Analysis and esign of Box-like Filters based on 3 2 Microring Resonator Arrays Xiaobei Zhang a *, Xinliang Zhang b and exiu Huang b a Key Laboratory of Specialty Fiber Optics and Optical Access Networks, School of Communication and Information Engineering, Shanghai University, Shanghai, 272, China b Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, 4374, China *hustzhang@gmail.com ABSTRACT This paper theoretically investigates spectral characteristics of the 3 2 microring resonator array, with its analytical model developed firstly. Simulation results show that the case of the ring-bus coupling coefficient smaller than the ringring coupling coefficient is suitable for box-like filters. After design principles are given, the optimization process are carried out by evaluating the side lobe rejection ratio, the passband ripple rejection ratio and the roll-off coefficient of the passband edge. The FWHM of the designed box-like filter can be adjustable in a flexible range. Keywords: microring resonator arrays; transfer matrix method; resonators; integrated optics devices. INTROUCTION Microring resonators are mainly designed and fabricated for optical filters since they were proposed in 969[]. The spectrum of a single microring resonator is Lorenz-like [2], which is far from practical applications of optical filters. Hence box-like filters based multiple microring resonators are quite attractive in recent years. Series cascaded forms have shown their properties with larger side lobe rejection ratios and smaller roll-off coefficients of the passband edges [3-6]. Parallel cascaded forms are also received considerable attentions due to their improvements of filtering characteristics [7-9]. The M N microring resonator array is formed by N times parallel connections of M series coupled microring resonators through lower and upper waveguides [, ]. They assemble properties of both series and parallel cascaded forms, and thus own various spectral characteristics for box-like filters. A particular microring resonator array model with the same coupling coefficients is proposed and studied []. Odd-row microring resonator arrays are mainly studied for optical filters, and the 3 8 microring resonator array shows a box-like filtering property []. Generally, more resonators result in better box-like properties. However, unavoidable losses and fabrication errors become significant as the number of resonators increases. And experiments of the 3 8 microring resonator array show that there are some notches in the passband due to fabrication errors and losses []. Ideal box-like filters have high a side lobe rejection ratio, a small passband ripple rejection ratio and a small roll-off coefficient of the passband edge. Through combing processes of several methods for improving spectral properties, an ideal box-like filter with minimal numbers of microring resonators and minimal variations of coupling coefficients can be designed. Firstly, the row number can be chosen as 3, which the lowest number of rows. Secondly, the column number can be chosen as 2, which is the lowest number of columns. Finally, the coupling coefficients can be optimized for box-like filters. For the first time to our knowledge, this paper presents the analysis and design of box-like filters based on odd-row microring resonator arrays with least number of resonators, i.e., 3 2 microring resonator arrays. 2. ANALYSIS OF SPECTRAL CHARACTERISTICS Firstly we present the analytical expression of the parallel cascaded dual microring resonators using the transfer matrix method. Fig. shows its schematic diagram. The normalized frequency is denoted as θ, while the round-trip transmission coefficient of each resonator is τ. As τ decreases, the loss increases. The ring-bus coupling and transmission coefficients are k and t respectively. Under lossless coupling, k 2 +t 2 =. The resonator radius is R and the waveguide Passive Components and Fiber-based evices VI, edited by Perry P. Shum, Proc. of SPIE-OSA-IEEE Asia Communications and Photonics, SPIE Vol. 763, 7637 29 SPIE-OSA-IEEE CCC code: 277-786X/9/$8 doi:.7/2.8528 Proc. of SPIE-OSA-IEEE/ Vol. 763 7637-
length between two columns is πr. After developing the model, we can obtain the analytical expression of the spectrum at the drop port as 4 = k τ 2 + τ + 2τ cosθ 2 2 2 2 ( 2 k ) 2( 2 k ) cos + τ τ θ () Figure. shows the schematic diagram of the 3 2 microring resonator array. Considering its symmetrical property, only two coefficients are adopted in this microring resonator array. The ring-bus and ring-ring coupling coefficients are assumed as k and k respectively. The radius and round-trip transmission coefficient of resonators are also R and τ respectively, with the waveguide length as πr. n eff is the effective index of waveguides. The normalized responses of series cascaded triple microring resonators are r and q for the through and drop ports respectively. Thus the total normalized response q all with r and q can be written as q all ( q r ) l = q 2 2 ql 2 2 2 (2) t ( t + tt + tt) p + ( t + tt + tt) p t p r = 2 2 2 2 2 4 6 2 2 2 2 4 2 6 (2 tt + t) p + (2 tt + tt ) p tp (3) kk p q = (4) (2 tt + t ) p + (2 tt + tt ) p t p 2 2 3 2 2 2 2 4 2 6 where p=τ /2 exp(iθ/2), θ=4π 2 n eff R/λ and λ as the wavelength. l represents the factor induced by the center cavity formed by two straight waveguides. When the center cavity is lossless, l=exp(iθ/2). The spectrum at the drop port can be written as = q all 2. Fig.(c) shows spectra of several structures under the lossless case and the coupling coefficient of.5. The spectrum of the single microring resonator (denoted as ) is a standard Lorenz shape. The peak is a single peak and the roll-off coefficient of the passband edge is not small. The spectrum of the parallel cascaded dual microring resonators (denoted as 2) has a wider full width at half magnitude (FWHM), compared to that of the single microring resonator. Similarly, the roll-off coefficient of the passband edge is still not small. The spectrum of the series-cascaded triple microring resonators (denoted as 3 ) has a smaller roll-off coefficient of the passband edge, while the passband ripple rejection ratio is large. The spectrum of the 3 2 microring resonator array shows an improved spectrum compared to those of previous three structures. Both the roll-off coefficient of the passband edge and the passband ripple rejection ratio are modified for better box like filtering characteristics. Through Through πr k n eff R τ k πr k q k k n eff R τ r k.8.6.4.2 3 3 2 2 - -.5.5 rop Input rop q all Input (c) Fig.. Schematic diagram of the parallel cascaded dual microring resonators and the 3 2 microring resonator array. (c)spectra for various configurations of microring resonators under k =k =k=.5 and τ=. When the ring-ring and ring-bus coupling coefficients k and k are the same, spectra are investigated as a function of k. Fig.2 shows the counter plot, while Fig.2 shows spectra for cases of k =k =.2,.5 and.8 respectively. As the coupling coefficient increases, the FWHM increases and the passband ripple rejection ratio decreases, while the roll-off coefficient of the passband edge decreases a little. When the ring-bus coupling coefficient k is fixed as.5, spectra are investigated as a function of k. Similarly, Fig.3 shows the counter plot, while Fig.3 shows spectra for cases of k =.2,.5 and.8 respectively. When the coupling coefficients Proc. of SPIE-OSA-IEEE/ Vol. 763 7637-2
satisfy k <k, zero transmissions will appear in the passband. And zeros are located at the outsides of outer peaks. When the coupling coefficients satisfy k =k, zero transmissions will not appear. However, when coupling coefficients satisfy k >k, zero transmissions will also appear again. And zeros are located at the insides of outer peaks. When the coupling coefficients satisfy k <k, the passband ripple rejection ratio is smaller and the roll-off coefficient of the passband edge is larger. Therefore, the case of coupling coefficients k <k is suitable for box like filters. However, the side lobe rejection ratio has to be suppressed..8 k=k.6.4.2 - -.5.5 Fig.2. When the ring-bus coupling coefficient k and the ring-ring coupling coefficient k are the same, spectra as a function of k, and for cases of k =k =.2,.5 and.8 respectively. (iii) (ii).8.6 (i) (ii) (iii).4 (i).2 - -.5.5 Fig.3. When the ring-bus coupling coefficient k is fixed as.5, spectra as a function of k, and for cases of k =.2,.5 and.8 respectively. 3. ESIGN OF BOX-LIKE FILTERS To achieve box-like filtering characteristics, the ring-ring coupling coefficient k should be smaller than the ring-bus coupling coefficient k. Thus in the process of optimizing box-like filtering characteristics, the ring-bus coupling coefficient k is a dynamic parameter, while the ratio k /k of coupling coefficients k to k is chosen to be studied. Fig.4 shows definitions of both global and detailed characterization parameters for the design of box-like filters. The intensity at the center frequency (θ=θ =) is. The intensity at the notch frequency (θ=θ 2 ) is 2. The intensity at the ripple frequency (θ=θ ) is. The intensity at the side lobe frequency (θ=θ 3 ) is is 3. When the intensity 7 = -., the frequency is θ 7. When the intensity 5 =.5, the frequency is θ 5. When the intensity 6 =., the frequency is θ 6. When the intensity is minimum 4, the frequency is θ 4. Obviously, = = and 4 = for lossless cases. Values of intensities - 7 are also labeled as shown in Fig.4, while corresponding frequencies θ -θ 7 are also shown along the horizontal axis. Numerical simulations are adopted for optimizing box-like filtering properties. The side lobe rejection ratio, the passband ripple rejection ratio and the roll-off coefficient of the passband edge are investigated as a function of the coupling coefficient ratio k /k. To achieve box-like properties, some principles are given as follows. () The side lobe rejection ratio η s =-log ( 3 / ) db. (2) The passband ripple rejection ratio η 2 =-log ( 2 / ).db. (3) The roll-off coefficient of the passband edge η b =-log (θ 7 /θ 6 ) db. Proc. of SPIE-OSA-IEEE/ Vol. 763 7637-3
(4) If previous three requirements can be achieved, the FWHM FWHM=2θ 5 can be adjustable by varying coupling coefficients. Fig.4. efinition of characterization parameters for the design of box-like filters, global and detailed parameters. By choosing the ring-bus coupling coefficient k as.2,.5 and.8 respectively, simulation results of the side lobe rejection ratio η s, the passband ripple rejection ratio η 2, the roll-off coefficient of the passband edge η b and FWHM as functions of the coupling coefficient ratio k /k are shown in Fig.5. Fig.5 shows that the side lobe rejection ratio η s decreases as the coupling coefficient ratio k /k increases. Thus a smaller k /k corresponds to a larger side lobe rejection ratio η s, while a larger ring-bus coupling coefficient k corresponds to a larger side lobe rejection ratio η s. Fig.5 shows that the passband ripple rejection ratio η 2 increases nearly linearly as the coupling coefficient ratio k /k increases, when k /k is larger than a specified value. Moreover, a larger ring-bus coupling coefficient k corresponds to a smaller passband ripple rejection ratio η 2. Fig.5(c) shows that the roll-off coefficient of the passband edge η b decreases rapidly as the coupling coefficient ratio k /k increases, while a smaller ring-bus coupling coefficient k corresponds to a smaller roll-off coefficient of the passband edge η b. Fig.5(d) shows that FWHM increases nearly linearly as the coupling coefficient ratio k /k increases and a larger ring-bus coupling coefficient k corresponds to a larger FWHM. From above simulation results, several conclusions can be made for the design process. A smaller k /k and a larger k are constructive for the side lobe rejection ratio η s and the passband ripple rejection ratio η 2. However, a larger k /k and a smaller k are constructive for the roll-off coefficient of the passband edge η b. Thus there should be some trade-off between the side lobe rejection ratio η s (or the passband ripple rejection ratio η 2 ) and the roll-off coefficient of the passband edge η b, during the practical design process. 8 6 4 k =.2 k =.5 k =.8 8 6 4 2 2 8.2.4.6.8 k /k (c).8.2.4.6.8 k /k (d) 6.6 4.4 2.2.2.4.6.8.2.4.6.8 k /k k /k Fig.5. Under k =.2,.5 and.8, the side lobe rejection ratio ηs, the passband ripple rejection ratio η 2, (c) the roll-off coefficient of the passband edge η b and (d) FWHM as functions of k /k. Proc. of SPIE-OSA-IEEE/ Vol. 763 7637-4
From above requirements in the design process, ranges of k /k for k =.2,.5 and.8 respectively can be calculated. In the simulation process, the coupling coefficient ratio k /k is discrete and the discrete step is adopted as.. When the coupling coefficient k =.2,.5 and.8 respectively, the calculated ranges of k /k are.7-.2,.9-.29 and.37-.42 respectively as shown in Fig.6. Minimal and maximal values of k /k correspond to requirements of η b and η s respectively. Fig.6 shows three cases of box-like filtering characteristics, with k /k chosen as the middle value of the calculated range. And the middle values of the calculated ranges are.95,.24 and.395 for the coupling coefficient k =.2,.5 and.8 respectively. Corresponding FWHMs are calculated as.85π,.6π and.32π. Similarly, corresponding side lobe rejection ratios are calculated as 3.49dB, 2.73dB and.33db, while passband ripple rejection ratios are calculated as.9db, db and db. Corresponding roll-off coefficients of the passband edges are calculated as.54db,.6db and.83db respectively. Through previous three cases, it is obvious that both the FWHM and the roll-off coefficient of the passband edge η b increase as the coupling coefficient k or the coupling coefficient ratio k /k increases, while the both the side lobe rejection ratio η s and the passband ripple rejection ratio η 2 decreases. FWHM/π.4.3.2. k =.2 k =.5 k =.8.8.6.4.2 k =.2 k /k =.95 k =.5 k /k =.24 k =.8 k /k =.395..2.3.4.5 k /k..2.3.4 Fig.6. Under k =.2,.5 and.8, ranges of k /k for box-like filters. Three cases of box-like filters with k /k as.95,.24 and.395 respectively. 4. CONCLUSION In conclusion, the spectrum of the 3 2 microring resonator array is investigated analytically. After developing the analytical model using the transfer matrix method, it is found that the passband ripple rejection ratio is smaller and the roll-off coefficient of the passband edge is smaller, if the ring-ring coupling coefficient is smaller than the ring-bus coupling coefficient. After the design process, box-like filters with the side lobe rejection ratio larger than db, the passband ripple rejection ratio smaller than.db and the roll-off coefficient of the passband edge smaller than db are achieved, with the FWHM adjustable in a flexible range. REFERENCES. Marcatili E. A. J. "Bends in optical dielectric guides", Bell Labs Technical Journal, 48, 23-232, 969. 2. Little B. E., Chu S. T., Haus H. A., et al. "Microring resonator channel dropping filters", Journal of Lightwave Technology, 5(6),998-5, 997. 3. Hryniewicz J. V., Absil P. P., Little B. E., et al. "Higher order filter response in coupled microring resonators", IEEE Photonics Technology Letters, 2(3),32-322, 2. 4. Little B. E., Chu S. T., Absil P. P., et al. "Very high-order microring resonator filters the WM applications", IEEE Photonics Technology Letters, 6(), 2263-2265, 24. 5. Popovic M. A., Barwicz T., Watts M. R., et al. "Multistage high-order microring-resonator add-drop filters", Optics Letters, 3(7), 257-2573,26. 6. Xia F., Rooks M., Sekaric L., et al. "Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects", Optics Express, 5(9), 934-94, 27 Proc. of SPIE-OSA-IEEE/ Vol. 763 7637-5
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