Characteristics of optical bandpass filters employing series-cascaded double-ring resonators q

Transcription

1 Otics Communications 8 (003) Characteristics of otical bandass filters emloying series-cascaded double-ring resonators q Jianyi Yang a,b, *, Qingjun Zhou b, Feng Zhao b, Xiaoqing Jiang a, Brie Howley b, Minghua Wang a, Ray T. Chen b a Deartment of Information Science and Electronics Engineering, Zhejiang University, 38 ZheDa Rd., Hangzhou 31007, China b Microelectronics Research Center/Deartment of Electrical and Comuter Engineering, University of Texas at Austin, Austin, TX 78758, USA Received 6 June 003; received in revised form 5 Setember 003; acceted 6 Setember 003 Abstract The filtering characteristics of a series-cascaded double-ring otical resonator (SDRR) are investigated. The roles of couling coefficients and the effects of otical loss are analyzed. The relationshis of the couling coefficients with the characteristics of the SDRR filter are exressed with simle analytical formulas. With the derived aroximate formulas, it is found that the bandwidth ratio of the SDRR filter is mainly deendent on the shae factor defined in this aer. The erformance of the SDRR filter, esecially the bandwidth ratio, is imroved significantly in comarison with that of the single-ring-resonator filter. The analytical results also indicate that the otical loss in the microrings has a strong influence on the characteristics of the SDRR filter. Ó 003 Elsevier B.V. All rights reserved. Keywords: Integrated otics; Otical filter; Otical waveguide; Otical microring resonator 1. Introduction With recent advances in lanar fabrication techniques, there has been an increased interest in microring-based otical waveguide resonators q This work is suorted in art by the Major State Basic Research Develoment Program under the contract G and by the National Natural Science Foundation under the contracts , and * Corresonding author. Tel.: ; fax: address: yangjy@zju.edu.cn (J. Yang). [1 7]. A lot of work has been erformed on otical filters incororating waveguide microrings as the building block elements [3 5,8 15]. Since the single-ring-resonator (SRR) filter has a simle Lorentzian resonse, high-order multile-ringresonator (MRR) filters, which can be realized in either a serial or arallel cascade configuration [8 15], were roosed to achieve flat-to, fast-rolloff, and large-stoband-rejection filtering bands. In LittleÕs recent reort [15], MRRs fabricated on the glass material called Hydex TM with low-insertion loss and box-like resonses were reorted. These high-order filters, however, require tight /$ - see front matter Ó 003 Elsevier B.V. All rights reserved. doi: /j.otcom

2 9 J. Yang et al. / Otics Communications 8 (003) fabrication control to ensure coincident resonances in all microrings and accurate ower couling coefficients of all couling regions. Many methods, such as the couled-mode method [8] and the transfer-matrix method [9,11], have been emloyed to analyze the characteristics of microring-based otical filters. An aroach to synthesize high-order otical filters has also been roosed [8,11]. However, to our knowledge, excet for the SRR filter, no ublished study has quantitatively exlained the exact roles all couling coefficients lay in determining the bandwidth, the bandwidth ratio, and the stoband rejection, and the ossible effects otical loss makes on the characteristics of the MRR filter. The aim of this aer is to exlore the feasibility of emloying a series-cascaded double-ring resonator (SDRR) to realize a high-erformance otical filter. One of the advantages of this second-order microring-based otical filter is that it has the simlest structure among high-order filters, and thus its fabrication is much easier and more ractical. Meanwhile, the size of the SDRR filter is ket small, which is the key feature of the microring-based otical comonent for highdensity otical integration. In this aer, the roles of couling coefficients and the effects of otical loss are analyzed and formulated for the SDRR filter. With the derived formulas, we can easily set and/or tune the bandwidth ratio of the SDRR, which mainly deends on the shae factor defined in the following sections, as well as the bandwidth. This makes designing high-erformance SDRR filters simle and is very useful to guide trimming in the fabrication of SDRR filters. We calculated the bandwidth ratio of the filtering band and found that the band shae of the SDRR filter is much better than that of the SRR filter. The aer is organized as follows. In Section, the transfer function of the SDRR bandass filter is resented. Based on the transfer function, in Section 3, the formulas for the filtering characteristics are derived. The erformance of the SDRR filter is comared with that of the SRR filter. The influence of otical loss on the filtering characteristics is also investigated in Section 3. The results are summarized in Section 4.. Transfer function The schematic diagram of the SDRR otical filter is deicted in Fig. 1. It consists of two mutually couled waveguide microrings (Ring 1 and Ring ) and two tangential straight waveguides (the bus and the droing channels) that serve as evanescent wave inut and outut coulers. All waveguides are monomode. The two rings are identical and have the same free sectral range (FSR). Using the transfer-matrix method [11], we get the transfer function of the SDRR bandass filter at the droing channel: DðhÞ jl 1= 1 L 1= K 1 K K 3 exð jhþ ¼ 1 ðl 1 T þ L T 3 T Þexð jhþþl 1 L T 3 exð jhþ ; ð1þ where, K 1 and K 3 are the two into-/out-of-ring amlitude couling coefficients and K is the ringto-ring amlitude couling coefficient, as shown in Fig. 1. Ti ¼ 1 Ki (i ¼ 1; ; 3). L i ¼ exð a i RÞ (i ¼ 1; ) is the round-tri amlitude attenuation in Ring i, where a i is the ower loss coefficient of Ring i and R is the radius of both rings. h is defined as the normalized frequency: Fig. 1. Schematic diagram of the otical filter emloying the series-cascaded double-ring resonator (SDRR).

3 h ¼ v ; ðþ FSR v i.e., the hase delay of the light with a frequency v in a round tri of a single waveguide ring. FSR v is the FSR in the frequency domain. Since the transfer function DðhÞ is eriodic, we just need to analyze the characteristics in the eriod of ð ; Þ. J. Yang et al. / Otics Communications 8 (003) Characteristic analysis In this section, we first consider the case that the otical loss in the microrings can be ignored, and the two into-/out-of-ring amlitude couling coefficients K 1 and K 3 are set identical. The influence of otical loss is analyzed later The resonances In the case L 1 ¼ L ¼ 1, the intensity transfer function of the SDRR filter can be obtained from Eq. (1) as follows: jdðhþj ðk 1 K K 3 Þ ¼ j1 ð T þ T 3 T Þexð jhþþ T 3 exð jhþj : ð3aþ Fig. illustrates the deendence of the intensity transfer function on the normalized frequency in one eriod of the normalized frequency. In the curve of K ¼ 0:01 or K ¼ 0:1, there are two oints giving the eak resonse. The two oints corresond to the two resonances. It means that every resonant oint of a single ring is slit into two when this ring is set couled with another identical ring. From Eq. (3a), it can be derived that the resonse at the resonance can reach its maximum value 1 only when K 3 ¼ K 1. In the condition of K 3 ¼ K 1, the intensity of transfer function becomes: jdðhþj K1 4 ¼ K j 1 T exð jhþþt1 exð jhþ : j ð3bþ The normalized resonant frequencies h res is determined by the following equation: Fig.. Tyical intensity resonses of the bandass SDRR filter. The emloyed arameters: K 1 ¼ K 3 ¼ 0:1. cos h res ¼ 1 þ T : ð4þ From Eq. (4), we know that the two resonances are degenerate only when T < =ð1 þ T1 Þ. If T P =ð1 þ T1 Þ, which also means: K 6 K 1 ð5aþ K1 the two resonant frequencies will merge back into the zero oint h ¼ 0 and the resonse at the resonance will decrease as K decreases, as shown in Fig.. When K is large enough to suort the two degenerate resonances, as is seen from the curves of K ¼ 0:01 and K ¼ 0:1 in Fig., the intensity resonse has a minimum at h ¼ 0 in the range between the two resonant frequencies. For a bandass filter, it is commonly required that the rile of the assband resonse should be minimized to an accetable value. Thus, for the SDRR filter, the resonse jdðhþj at h ¼ 0 should be above a certain value f 0. For examle, f 0 is about 0.9 if the rile is required to be below 0.5 db. Since generally Ki 1 (i ¼ 1; ), to meet the condition jdð0þj P f 0 it can be derived from Eq. (3b) that K must be in the following range: 1 ffiffiffiffiffiffiffiffiffiffiffiffi 1 f ffiffiffiffi 0 K1 6 K f 0 K1 6 1 þ ffiffiffiffiffiffiffiffiffiffiffiffi 1 f K ffiffiffiffi 0 1 : f 0 K1 ð5bþ

4 94 J. Yang et al. / Otics Communications 8 (003) In fact, Eq. (5b) includes the conditions that the two resonances are merged. Combining Eqs. (5a) and (5b), to satisfy both jdðh res Þj ¼ 1 and jdð0þj P f 0, we find that the ring-to-ring couling coefficient K needs to be tuned as K ¼ q K 1 ; ð6þ K1 where q ¼ð1þ ffiffiffiffiffiffiffiffiffiffi 1 fþ= ffiffi f. Here, we define a shae factor f, which can be used not only to indicate the resonse at h ¼ 0, but also to control the bandwidth ratio as analyzed in Section 3.3. The shae factor f is in the range of f 0 6 f 6 1. Thus, if K is controlled to be the value given by Eq. (6), we can obtain a good assband shae in which the two resonances are ket degenerate and the resonse at the zero oint is f. It should be noted that K is of the same order of magnitude as K1. The resonant frequencies have the following aroximate exression: h res sin h res K K4 1 =4 1 K 1 ðq 1Þ 4 Here, it is assumed that K 1 1. K1 4 T1 : ð7þ BW 1=g q ffiffiffiffiffiffiffiffiffiffiffi g 1 þ q 1 1= K 1 : ð10þ Fig. 3 demonstrates that Eq. (10) is accurate enough to give the 1 db (1=g 0:8), 3 db (1=g 0:5), 0 db (1=g ¼ 0:01) and 30 db (1=g ¼ 0:001) bandwidths when K1 is small. From Eq. (10), it can be found that the bandwidth of the SDRR filter is almost roortional to the ower couling coefficient K1. With this bandwidth formula, it is very easy to calculate the bandwidth ratio which will be analyzed in the following section. 3.. The bandwidth To analyze the bandwidth of the SDRR filter, we begin with the denominator in the right-hand side of Eq. (3b). For the normalized frequencies near the zero oint, considering the two resonant frequencies exressed by Eq. (7), we can aroximate the denominator dðhþ as: dðhþ 1 K 1 h K K 4 1 =4 þ K 4 1 K : ð8þ Since the bandwidth is generally far smaller than the FSR, from Eq. (8), the normalized bandwidth of the bandass SDRR filter is given by: ffiffiffiffiffiffiffiffiffiffiffi g 1K1 BW 1=g ¼ K þ K K4 1 =4 1= : ð9þ 1 K1 Here, g is defined as the bandwidth factor and Eq. (9) gives the bandwidth in which the resonse is within 10 log g (db) of the eak. With Eq. (6), the bandwidth from Eq. (9) can be rewritten as: Fig. 3. Comarison between the aroximate formula (11a) and (11b) and the accurate bandwidth values: (a) f ¼ 1 and (b) f ¼ 0:9. The solid line reresents the accurate bandwidths; the dash line reresents the bandwidths given by Eqs. (11a) and (11b).

5 J. Yang et al. / Otics Communications 8 (003) Comarison with the SRR filter In a bandass filter, a large stoband rejection is critical to minimize crosstalk from neighboring signal channels. Here, we use the maximum extinction ratio EX max ¼ 10 logðjdðhþj max =jdðhþj min Þ (db) to evaluate the stoband rejection and comare the difference between the SDRR filter and the SRR filter.for the SDRR filter, the eak resonse is at the resonant oints: jdðhþj max ¼jDðh resþj ¼ 1 ð11aþ Fig. 4. Curves of the maximum extinction ratio versus the ower couling coefficient K1. The solid line reresents the accurate curves; the dash line reresents the aroximate curves given by Eq. (9). and the minimum resonse is at the oints h ¼: jdðhþj min ¼jDðÞj ¼ K 4 1 K ð1 þ T þ T1 Þ K8 q 1 64 ; ð11bþ where the aroximation can be obtained by assuming K1 1. Thus the maximum extinction ratio of SDRR filter can be aroximated by: EX max 10 log 64 ðdbþ: ð1þ q K1 8 Fig. 4 shows the curves of the maximum extinction ratio versus the ower couling coefficient K1. It can be found that the maximum extinction ratio given by Eq. (1) is accurate if K1 is small. Additionally, the maximum extinction ratio of the SRR filter is EX max 10 logð4=k1 4 Þ [16] if its two into-/out-of-ring couling coefficients are both small values of K 1 Since q cannot be larger than (when f is 0.9, q is about 1.39), the maximum extinction ratio of the SDRR filter is more than twice that of the SRR filter. This means the SDRR filter can afford larger stoband rejection and lower crosstalk comared to the SRR filter. The bandwidth ratio is another key roerty for a bandass filter. When f ¼ 1, we have q ¼ 1 and BW 1=g ¼ð ffiffiffiffiffiffiffiffiffiffiffi g 1Þ 1= ðk1 =Þ for the lossless SDRR filter according to the analysis in Section 3.. Table 1 lists the bandwidth formulas of the Table 1 Formulas of commonly defined bandwidths (normalized) of the SDRR (f ¼ 1) and SRR filters and ratios between them Filter BW 1=g Formula Bandwidth ratio BW 1=g =BW 1dB BW 1=g =BW 3dB BW 1=g =BW 0 db SDRR f ¼ 1 BW 1dB K1 1 ffiffiffi BW K 3dB ffiffiffiffiffi K BW 0 db ffiffiffiffiffiffiffiffiffi K BW 30 db 63: SRR BW 1dB K1 1 BW 3dB K 1 1 BW 0 db 0 K BW 30 db 63: K

6 96 J. Yang et al. / Otics Communications 8 (003) SDRR (f ¼ 1) filter with various values of g and the ratios between them. The bandwidth formulas and the related bandwidth ratios of the SRR filter [16], in which the two into-/out-of-ring couling coefficients are assumed to have the same value K 1 are also resented in Table 1 for comarison. It should be ointed out that the bandwidth ratios listed in Table 1 do not deend on the ower couling coefficient K1 or any other arameter of the SDRR filter once f is chosen. Only when K1 is not small enough and the aroximation for Eqs. (8) (10) is not accurate enough will the bandwidth ratio become larger. This is esecially true for the ratio between the 1 db (or 3 db) and 0 db (or 30 db) bandwidths. Table 1 shows that the imrovement of the band shae is significant if the SDRR is used as an otical filter instead of the SRR. If the shae factor f of the SDRR filter is controlled to be smaller than 1, q becomes greater than 1 and the coefficient ðq ffiffiffiffiffiffiffiffiffiffiffi g 1 þ q 1Þ 1= in Eq. (10) increases. This increase of the coefficient ðq ffiffiffiffiffiffiffiffiffiffiffi g 1 þ q 1Þ 1= is greater for g of a smaller value (e.g., 1.5 or for the 1 or 3 db bandwidth, resectively) than that of a larger value (e.g., 100 or 1000 for the 0 or 30 db bandwidth, resectively). Therefore, the decrease of f can result in the decrease of the bandwidth ratio and greater imrovement of the band shae of the SDRR filter. Fig. 5 illustrates the changes of bandwidth ratios introduced by altering f. It should be noted that the decrease of f must meet the rile requirement. Fig. 5. Curves of the bandwidth ratios versus the shae factor f The influence of otical loss The otical loss in microrings is unavoidable. It means we always have L 1 < 1 and L < 1 for Eq. (1). Introducing the following definitions: T 0 1 ¼ L 1L T 3 ; ð13aþ K ¼ 1 T1 ; ð13bþ c 1 T1 0 ¼ L 1 ; c T1 0 ¼ L ð13cþ T 3 ; C ¼ c 1 þ c ; ð13dþ T 0 ¼ CT ; ð13eþ K 0 ¼ 1 T 0 ð13fþ we have the intensity transfer function of the bandass SDRR filter from Eq. (1): jdðhþj ¼ L 1L K 1 K K 3 K1 04K0 K1 04K0 j 1 T1 0T 0 0 exð jhþþt1 exð jhþ : j ð14þ The second term of the right-hand side of Eq. (14) is the intensity transfer function of a lossless SDRR filter, in which the two into-/out-of-ring couling coefficients are both K1 0 and the ring-toring couling coefficient is K 0. The first term is indeendent of the normalized frequency. Therefore, the influence of otical loss on the band shae is only resented by the second term and the first term gives the attenuation caused by otical loss. Eq. (14) can also be used to analyze the characteristics when K 3 is not set equal to K 1 in the lossless condition. Fig. 6 demonstrates the change of the resonse given by the second term of the righthand side of Eq. (14). As otical loss increases, the bandwidth ratio increases and the maximum extinction ratio decreases. However, the change of the bandwidth is little when otical loss is very small and the two resonances are still ket degenerate, which can be exlained by Eqs. (6), (10) and (13a) (13f). If otical loss is severe, for examle L ¼ 0:5 as seen in Fig. 6, the band shae of the SDRR filter degrades dramatically.

7 J. Yang et al. / Otics Communications 8 (003) outut at the normalized frequencies given by the equation cos h ¼ððL 1 þ T 1 Þ=ðL 1 ÞÞT. However, the filtered outut from the droing channel is far away from its maximum value, which can be reached by setting T 3 to the value around L 1 L. Assuming L 1 ¼ L ¼ L, Fig. 7 comares the three cases that T 3 is controlled to be T 3 ¼ =L, T 3 ¼, and T 3 ¼ L. 4. Summary Fig. 6. Change of the band shae caused by otical loss. The emloyed arameters: K1 ¼ K 3 ¼ 0:1 and f ¼ 0:9. The bandwidth is much larger than exected, the bandwidth ratio is no longer as good as the data given in Table 1, the maximum extinction ratio is lower than )30 db, and more attenuation of the resonse is introduced. The main influence of otical loss is the attenuation of the outut amlitude. Fig. 7 shows the influence of otical loss on the filtering resonse at the resonant frequency. We notice that even low loss will result in a raid decrease of the outut in the droing channel. If is given and T 3 is set to be T 3 ¼ =ðl 1 L Þ, the bus channel of the SDRR filter never has an Resonse at Resonance T 3 =L T 3 = T 3 = /L Fig. 7. Influence of otical loss on the outut of the droing channel at the resonant oint. The emloyed arameters: K1 ¼ 0:1 and f ¼ 1. In the above sections, the bandass characteristics of the SDRR filter are studied in details. A set of analytical formulas is derived. These formulas give the relationshi between the bandass characteristics and the couling coefficients, and can be emloyed to design high-erformance SDRR filters. The analytical results indicate that the ring-toring couling coefficient should follow Eq. (6) to generate a good band shae for the SDRR filter, and then the bandwidth is mainly determined by the into-/out-of-ring couling coefficients. The Bandwidth ratio of the SDRR filter can be calculated very easily with a given shae factor f, and is almost indeendent of any other arameter of the SDRR filter. Comared to the characteristics of the SRR filter, those of the SDRR filter are imroved significantly, esecially the bandwidth ratio. The influence of the otical loss in microrings is also formulated and analyzed. To get a SDRR filter with good filtering erformance, it is very imortant that the otical loss in microrings is controlled to be as low as ossible. Otherwise, otical loss will not only cause serious attenuation of the bandass resonse, but also lead to deterioration of the bandwidth ratio, the maximum extinction ratio, and even the bandwidth. References [1] J.P. Zhang, D.Y. Chu, S.L. Wu, S.T. Ho, W.G. Bi, C.W. Tu, R.C. Tiberio, Phys. Rev. Lett. 75 (14) (1995) 678. [] M. Fujita, T. Baba, Al. Phys. Lett. 80 (1) (00) 051. [3] B. Little, J. Foresi, G. Steinmeyer, E. Thoen, S. Chu, H. Haus, E. Ien, L. Kimerling, W. Greene, IEEE Photon. Technol. Lett. 10 (4) (1998) 549. [4] J. Hryniewicz, P. Absil, B. Little, R. Wilson, P. Ho, IEEE Photon. Technol. Lett. 1 (3) (000) 30.

8 98 J. Yang et al. / Otics Communications 8 (003) [5] T. Kato, S. Suzuki, Y. Kokubun, CLEO/Pacific Rim 1 (001) I_398. [6] P. Rabiei, W. Steier, C. Zhang, L. Dalton, OFC TuF6 (00) 31. [7] H. Haus, C. Manolatou, OFC 3 (000) 16. [8] B. Little, S. Chu, H. Haus, J. Foresi, J. Laine, J. Lightwave Technol. 15 (6) (1997) 998. [9] G. Griffel, IEEE Photon. Technol. Lett. 1 (7) (000) 810. [10] G. Griffel, IEEE Photon. Technol. Lett. 1 (1) (000) 164. [11] R. Orta, P. Savi, R. Tascone, D. Trinchero, IEEE Photon. Technol. Lett. 7 (1) (1995) [1] B. Little, S. Chu, J. Hryniewicz, P. Absil, Ot. Lett. 5 (5) (000) 344. [13] B. Little, S. Chu, W. Pan, D. Riin, T. Kaneko, Y. Kokubun, E. Ien, IEEE Photon. Technol. Lett. 11 () (1999) 15. [14] S. Suzuki, Y. Hatakeyama, Y. Kokubun, S. Chu, J. Lightwave Technol. 0 (4) (00) 745. [15] B. Little, Advances in Microring Resonators, OSA Integrated Photonics Research Conference (IPR 003), ITuE6, Jun. 003, Washington, DC. [16] J. Yang, X. Jiang, M. Wang, Q. Zhou, R. Chen, J. Otoelectron. Lasers 14 (1) (003) 1.