- Email: [email protected]
Independently tunable optical notch filter based on double ring resonator structure
Optik 124 (2013) 1307–1310
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Independently tunable optical notch filter based on double ring resonator structure Juan Zhang ∗ , Sen Guo, Xue Li Key Laboratory of Specialty Fiber Optics and Optical Access Networks, School of Communication and Information Engineering, Shanghai University, 160#, No.149, Yanchang Road, Shanghai 200072, PR China
a r t i c l e
i n f o
Article history: Received 29 October 2011 Accepted 14 March 2012
Keywords: Optical notch filter Mach–Zehnder interferometer Optical allpass filter Ring resonator Z-transform
a b s t r a c t A novel technique of designing optical notch filter is proposed from digital filter perspective for Mach–Zehnder interferometer with double ring resonator coupled structure. It is based on Z-transform method. The key advantage of this structure is that it removes phase shifters, which is usually used in existing notch filter with ring resonator structure. So the proposed notch filter has high stability and good reliability. The proposed design method to obtain the structural parameters of the notch filter is not only effective and simple, but also can be used to implement the optical notch filter with arbitrary notch frequency, 3 dB rejection bandwidth and notch point number. One design example of notch filter with two notch frequencies and different 3 dB rejection bandwidth is presented in the paper in detail. The change of the intensity response of the design example is also investigated for each design parameters deviating from ideal value. The results show that the notch frequency and 3 dB rejection bandwidth of each notch point can be tuned independently. © 2012 Elsevier GmbH. All rights reserved.
1. Introduction Optical filters are key components of optical systems. Among them, notch filters which can block undesired signals but do nothing to the useful signals are very useful. These filters are widely used in communication, biomedical engineering and other fields [1–4]. Up to now, some schematics have been developed to realize optical notch filters. By cascading birefringent crystals, a notch response can be realized [5]. However, a large number of crystals are needed to obtain notch filters with narrower 3 dB rejection bandwidth. Thus the whole structure is complicated and the cost is expensive. You et al. [6] proposed a topology for a tunable microwave photonic notch filter. But the output spectrum of the notch filter has a poor performance (sine shape). Based on Mach–Zehnder interferometer (MZI) with cascaded single ring resonators and phase shifter, Rasras [7] demonstrated a tunable optical notch filter. The whole structure is simple and compact. In this paper, from digital filter perspective using Z-tansform, a novel tunable optical notch filter is presented. It is composed of a MZI with cascaded double ring resonator structure in one of its arms. Each double ring resonator structure will introduce a notch at its resonance frequency. Optical notch filter with arbitrary notch frequency, 3 dB rejection bandwidth and notch point
∗ Corresponding author. Tel.: +86 21 56332285; fax: +86 21 56332285. E-mail address: [email protected] (J. Zhang). 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.03.034
number can be accomplished by appropriately choosing the coupling coefficients of double ring resonator structure. As a detailed design example, a notch filter with two notch frequencies and different 3 dB rejection bandwidth is given in the paper. In practical applications, the structural parameters cannot always be accurate as the designed value, so the influence of design parameters on the output spectrum is discussed. The results showed that the notch frequency and 3 dB rejection bandwidth of each notch point could be tuned independently. 2. Design principle It was showed in Ref. [8] that in frequency domain the design of notch filter can be transformed into allpass filter. The theory can also be applied to design notch filters in optical domain. The basic principle of design optical notch filter is given in Fig. 1. From Fig. 1, the transfer function of notch filter can be obtained as H(z) =
1 [1 + Hallpass (z)], 2
(1)
where Hallpass (z) is the transfer function of optical allpass filter (OAF). For a 2N-order allpass filter, the transfer function can be defined as [9] Hallpass (z) =
a2N + · · · + a1 z −2N+1 + z −2N , 1 + a1 z −1 + · · · + a2N z −2N
(2)
1308
J. Zhang et al. / Optik 124 (2013) 1307–1310
From Eqs. (3)–(7), we can derive that the transfer function of notch filter in Fig. 2 is given by
1 H(z) = 2
1+
N ci2 − ci1 (ci2 + 1)z −1 + z −2 i=1
Fig. 1. The design principle of optical notch filter based on optical allpass filter.
1 − ci1 (1 + ci2 )z −1 + ci2 z −2
.
(8)
It can be seen that Eq. (8) is in accordance with Eq. (1), so the filter of this structure can get a notch response. Compared with the structure in Ref. [7], the filter structure shown in Fig. 2 has higher environmental stability and better reliability for removing the influence of phase shifter. 3. Design method Fig. 2. The physical structure to realize a notch filter.
Fig. 3. The (a) physical structure; (b) digital model of double ring resonator.
where a1 ,. . .,a2N are filter coefficients of allpass filter respectively. Fig. 2 is the structure of our proposed notch filter. In Fig. 2, the OAF is composed of N cascaded double ring resonators. The length of up and down arm of the MZI is equal. The couplers at input and output ports of MZI have a power coupling ratio of 0.5. The power coupling ratios of the ith double ring resonator are ki1 , ki2 (i = 1, 2,. . .,N) respectively. Fig. 3(a) shows the structure of the ith double ring resonator. representation is shown in Fig. 3(b). The Z-transform Where ci1 = 1 − ki1 , ci2 = 1 − ki2 , si1 = ki1 , and si2 = ki2 , and ki1 , ki2 are the power coupling ratio of the first and the second coupler in the ith double ring resonator respectively. From Fig. 3(b), the transfer function of the ith double ring resonator can be deduced as follows: i Hdou (z) =
z−1
ci2 − ci1 (ci2 + 1)z −1 + z −2 , 1 − ci1 (1 + ci2 )z −1 + ci2 z −2
(3)
where f is the frequency of light, C is the speed of light in the vacuum, n and L are the refractive index and length of ring resonator respectively. The transfer function of the N cascaded double ring resonators can be expressed as (4)
The transfer matrix of the coupler at the input and output of MZI can be expressed as √ √ 1√ − k −j k out √ ˚in = ˚ = . (5) cplr cplr −j k 1−k where k = 0.5 is the power coupling ratio of the coupler. The delay matrix is written for both arms as follows
˚delay =
Hallpass (z) 0
0 1
.
(6)
According to the theory of transfer matrix, the overall transfer matrix for the filter is the product of matrices H(z) = ˚out × ˚delay × ˚in . cplr cplr
(1 − zi z −1 )(1 − zi∗ z −1 ) = 1 − 2ri cos i z −1 + (ri )2 z −2 .
(7)
(9)
i Let Eq. (9) equal to the denominator of Hdou (z), we obtain
⎧ ⎨ ci2 = (ri )2 ⎩ ci1 =
= e−j2fLn/C ,
1 2 N Hallpass (z) = Hdou (z) × Hdou (z) × · · · × Hdou (z).
In order to obtain notch response with different spectral characteristics, the number of cascaded double ring resonators and the power coupling ratios of all the double ring resonators need to be determined. According to the spectral characteristics of the desired optical notch filter, we can calculate the notch filter’s transfer function in frequency domain shown in Eq. (2) utilizing the method described in Ref. [9]. Thus the order of the notch filter can be known. On this basis, the number of cascaded double ring resonator can be determined. It is equal to the half of the order of the notch filter. Then the optical structure of the notch filter can be determined and its transfer function can be obtained as Eq. (8). The power coupling ratios of all the double ring resonators can be obtained by making Eq. (2) equal to Eq. (8). In signal processing technology when the transfer function of a filter is known, it is usually straightforward to obtain all the values of poles of the filter. Because Hallpass (z), the transfer function of allpass filter in Eq. (2), is a function of z with real coefficients, its poles/zeros are symmetric about the real axis. If one of the poles of Hallpass (z) is expressed as zi = ri eji , ri and (i are the modulus and argument of zi respectively, zi∗ = ri e−ji must be another pole of Hallpass (z). Therefore the denominator of Hallpass (z) must include the item as follows:
2ri cos i .
(10)
1 + (ri )2
The above is the case for single pole. The power coupling ratios of all the double ring resonators can be obtained similarly. The proposed method to obtain the design parameters of double ring resonator is much simpler than directly solving Eqs. (2) and (8) simultaneously, especially for the case that have many polzes. 4. Design example As an example, a notch filter with two notch points in one spectral period is designed. The locations of two notch points are 1 = 1550.2 nm, 2 = 1550.3 nm, and the 3 dB rejection bandwidth of two notch points are 1 = 0.008 nm, 2 = 0.016 nm respectively. Spectral period (FSR) equals 0.8 nm. Firstly, according to the desired spectral characteristics and the method described in Ref. [9], the filter coefficients of allpasss filter can be obtained as follows: a1 = −1.6491, a2 = 2.1008, a3 = −1.5287 and a4 = 0.8211. Therefore the transfer function of the filter to be designed in frequency domain equals: H(z) =
1 2
1+
0.8211 − 1.5287z −1 + 2.1008z −2 − 1.6491z −3 + z −4 1 − 1.6491z −1 + 2.1008z −2 − 1.5287z −3 + 0.8211z −4
.
(11)
J. Zhang et al. / Optik 124 (2013) 1307–1310 1
1309
1.0
0.8
0.8
0.4
Intensity response
Imaginary Part
0.6
0.2 0 -0.2 -0.4 -0.6
0.6
0.4
0 shift +1% shift -1% shift
(a)
0.2
-0.8 -1
-1.5
-1
-0.5
0
0.5
1
1.5
0.0
Real Part
1550.1
1550.2
Fig. 4. Pole/zero plot.
1550.3
1550.4
Wavelength (nm) 1.0
1.0
0.8
Intensity response
Intensity Response
0.8
0.6
0.4
0.6
0.4
0.2
0.2
0.0
0.0
1550.10
1550.15
1550.20
1550.25
1550.30
1550.35
1550.40
0 shift +20% shift -20% shift
(b)
1550.1
1550.2
Wavelength (nm)
Fig. 7. The notch response when parameters (a) k11 ; (b) k12 shift respectively.
From Eq. (11), we know that the order of the allpass filter equals 4. So two double ring resonators are needed to construct such an optical notch filter. The transfer function of the notch filter can be calculated from Eq. (8) as follows: 1 2
1+
c12 − c11 (c12 + 1)z −1 + z −2 c22 − c21 (c22 + 1)z −1 + z −2 × 1 − c11 (1 + c12 )z −1 + c12 z −2 1 − c21 (1 + c22 )z −1 + c22 z −2
1.0
0.8
.
(12) Fig. 4 is the pole/zero plot of allpass filter shown in Eq. (11). We can obtain that the values of the poles are p1,2 = 0.09915617√± 0.9610582j and p3,4 = 0.7253938 ± 0.5944996j respectively, j = −1. The design parameters can then be easily obtained using Eq. (10) as c12 = 0.9335, c11 = 0.1026, c22 = 0. 8796, c21 = 0.7719. The intensity response of the filter corresponding to these structural parameters is shown in Fig. 5. From Fig. 5, it can be seen that the design result is in good agreement with the target filter.
Intensity response
H(z) =
1550.4
Wavelength (nm)
Fig. 5. The intensity response of notch filter with unequal 3 dB rejection bandwidth.
1550.3
0.6
0 shift +10% shift -10% shift
0.4
(a) 0.2
0.0 1550.1
1550.2
1550.3
1550.4
Wavelength (nm) 1.0
1.0
Intensity response
0.8
Intensity response
0.8 0.6 0.4
0.4
0 shift +20% shift -20% shift
(b)
0.2
0.2
0.0
0.0 1550.1
0.6
1550.1
1550.2
1550.3
1550.4
Wavelength(nm) Fig. 6. The intensity response of notch filter with equal 3 dB rejection bandwidth.
1550.2
1550.3
1550.4
Wavelength (nm)
Fig. 8. The notch response when parameters (a) k21 ; (b) k22 shift respectively.
1310
J. Zhang et al. / Optik 124 (2013) 1307–1310
1.0
Intensity response
0.8
0.6
0.4
(a)
0.2
0.0 1550.10
1550.15
1550.20
1550.25
1550.30
1550.35
1550.40
Wavelength (nm) 1.0
Intensity response
0.8
0.6
0.4
(b)
6. Conclusions
0.2
0.0 1550.10
The 3 dB rejection bandwidth of the first and the second notch point is controlled by k12 and k22 respectively and the 3 dB rejection bandwidth increase with the increasing of the k12 and k22. The notch location of the first and second notch point is determined by k11 and k21 respectively and with the decreasing of k11 and k21 , the notch location of notch filter shifts to the long wavelength. Comparing Figs. 7 and 8, it can be seen that the filter is most sensitive to k11 , and then are the k21 , k22 , and k12 in sequence. Furthermore, when k11 is larger than the ideal value, the change of notch location of the first notch point is faster than that when k11 is smaller than the ideal value. So in practical applications, we would rather k11 smaller than larger. While for other three parameters, i.e. k21 , k22 , and k12 , the relations between the change of spectral characteristics and the shift of parameters are linear. From the analysis above, it can be seen that the notch frequency and 3 dB rejection bandwidth of each notch point can be independently tuned. The tunability of notch frequency and 3 dB rejection bandwidth of the ith notch point can be realized through adjusting parameters ki1 and ki2 respectively.
1550.15
1550.20
1550.25
1550.30
1550.35
1550.40
Wavelength (nm)
Fig. 9. The intensity response of notch filter with notch location at (a) 1550.2 nm; (b) 1550.3 nm.
Optical notch filters with other spectral characteristics can also be designed using the method similarly. Fig. 6 shows the intensity response of the notch filter with equal 3 dB rejection bandwidth. When only one double ring resonator is used in the structure, the notch filter with single notch point can be realized. 5. Independently tunable characteristics The tunable characteristics of notch filter can be obtained from analyzing the design parameters. Hereinafter taking the design example above as an example, we will discuss the change of output spectrum when the design parameters shift from the designed value. Because power coupling ratios are often used in practical applications, design parameters c12 = 0.9335, c11 = 0.1026, c22 = 0. 8796, c21 = 0.7719 are transformed into power coupling ratios as k12 = 0.1286, k11 = 0.9895, k22 = 0.2263, and k21 = 0.4042. Figs. 7(a and b) and 8(a and b) show the intensity spectral of the filter with k11 , k12 , k21 and k22 shift respectively. From the two figures we can see that parameter k12 , k11 determine the characteristics of the first notch point (1550.2 nm), and parameter k22 , k21 determine the characteristics of the second notch point (1550.3 nm). That is to say, each double ring resonator introduces a notch at its resonance frequency. If single double ring resonator, such as the first double ring resonator is only used in the structure, single notch point notch filter with notch location at 1550.2 nm would be obtained. The intensity response is shown in Fig. 9(a). Similarly, when the second double ring resonator is only used in the structure, the notch filter with notch location at 1550.3 nm would be obtained. Its intensity response is shown in Fig. 9(b).
From digital filter perspective using Z-tansform, a novel method of designing optical notch filter is proposed for MZI with cascaded double ring resonators coupled structure. Optical notch filter with arbitrary notch frequency, 3 dB rejection bandwidth and notch point number can be designed by the method conveniently. A notch filter with two notch frequencies and different 3 dB rejection bandwidth is designed in the paper in detail. The change of output intensity spectrum is investigated for each design parameters deviating from ideal value. It shows that the notch frequency and 3 dB rejection bandwidth of each notch point can be independently tuned. Acknowledgments This work was supported by the National Natural Science Foundation of China under the grant no. 10804070, Innovation Program of Shanghai Municipal Education Commission under the grant no. 09YZ06 and Shanghai Leading Academic Discipline Project under the grant no. S30108. References [1] A. Vallese, A. Bevilacqua, Analysis and design of an integrated notch filter for the rejection of interference in UWB systems, IEEE J. Solid State Circuits 44 (2009) 331–343. [2] C.D. McManus, D. Neubert, E. Cramer, Characterization and elimination of AC noise in electrocardiograms: a comparison of digital filtering methods, Comput. Biomed. Res. 26 (1993) 48–67. [3] D.V.B. Rao, S.Y. Kung, Adaptive notch filtering for the retrieval of sinusoids in noise, IEEE Trans. Acoust. Speech Signal Process. ASSP-32 (1984) 791–802. [4] S.-C. Pei, C.-C. Tseng, A novel structure for cascade form adaptive notch filters, Signal Process. 33 (1993) 95–110. [5] S. Nikitin, C. Manka, J. Grun, Modified Sˇ olc notch filter for deep ultraviolet applications, Opt. Soc. Am. 48 (2009) 1184–1189. [6] N. You, R.A. Minasian, A novel tunable microwave optical notch filter, IEEE Trans. Microw. Theory Tech. 49 (2001) 2002–2005. [7] M.S. Rasras, K.Y. Tu, D.M. Gill, et al., Demonstration of a tunable microwavephotonic notch filter using low-loss silicon ring resonators, J. Lightwave Technol. 27 (2009) 2105–2110. [8] C.K. Madsen, Jian H. Zhao, Optical Filter Design and Analysis, John wiley & Sons, Inc., New York, 1999. [9] S.-C. Pei, C.-C. Tseng, IIR multiple notch filter design based on allpass filter, IEEE Trans. Circuits Syst. II: Analog Digital Signal Process. 44 (1997) 133–136.
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Independently tunable optical notch filter based on double ring resonator structure Juan Zhang ∗ , Sen Guo, Xue Li Key Laboratory of Specialty Fiber Optics and Optical Access Networks, School of Communication and Information Engineering, Shanghai University, 160#, No.149, Yanchang Road, Shanghai 200072, PR China
a r t i c l e
i n f o
Article history: Received 29 October 2011 Accepted 14 March 2012
Keywords: Optical notch filter Mach–Zehnder interferometer Optical allpass filter Ring resonator Z-transform
a b s t r a c t A novel technique of designing optical notch filter is proposed from digital filter perspective for Mach–Zehnder interferometer with double ring resonator coupled structure. It is based on Z-transform method. The key advantage of this structure is that it removes phase shifters, which is usually used in existing notch filter with ring resonator structure. So the proposed notch filter has high stability and good reliability. The proposed design method to obtain the structural parameters of the notch filter is not only effective and simple, but also can be used to implement the optical notch filter with arbitrary notch frequency, 3 dB rejection bandwidth and notch point number. One design example of notch filter with two notch frequencies and different 3 dB rejection bandwidth is presented in the paper in detail. The change of the intensity response of the design example is also investigated for each design parameters deviating from ideal value. The results show that the notch frequency and 3 dB rejection bandwidth of each notch point can be tuned independently. © 2012 Elsevier GmbH. All rights reserved.
1. Introduction Optical filters are key components of optical systems. Among them, notch filters which can block undesired signals but do nothing to the useful signals are very useful. These filters are widely used in communication, biomedical engineering and other fields [1–4]. Up to now, some schematics have been developed to realize optical notch filters. By cascading birefringent crystals, a notch response can be realized [5]. However, a large number of crystals are needed to obtain notch filters with narrower 3 dB rejection bandwidth. Thus the whole structure is complicated and the cost is expensive. You et al. [6] proposed a topology for a tunable microwave photonic notch filter. But the output spectrum of the notch filter has a poor performance (sine shape). Based on Mach–Zehnder interferometer (MZI) with cascaded single ring resonators and phase shifter, Rasras [7] demonstrated a tunable optical notch filter. The whole structure is simple and compact. In this paper, from digital filter perspective using Z-tansform, a novel tunable optical notch filter is presented. It is composed of a MZI with cascaded double ring resonator structure in one of its arms. Each double ring resonator structure will introduce a notch at its resonance frequency. Optical notch filter with arbitrary notch frequency, 3 dB rejection bandwidth and notch point
∗ Corresponding author. Tel.: +86 21 56332285; fax: +86 21 56332285. E-mail address: [email protected] (J. Zhang). 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.03.034
number can be accomplished by appropriately choosing the coupling coefficients of double ring resonator structure. As a detailed design example, a notch filter with two notch frequencies and different 3 dB rejection bandwidth is given in the paper. In practical applications, the structural parameters cannot always be accurate as the designed value, so the influence of design parameters on the output spectrum is discussed. The results showed that the notch frequency and 3 dB rejection bandwidth of each notch point could be tuned independently. 2. Design principle It was showed in Ref. [8] that in frequency domain the design of notch filter can be transformed into allpass filter. The theory can also be applied to design notch filters in optical domain. The basic principle of design optical notch filter is given in Fig. 1. From Fig. 1, the transfer function of notch filter can be obtained as H(z) =
1 [1 + Hallpass (z)], 2
(1)
where Hallpass (z) is the transfer function of optical allpass filter (OAF). For a 2N-order allpass filter, the transfer function can be defined as [9] Hallpass (z) =
a2N + · · · + a1 z −2N+1 + z −2N , 1 + a1 z −1 + · · · + a2N z −2N
(2)
1308
J. Zhang et al. / Optik 124 (2013) 1307–1310
From Eqs. (3)–(7), we can derive that the transfer function of notch filter in Fig. 2 is given by
1 H(z) = 2
1+
N ci2 − ci1 (ci2 + 1)z −1 + z −2 i=1
Fig. 1. The design principle of optical notch filter based on optical allpass filter.
1 − ci1 (1 + ci2 )z −1 + ci2 z −2
.
(8)
It can be seen that Eq. (8) is in accordance with Eq. (1), so the filter of this structure can get a notch response. Compared with the structure in Ref. [7], the filter structure shown in Fig. 2 has higher environmental stability and better reliability for removing the influence of phase shifter. 3. Design method Fig. 2. The physical structure to realize a notch filter.
Fig. 3. The (a) physical structure; (b) digital model of double ring resonator.
where a1 ,. . .,a2N are filter coefficients of allpass filter respectively. Fig. 2 is the structure of our proposed notch filter. In Fig. 2, the OAF is composed of N cascaded double ring resonators. The length of up and down arm of the MZI is equal. The couplers at input and output ports of MZI have a power coupling ratio of 0.5. The power coupling ratios of the ith double ring resonator are ki1 , ki2 (i = 1, 2,. . .,N) respectively. Fig. 3(a) shows the structure of the ith double ring resonator. representation is shown in Fig. 3(b). The Z-transform Where ci1 = 1 − ki1 , ci2 = 1 − ki2 , si1 = ki1 , and si2 = ki2 , and ki1 , ki2 are the power coupling ratio of the first and the second coupler in the ith double ring resonator respectively. From Fig. 3(b), the transfer function of the ith double ring resonator can be deduced as follows: i Hdou (z) =
z−1
ci2 − ci1 (ci2 + 1)z −1 + z −2 , 1 − ci1 (1 + ci2 )z −1 + ci2 z −2
(3)
where f is the frequency of light, C is the speed of light in the vacuum, n and L are the refractive index and length of ring resonator respectively. The transfer function of the N cascaded double ring resonators can be expressed as (4)
The transfer matrix of the coupler at the input and output of MZI can be expressed as √ √ 1√ − k −j k out √ ˚in = ˚ = . (5) cplr cplr −j k 1−k where k = 0.5 is the power coupling ratio of the coupler. The delay matrix is written for both arms as follows
˚delay =
Hallpass (z) 0
0 1
.
(6)
According to the theory of transfer matrix, the overall transfer matrix for the filter is the product of matrices H(z) = ˚out × ˚delay × ˚in . cplr cplr
(1 − zi z −1 )(1 − zi∗ z −1 ) = 1 − 2ri cos i z −1 + (ri )2 z −2 .
(7)
(9)
i Let Eq. (9) equal to the denominator of Hdou (z), we obtain
⎧ ⎨ ci2 = (ri )2 ⎩ ci1 =
= e−j2fLn/C ,
1 2 N Hallpass (z) = Hdou (z) × Hdou (z) × · · · × Hdou (z).
In order to obtain notch response with different spectral characteristics, the number of cascaded double ring resonators and the power coupling ratios of all the double ring resonators need to be determined. According to the spectral characteristics of the desired optical notch filter, we can calculate the notch filter’s transfer function in frequency domain shown in Eq. (2) utilizing the method described in Ref. [9]. Thus the order of the notch filter can be known. On this basis, the number of cascaded double ring resonator can be determined. It is equal to the half of the order of the notch filter. Then the optical structure of the notch filter can be determined and its transfer function can be obtained as Eq. (8). The power coupling ratios of all the double ring resonators can be obtained by making Eq. (2) equal to Eq. (8). In signal processing technology when the transfer function of a filter is known, it is usually straightforward to obtain all the values of poles of the filter. Because Hallpass (z), the transfer function of allpass filter in Eq. (2), is a function of z with real coefficients, its poles/zeros are symmetric about the real axis. If one of the poles of Hallpass (z) is expressed as zi = ri eji , ri and (i are the modulus and argument of zi respectively, zi∗ = ri e−ji must be another pole of Hallpass (z). Therefore the denominator of Hallpass (z) must include the item as follows:
2ri cos i .
(10)
1 + (ri )2
The above is the case for single pole. The power coupling ratios of all the double ring resonators can be obtained similarly. The proposed method to obtain the design parameters of double ring resonator is much simpler than directly solving Eqs. (2) and (8) simultaneously, especially for the case that have many polzes. 4. Design example As an example, a notch filter with two notch points in one spectral period is designed. The locations of two notch points are 1 = 1550.2 nm, 2 = 1550.3 nm, and the 3 dB rejection bandwidth of two notch points are 1 = 0.008 nm, 2 = 0.016 nm respectively. Spectral period (FSR) equals 0.8 nm. Firstly, according to the desired spectral characteristics and the method described in Ref. [9], the filter coefficients of allpasss filter can be obtained as follows: a1 = −1.6491, a2 = 2.1008, a3 = −1.5287 and a4 = 0.8211. Therefore the transfer function of the filter to be designed in frequency domain equals: H(z) =
1 2
1+
0.8211 − 1.5287z −1 + 2.1008z −2 − 1.6491z −3 + z −4 1 − 1.6491z −1 + 2.1008z −2 − 1.5287z −3 + 0.8211z −4
.
(11)
J. Zhang et al. / Optik 124 (2013) 1307–1310 1
1309
1.0
0.8
0.8
0.4
Intensity response
Imaginary Part
0.6
0.2 0 -0.2 -0.4 -0.6
0.6
0.4
0 shift +1% shift -1% shift
(a)
0.2
-0.8 -1
-1.5
-1
-0.5
0
0.5
1
1.5
0.0
Real Part
1550.1
1550.2
Fig. 4. Pole/zero plot.
1550.3
1550.4
Wavelength (nm) 1.0
1.0
0.8
Intensity response
Intensity Response
0.8
0.6
0.4
0.6
0.4
0.2
0.2
0.0
0.0
1550.10
1550.15
1550.20
1550.25
1550.30
1550.35
1550.40
0 shift +20% shift -20% shift
(b)
1550.1
1550.2
Wavelength (nm)
Fig. 7. The notch response when parameters (a) k11 ; (b) k12 shift respectively.
From Eq. (11), we know that the order of the allpass filter equals 4. So two double ring resonators are needed to construct such an optical notch filter. The transfer function of the notch filter can be calculated from Eq. (8) as follows: 1 2
1+
c12 − c11 (c12 + 1)z −1 + z −2 c22 − c21 (c22 + 1)z −1 + z −2 × 1 − c11 (1 + c12 )z −1 + c12 z −2 1 − c21 (1 + c22 )z −1 + c22 z −2
1.0
0.8
.
(12) Fig. 4 is the pole/zero plot of allpass filter shown in Eq. (11). We can obtain that the values of the poles are p1,2 = 0.09915617√± 0.9610582j and p3,4 = 0.7253938 ± 0.5944996j respectively, j = −1. The design parameters can then be easily obtained using Eq. (10) as c12 = 0.9335, c11 = 0.1026, c22 = 0. 8796, c21 = 0.7719. The intensity response of the filter corresponding to these structural parameters is shown in Fig. 5. From Fig. 5, it can be seen that the design result is in good agreement with the target filter.
Intensity response
H(z) =
1550.4
Wavelength (nm)
Fig. 5. The intensity response of notch filter with unequal 3 dB rejection bandwidth.
1550.3
0.6
0 shift +10% shift -10% shift
0.4
(a) 0.2
0.0 1550.1
1550.2
1550.3
1550.4
Wavelength (nm) 1.0
1.0
Intensity response
0.8
Intensity response
0.8 0.6 0.4
0.4
0 shift +20% shift -20% shift
(b)
0.2
0.2
0.0
0.0 1550.1
0.6
1550.1
1550.2
1550.3
1550.4
Wavelength(nm) Fig. 6. The intensity response of notch filter with equal 3 dB rejection bandwidth.
1550.2
1550.3
1550.4
Wavelength (nm)
Fig. 8. The notch response when parameters (a) k21 ; (b) k22 shift respectively.
1310
J. Zhang et al. / Optik 124 (2013) 1307–1310
1.0
Intensity response
0.8
0.6
0.4
(a)
0.2
0.0 1550.10
1550.15
1550.20
1550.25
1550.30
1550.35
1550.40
Wavelength (nm) 1.0
Intensity response
0.8
0.6
0.4
(b)
6. Conclusions
0.2
0.0 1550.10
The 3 dB rejection bandwidth of the first and the second notch point is controlled by k12 and k22 respectively and the 3 dB rejection bandwidth increase with the increasing of the k12 and k22. The notch location of the first and second notch point is determined by k11 and k21 respectively and with the decreasing of k11 and k21 , the notch location of notch filter shifts to the long wavelength. Comparing Figs. 7 and 8, it can be seen that the filter is most sensitive to k11 , and then are the k21 , k22 , and k12 in sequence. Furthermore, when k11 is larger than the ideal value, the change of notch location of the first notch point is faster than that when k11 is smaller than the ideal value. So in practical applications, we would rather k11 smaller than larger. While for other three parameters, i.e. k21 , k22 , and k12 , the relations between the change of spectral characteristics and the shift of parameters are linear. From the analysis above, it can be seen that the notch frequency and 3 dB rejection bandwidth of each notch point can be independently tuned. The tunability of notch frequency and 3 dB rejection bandwidth of the ith notch point can be realized through adjusting parameters ki1 and ki2 respectively.
1550.15
1550.20
1550.25
1550.30
1550.35
1550.40
Wavelength (nm)
Fig. 9. The intensity response of notch filter with notch location at (a) 1550.2 nm; (b) 1550.3 nm.
Optical notch filters with other spectral characteristics can also be designed using the method similarly. Fig. 6 shows the intensity response of the notch filter with equal 3 dB rejection bandwidth. When only one double ring resonator is used in the structure, the notch filter with single notch point can be realized. 5. Independently tunable characteristics The tunable characteristics of notch filter can be obtained from analyzing the design parameters. Hereinafter taking the design example above as an example, we will discuss the change of output spectrum when the design parameters shift from the designed value. Because power coupling ratios are often used in practical applications, design parameters c12 = 0.9335, c11 = 0.1026, c22 = 0. 8796, c21 = 0.7719 are transformed into power coupling ratios as k12 = 0.1286, k11 = 0.9895, k22 = 0.2263, and k21 = 0.4042. Figs. 7(a and b) and 8(a and b) show the intensity spectral of the filter with k11 , k12 , k21 and k22 shift respectively. From the two figures we can see that parameter k12 , k11 determine the characteristics of the first notch point (1550.2 nm), and parameter k22 , k21 determine the characteristics of the second notch point (1550.3 nm). That is to say, each double ring resonator introduces a notch at its resonance frequency. If single double ring resonator, such as the first double ring resonator is only used in the structure, single notch point notch filter with notch location at 1550.2 nm would be obtained. The intensity response is shown in Fig. 9(a). Similarly, when the second double ring resonator is only used in the structure, the notch filter with notch location at 1550.3 nm would be obtained. Its intensity response is shown in Fig. 9(b).
From digital filter perspective using Z-tansform, a novel method of designing optical notch filter is proposed for MZI with cascaded double ring resonators coupled structure. Optical notch filter with arbitrary notch frequency, 3 dB rejection bandwidth and notch point number can be designed by the method conveniently. A notch filter with two notch frequencies and different 3 dB rejection bandwidth is designed in the paper in detail. The change of output intensity spectrum is investigated for each design parameters deviating from ideal value. It shows that the notch frequency and 3 dB rejection bandwidth of each notch point can be independently tuned. Acknowledgments This work was supported by the National Natural Science Foundation of China under the grant no. 10804070, Innovation Program of Shanghai Municipal Education Commission under the grant no. 09YZ06 and Shanghai Leading Academic Discipline Project under the grant no. S30108. References [1] A. Vallese, A. Bevilacqua, Analysis and design of an integrated notch filter for the rejection of interference in UWB systems, IEEE J. Solid State Circuits 44 (2009) 331–343. [2] C.D. McManus, D. Neubert, E. Cramer, Characterization and elimination of AC noise in electrocardiograms: a comparison of digital filtering methods, Comput. Biomed. Res. 26 (1993) 48–67. [3] D.V.B. Rao, S.Y. Kung, Adaptive notch filtering for the retrieval of sinusoids in noise, IEEE Trans. Acoust. Speech Signal Process. ASSP-32 (1984) 791–802. [4] S.-C. Pei, C.-C. Tseng, A novel structure for cascade form adaptive notch filters, Signal Process. 33 (1993) 95–110. [5] S. Nikitin, C. Manka, J. Grun, Modified Sˇ olc notch filter for deep ultraviolet applications, Opt. Soc. Am. 48 (2009) 1184–1189. [6] N. You, R.A. Minasian, A novel tunable microwave optical notch filter, IEEE Trans. Microw. Theory Tech. 49 (2001) 2002–2005. [7] M.S. Rasras, K.Y. Tu, D.M. Gill, et al., Demonstration of a tunable microwavephotonic notch filter using low-loss silicon ring resonators, J. Lightwave Technol. 27 (2009) 2105–2110. [8] C.K. Madsen, Jian H. Zhao, Optical Filter Design and Analysis, John wiley & Sons, Inc., New York, 1999. [9] S.-C. Pei, C.-C. Tseng, IIR multiple notch filter design based on allpass filter, IEEE Trans. Circuits Syst. II: Analog Digital Signal Process. 44 (1997) 133–136.