frequency 0.37404 (2πc/a). The center frequency difference from the theoretical result can be addressed to the numerical approximation of the coupling coefficients in the transfer-matrix formalism. The optimized spectrum denoted by the dashed line has a flat-top passband ranging from the normalized frequency 0.3738 to 0.3743 (2πc/a) with ripples less than 0.1dB.
3. Filter performance analysis
Figure 6 shows the transmission spectra of three phase-shifted PC waveguide gratings for different L out(L in). The transmission band shows a reducing bandwidth with increasing length of waveguide gratings. For a lattice constant a=0.58µm, grating lengths L out=19Λ (L in=20Λ) and r 0=0.18a, as shown in Fig. 6, the transmission band of the filter has a center frequency at 193.33THz, and a flat bandwidth of about 50GHz. The ripples in the passband are less than 0.1dB. These characteristics of the designed filter are suitable for use in DWDM optical communication systems with a 100-GHz channel spacing.
The center frequency of the transmission band can be modified by altering the magnitude of the phase-shift . As mentioned in section 2, the phase-shift is mainly affected by the size of the border rods in the phase-shift region. Therefore, the center frequency can be changed by altering radius r 0. Figure 7 shows the transmission spectra of the bandpass filter for different radius r 0 of border rods in the center phase-shift region. As the radius r 0 is increased by 0.005a, the center frequency shift is about 35GHz. Accordingly, the transmission bands of the filters with the radius r 0=0.18a and 0.195a are centered at frequency 193.33THz and 193.23THz, respectively, with the frequency spacing equal to 100-GHz, which indicates that the center frequency shows sensitive to the radius r 0. If strict center frequency precision is needed in the system, the fabrication accuracy is required rigorously. But if it is not finely required, the effect of the variation of the radius r 0 is limited. The value of ripples (dB) in the passband for different values of r 0 is shown in Fig. 8. It can be noted that the value of ripples (dB) is slowly varying near the optimum at r 0=0.18a and less that 0.25dB, making the performance of filter less sensitive to phase shift variation.
Phase-shifted PC waveguide gratings can be constructed with the system of holes as well as that of rods, the system of hole-type PC waveguide gratings is particularly discussed in .
In this paper, a bandpass transmission filter based on phase-shifted PC waveguide gratings is reported. Phase-shift regions each consisting of one period of PC waveguide are inserted into the PC waveguide gratings. The π/2 phase-shift is achieved by properly choosing the radius of border rods in the phase-shift regions. The designed filter has a flat-top and sharp-roll response with a narrow bandwidth. The ripples in the passband are reduced to less than 0.1dB by slightly adjusting the length of the outer waveguide gratings. The center frequency of the transmission band can be modified by altering the size of border rods in the center phase-shift region with little filter performance deterioration. We believe that the phase-shifted PC waveguide grating structure will become a platform for channel-dropping filter based on PCs in DWDM optical communication systems.
This work was supported in part by the National Natural Science Foundation of China under Contract 60577033, 90401025 and in part by the Program for New Century Excellent Talents in University under Contract NCET-06-0093.
References and links
3. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).
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. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H.A. Haus, “Channel drop filters in photonic crystals” Opt. Express 3, 4–11 (1998). http://www.opticsinfobase.org/abstract.cfm?URI=oe-3-1-4. [CrossRef] [PubMed]
6. R. Costa, A. Melloni, and M. Martinelli, “Bandpass resonant filters in photonic-crystal waveguides,” IEEE Photon. Technol. Lett. 15, 401–403 (2003). [CrossRef]
7. D. Park, S. Kim, I. Park, and H. Lim, “Higher order optical resonant filters based on coupled defect resonators in photonic crystals,” J. Lightwave Technol. 23, 1923–1928 (2005). [CrossRef]
8. G. P. Agrawal and S. Radic, “Phase-shifted fiber Bragg gratings and their application for wavelength demultiplexing,” IEEE Photon. Technol. Lett. 6, 995–997 (1994). [CrossRef]
9. R. Zengerle and O. Leminger, “Phase-shifted Bragg-gratings filters with improved transmission characteristics” J. Lightwave Technol. 13, 2354–2358 (1995). [CrossRef]
11. N. Yokoi, T. Fujisawa, K. Saitoh, and M. Koshiba, “Apodized photonic crystal waveguide gratings,” Opt. Express 14, 4459–4468 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-10-4459. [CrossRef] [PubMed]
12. C. Chen, X. Li, K. Xu, J. Wu, and J. Lin, “Photonic crystal waveguide sampled gratings,” Opt. Comm. 276, 237–241 (2007). [CrossRef]
13. H. A. Haus, Waves and Fields in Optoelectronics (Englewood Cliffs, NJ: Prentice-Hall, 1984), pp. 235–253.
14. A. Melloni and M. Martinelli, “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. 20, 296–303 (2002). [CrossRef]
15. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973) [CrossRef]
16. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997). [CrossRef]
17. A. Adibi, R. K. Lee, Y. Xu, A. Yariv, and A. Scherer, “Design of photonic crystal optical waveguides with singlemode propagation in the photonic bandgap,” Electron. Lett. 36, 1376–1378 (2000). [CrossRef]