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Analytical modelling of trio-optical asymmetrical microring resonator with vertical coupling
Perspectives in Science (2016) 8, 479—481
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/pisc
Analytical modelling of trio-optical asymmetrical microring resonator with vertical coupling夽 Suchita Lakra ∗, Sanjoy Mandal Indian School of Mines, Dhanbad 826004, India Received 11 February 2016; accepted 31 May 2016 Available online 9 July 2016
KEYWORDS Free spectral range; Group delay; Dispersion
Summary Modelling of vertically coupled trio-optical microring resonator (VCTOMRR) with asymmetrical rings is developed in theory by applying delay line signal processing method. The frequency simulated response characteristic and its dispersion and group delay characteristics are advanced using MATLAB environment. The effectiveness of VCTOMRR with regard to free spectral range (FSR) and crosstalk is obtained from the different performance graphs. This proposed VCTOMRR can provide very wide FSR of 50.1 nm range with extremely low spurious transmission (crosstalk) between two resonant peaks. © 2016 The Author(s). Published by Elsevier GmbH. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Introduction Designing of assorted side coupled multiple ring resonators modelled in Z-domain (Mandal and Dasgupta, 2006; Dey and Mandal, 2011) has been achieved by Mandal et al. The effectiveness of varied waveguide established microring resonators with vertical coupling have been reported in several published papers (Little and Chu, 1999; Ogata and Yoda, 2005). This paper deals with determination of overall transmittance of VCTOMRR with trio-asymmetrical rings in
夽 This article belongs to the special issue on Engineering and Material Sciences. ∗ Corresponding author. Tel.: +91 8986710028. E-mail addresses: [email protected] (S. Lakra), sanjoy [email protected] (S. Mandal).
Z-domain, applying Mason’s gain formula. Digital as well as optical filters are connected to each other and interpreted as LTI systems. Filter in optics involves delay at every single stage which is an integer product of a unit delay and is given in a manner of discrete sequel. Discrete-time signal in the form of Z-transform is illustrated by D(Z) could be asserted as (Proakis and Manolakis, 2006): D(Z) =
∞
{d(n)}Z −n
(1)
n=−∞
where complex-variable is indicated as ‘‘Z’’ and unit delay represented in the form of Z-transform as Z−1 . The analytical connection of free spectral range (FSR) and unit delay is to be established as (Madsen and Zhao, 1999): FSR =
1 c = Lunit Tu nref
(2)
http://dx.doi.org/10.1016/j.pisc.2016.05.002 2213-0209/© 2016 The Author(s). Published by Elsevier GmbH. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
480
S. Lakra, S. Mandal
where ‘‘Tu ’’ is unit delay, ‘‘Lunit ’’ is indicated for unit delay length, ‘‘cv ’’ is known to be light’s velocity and ‘‘nref ’’ is indicated for refractive-index. Dispersion and group delay characteristic of VCTOMRR are attained in this paper.
Formulistic description of optical microring resonator Directional coupler is an eminent passive device employed for the designing of the filters in optical signal processing, shown in (Madsen and Zhao, 1999). The ring resonators with series coupling have an association between input and output and can be described as 2 × 2 transfer matrix displayed as (Barbarossa et al., 1995):
0 I1
I20
C =q −jS
−jS C
i I1 I2i
(3)
where I1i , I2i represent input and I1o , I2o represent output coupler respectively.
Modelling of VCTOMRR Mason’s rule The association between input and output are obtained from signal flow graph (SFG) using Mason’s gain method (Mason, 1956), overall transmittance of optical circuit is exhibited as: 1 Fp p i
G=
(4)
p=1
Analytical model of VCTOMRR Model schematic of Z-domain VCTOMRR with three asymmetrical rings and correspondingly SFG modelling are displayed in Fig. 1(a) and (b) subsequently. In Fig. 1(b), I(Z) and O(Z), treated as input—output node subsequently. The gain of the forward-pathway through the SFG, shown as: √ √ √ F1 = (−jS1 ) Z −A (−jS2 ) Z −(3B/2) (−jS3 ) Z −C (−jS4 ) √ = S1 S2 S3 S4 Z −(A+3B/2+C) (5)
Figure 1 (a)—(b) Schematics of VCTOMRR and Z-transform (SFG) representation of the VCTOMRR. 0 -10 Transmittance (dB)
where ‘‘G’’ is the overall transfer function associated with input and output port together. "Fp " is the pth forward path gain, "i" is the entire forward-paths and "" is known to be the determinant of SFG. "p ", is denoted for the part of for those portion of the SFG for untouched the pth forward path.
-20 -30 -40 -50 -60
1
Figure 2
1.5 2 Frequency (Hz)
2.5 x 10
15
Frequency characteristic of VCTOMRR.
Simulation results
where A, B and √ C are natural as well as co-prime numbers, and Si = ki , i = 1, 2, 3, 4. Therefore, with the help VCTOMRR structure with three asymmetrical rings havof Mason’s method, the entire transfer function of VCTOMRR ing ring radii as 21.09 m, 27.12 m and 30.13 m are assuming the ring loss is illustrated as: √ q 4 1 2 3 k1 k2 k3 k4 Z −(A+3B/2+C) G(Z) = 1 − q 2 C1 C2 1 Z −A − q 2 C2 C3 2 Z −B − q 2 C3 C4 3 Z −C + q 2 C1 C3 1 2 Z −(A+B) + q 2 C2 C4 2 3 Z −(B+C) + q 4 C1 C2 C3 C4 1 3 Z −(C+D) − q 2 C1 C4 1 2 3 Z (A+B+C)
(6)
Coupling coefficient (k3 )
Analytical modelling of trio-optical asymmetrical microring resonator
x 10 7
6
5
4 1 0.95 Co upl ing 0.9 coe 0.85 ffic int (k 1)
Figure 3
-22 -24 -26 -28
Cross
talk (d
B)
Conclusion
Normalized Dispersion
This Z-domain analytical model of VCTOMRR structure is studied and designed and corresponding characteristic of frequency response is determined. FSR achieved is 50.1 nm. The crosstalk obtained is −24.51 dB with −0.9091 dB. Other performance analysis was also done. Dispersion and Group delay characteristics are demonstrated in the current article.
0.5
0
-0.5
Conflict of interest -1
Figure 4
1
1.5 2 Frequency (Hz)
2.5 15
Normalized dispersion characteristic of VCTOMRR.
0.8 0.6 0.4 0.2 0
1
1.5 2 Frequency (Hz)
None declared.
x 10
1 Normalized Group delay
and Takato, 1991), along with coefficient of amplitudetransmission as put-on 0.999 (Schwelb, 2014) for plotting the responses. The simulated consideration of unit delay length is Lunit = 3.013 m and resonant numeral using unit delay is calculated as A = 7, B = 9 and C = 10. FSR achieved out of the frequency simulated response is 50.1 nm. The highest crosstalk, alternatively spurious transmission in the middle of two resonant frequency peaks in VCTOMRR is −24.51 dB. Resonance loss is −0.9091 dB. Also full width half maximum is 0.36 nm and finesse is 139.1666. Dispersion and Group delay characteristics demonstrated are in unit in Figs. 4 and 5 subsequently.
Graph of crosstalk vs. coupling ratios of VCTOMRR. 1
Figure 5
481
2.5 15
x 10
Normalized characteristic group delay of VCTOMRR.
considered. Frequency response characteristic in MATLAB environment is plotted in Fig. 2. The optimum coupling coefficient values for optimum transmission considered here are k1 = k2 = 0.95 and k3 = k4 = 0.0055, which is obtained from Fig. 3 by plotting curves between k1 , k3 and crosstalk. Refractive index 1.7825 (Little and Chu, 1999), loss of the micro ring resonators are assumed to be ˛ = 0.1 dB/cm (Oda
References Barbarossa, G., Matteo, A.M., Armenise, M.N., 1995. Theoretical analysis of triple-coupler ring-based optical guided-wave resonator. J. Lightwave Technol. 13 (February (2)), 148—157. Dey, S., Mandal, S., 2011. Wide free-spectral-range triple ring resonator as optical filter. Opt. Eng. (SPIE) 50 (August (8)), 1—9. Little, B.E., Chu, S.T., 1999. Vertically coupled glass microring resonator channel dropping filters. IEEE Photonics Technol. Lett. 11 (February (2)), 215—217. Madsen, C.K., Zhao, J.H., 1999. Optical Filter Design and Analysis: A signal Processing Approach. John Wiley & Sons, Inc, New York. Mandal, S., Dasgupta, K., 2006. A generalized approach for modeling and analysis of ring-resonator performance as optical filter. Opt. Commun. 264 (August (1)), 97—104. Mason, S.J., 1956. Feedback theory — further properties of signal flow. Proc. IRE 44 (July (7)), 920—926. Oda, K., Takato, N., 1991. A wide-FSR waveguide double-ring resonator for optical FDM transmission systems. J. Lightwave Technol. 9 (June (6)), 728—736. Ogata, M., Yoda, Y., 2005. Ultracompact vertically coupled microring resonator with buried vacuum cladding structure. IEEE Photonics Technol. Lett. 17 (January (1)), 103—105. Proakis, J.G., Manolakis, D.G., 2006. Digital Signal Processing: Principles, Algorithms and Applications. Pearson Printice Hall. Schwelb, O., 2014. Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters — a tutorial overview. J. Lightwave Technol. 22 (May (5)), 1380—1394.
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/pisc
Analytical modelling of trio-optical asymmetrical microring resonator with vertical coupling夽 Suchita Lakra ∗, Sanjoy Mandal Indian School of Mines, Dhanbad 826004, India Received 11 February 2016; accepted 31 May 2016 Available online 9 July 2016
KEYWORDS Free spectral range; Group delay; Dispersion
Summary Modelling of vertically coupled trio-optical microring resonator (VCTOMRR) with asymmetrical rings is developed in theory by applying delay line signal processing method. The frequency simulated response characteristic and its dispersion and group delay characteristics are advanced using MATLAB environment. The effectiveness of VCTOMRR with regard to free spectral range (FSR) and crosstalk is obtained from the different performance graphs. This proposed VCTOMRR can provide very wide FSR of 50.1 nm range with extremely low spurious transmission (crosstalk) between two resonant peaks. © 2016 The Author(s). Published by Elsevier GmbH. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Introduction Designing of assorted side coupled multiple ring resonators modelled in Z-domain (Mandal and Dasgupta, 2006; Dey and Mandal, 2011) has been achieved by Mandal et al. The effectiveness of varied waveguide established microring resonators with vertical coupling have been reported in several published papers (Little and Chu, 1999; Ogata and Yoda, 2005). This paper deals with determination of overall transmittance of VCTOMRR with trio-asymmetrical rings in
夽 This article belongs to the special issue on Engineering and Material Sciences. ∗ Corresponding author. Tel.: +91 8986710028. E-mail addresses: [email protected] (S. Lakra), sanjoy [email protected] (S. Mandal).
Z-domain, applying Mason’s gain formula. Digital as well as optical filters are connected to each other and interpreted as LTI systems. Filter in optics involves delay at every single stage which is an integer product of a unit delay and is given in a manner of discrete sequel. Discrete-time signal in the form of Z-transform is illustrated by D(Z) could be asserted as (Proakis and Manolakis, 2006): D(Z) =
∞
{d(n)}Z −n
(1)
n=−∞
where complex-variable is indicated as ‘‘Z’’ and unit delay represented in the form of Z-transform as Z−1 . The analytical connection of free spectral range (FSR) and unit delay is to be established as (Madsen and Zhao, 1999): FSR =
1 c = Lunit Tu nref
(2)
http://dx.doi.org/10.1016/j.pisc.2016.05.002 2213-0209/© 2016 The Author(s). Published by Elsevier GmbH. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
480
S. Lakra, S. Mandal
where ‘‘Tu ’’ is unit delay, ‘‘Lunit ’’ is indicated for unit delay length, ‘‘cv ’’ is known to be light’s velocity and ‘‘nref ’’ is indicated for refractive-index. Dispersion and group delay characteristic of VCTOMRR are attained in this paper.
Formulistic description of optical microring resonator Directional coupler is an eminent passive device employed for the designing of the filters in optical signal processing, shown in (Madsen and Zhao, 1999). The ring resonators with series coupling have an association between input and output and can be described as 2 × 2 transfer matrix displayed as (Barbarossa et al., 1995):
0 I1
I20
C =q −jS
−jS C
i I1 I2i
(3)
where I1i , I2i represent input and I1o , I2o represent output coupler respectively.
Modelling of VCTOMRR Mason’s rule The association between input and output are obtained from signal flow graph (SFG) using Mason’s gain method (Mason, 1956), overall transmittance of optical circuit is exhibited as: 1 Fp p i
G=
(4)
p=1
Analytical model of VCTOMRR Model schematic of Z-domain VCTOMRR with three asymmetrical rings and correspondingly SFG modelling are displayed in Fig. 1(a) and (b) subsequently. In Fig. 1(b), I(Z) and O(Z), treated as input—output node subsequently. The gain of the forward-pathway through the SFG, shown as: √ √ √ F1 = (−jS1 ) Z −A (−jS2 ) Z −(3B/2) (−jS3 ) Z −C (−jS4 ) √ = S1 S2 S3 S4 Z −(A+3B/2+C) (5)
Figure 1 (a)—(b) Schematics of VCTOMRR and Z-transform (SFG) representation of the VCTOMRR. 0 -10 Transmittance (dB)
where ‘‘G’’ is the overall transfer function associated with input and output port together. "Fp " is the pth forward path gain, "i" is the entire forward-paths and "" is known to be the determinant of SFG. "p ", is denoted for the part of for those portion of the SFG for untouched the pth forward path.
-20 -30 -40 -50 -60
1
Figure 2
1.5 2 Frequency (Hz)
2.5 x 10
15
Frequency characteristic of VCTOMRR.
Simulation results
where A, B and √ C are natural as well as co-prime numbers, and Si = ki , i = 1, 2, 3, 4. Therefore, with the help VCTOMRR structure with three asymmetrical rings havof Mason’s method, the entire transfer function of VCTOMRR ing ring radii as 21.09 m, 27.12 m and 30.13 m are assuming the ring loss is illustrated as: √ q 4 1 2 3 k1 k2 k3 k4 Z −(A+3B/2+C) G(Z) = 1 − q 2 C1 C2 1 Z −A − q 2 C2 C3 2 Z −B − q 2 C3 C4 3 Z −C + q 2 C1 C3 1 2 Z −(A+B) + q 2 C2 C4 2 3 Z −(B+C) + q 4 C1 C2 C3 C4 1 3 Z −(C+D) − q 2 C1 C4 1 2 3 Z (A+B+C)
(6)
Coupling coefficient (k3 )
Analytical modelling of trio-optical asymmetrical microring resonator
x 10 7
6
5
4 1 0.95 Co upl ing 0.9 coe 0.85 ffic int (k 1)
Figure 3
-22 -24 -26 -28
Cross
talk (d
B)
Conclusion
Normalized Dispersion
This Z-domain analytical model of VCTOMRR structure is studied and designed and corresponding characteristic of frequency response is determined. FSR achieved is 50.1 nm. The crosstalk obtained is −24.51 dB with −0.9091 dB. Other performance analysis was also done. Dispersion and Group delay characteristics are demonstrated in the current article.
0.5
0
-0.5
Conflict of interest -1
Figure 4
1
1.5 2 Frequency (Hz)
2.5 15
Normalized dispersion characteristic of VCTOMRR.
0.8 0.6 0.4 0.2 0
1
1.5 2 Frequency (Hz)
None declared.
x 10
1 Normalized Group delay
and Takato, 1991), along with coefficient of amplitudetransmission as put-on 0.999 (Schwelb, 2014) for plotting the responses. The simulated consideration of unit delay length is Lunit = 3.013 m and resonant numeral using unit delay is calculated as A = 7, B = 9 and C = 10. FSR achieved out of the frequency simulated response is 50.1 nm. The highest crosstalk, alternatively spurious transmission in the middle of two resonant frequency peaks in VCTOMRR is −24.51 dB. Resonance loss is −0.9091 dB. Also full width half maximum is 0.36 nm and finesse is 139.1666. Dispersion and Group delay characteristics demonstrated are in unit in Figs. 4 and 5 subsequently.
Graph of crosstalk vs. coupling ratios of VCTOMRR. 1
Figure 5
481
2.5 15
x 10
Normalized characteristic group delay of VCTOMRR.
considered. Frequency response characteristic in MATLAB environment is plotted in Fig. 2. The optimum coupling coefficient values for optimum transmission considered here are k1 = k2 = 0.95 and k3 = k4 = 0.0055, which is obtained from Fig. 3 by plotting curves between k1 , k3 and crosstalk. Refractive index 1.7825 (Little and Chu, 1999), loss of the micro ring resonators are assumed to be ˛ = 0.1 dB/cm (Oda
References Barbarossa, G., Matteo, A.M., Armenise, M.N., 1995. Theoretical analysis of triple-coupler ring-based optical guided-wave resonator. J. Lightwave Technol. 13 (February (2)), 148—157. Dey, S., Mandal, S., 2011. Wide free-spectral-range triple ring resonator as optical filter. Opt. Eng. (SPIE) 50 (August (8)), 1—9. Little, B.E., Chu, S.T., 1999. Vertically coupled glass microring resonator channel dropping filters. IEEE Photonics Technol. Lett. 11 (February (2)), 215—217. Madsen, C.K., Zhao, J.H., 1999. Optical Filter Design and Analysis: A signal Processing Approach. John Wiley & Sons, Inc, New York. Mandal, S., Dasgupta, K., 2006. A generalized approach for modeling and analysis of ring-resonator performance as optical filter. Opt. Commun. 264 (August (1)), 97—104. Mason, S.J., 1956. Feedback theory — further properties of signal flow. Proc. IRE 44 (July (7)), 920—926. Oda, K., Takato, N., 1991. A wide-FSR waveguide double-ring resonator for optical FDM transmission systems. J. Lightwave Technol. 9 (June (6)), 728—736. Ogata, M., Yoda, Y., 2005. Ultracompact vertically coupled microring resonator with buried vacuum cladding structure. IEEE Photonics Technol. Lett. 17 (January (1)), 103—105. Proakis, J.G., Manolakis, D.G., 2006. Digital Signal Processing: Principles, Algorithms and Applications. Pearson Printice Hall. Schwelb, O., 2014. Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters — a tutorial overview. J. Lightwave Technol. 22 (May (5)), 1380—1394.