Abstract

In this paper, the bifurcation behavior of light in the PANDA ring resonator is investigated using the signal flow graph (SFG) method, where the optical transfer function for the through and drop ports of the PANDA Vernier system are derived. The optical nonlinear phenomena, such as bistability, Ikeda instability, and dynamics of light in the silicon-on-insulator (SOI) PANDA ring resonator with four couplers are studied. The transmission curves for bistability and instability as a function of the resonant mode numbers and coupling coefficients for the coupler are derived by the SFG method and simulated. The proposed system has an advantage as no optical pumping component is required. Simulated results show that closed-loop bistable switching can be generated and achieved by varying mode resonant numbers in the SOI-PANDA Vernier resonator, where a smooth and closed-loop bistable switching with low relative output/input power can be obtained and realized. The minimum through-port switching time of 1.1 ps for resonant mode numbers of 5;4;4 and minimum drop port switching time of 1.96 ps for resonant mode numbers of 9;7;7 of the PANDA Vernier resonator are achieved, which makes the PANDA Vernier resonator an operative component for optical applications, such as optical signal processing and a fast switching key in photonics integrated circuits.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  37. S. Mandal, K. Dasgupta, T. Basak, and S. Ghosh, “A generalized approach for modeling and analysis of ring-resonator performance as optical filter,” Opt. Commun. 264, 97–104 (2006).
    [CrossRef]
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    [CrossRef]
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2012 (6)

N. Zou, W. Li, B. Huang, Z. Xu, S. Xu, and C. Yang, “An optical continuous phase FSK modulation scheme with an arbitrary modulation index over long-haul transmission fiber link,” Opt. Commun. 285, 2591–2595 (2012).
[CrossRef]

A. Maurente, F. H. R. França, K. Miki, and J. R. Howell, “Application of approximations for joint cumulative k-distributions for mixtures to FSK radiation heat transfer in multi-component high temperature non-LTE plasmas,” J. Quant. Spectrosc. Radiat. Transfer 113, 1521–1535 (2012).
[CrossRef]

M. Bahadoran, A. Afroozeh, J. Ali, and P. P. Yupapin, “Slow light generation using microring resonators for optical buffer application,” Opt. Eng. 51, 044601 (2012).
[CrossRef]

C. Sirawattananon, M. Bahadoran, J. Ali, S. Mitatha, and P. P. Yupapin, “Analytical Vernier effects of a PANDA ring resonator for micro force sensing application,” IEEE Trans. Nanotech. 11, 707–712 (2012).
[CrossRef]

M. S. Aziz, S. Daud, M. Bahadoran, J. Ali, and P. P. Yupapin, “Light pulse in a modified add–drop optical filter for optical tweezers generation,” J. Nonlinear Opt. Phys. Mater. 21, 1250047 (2012).
[CrossRef]

S. Dey and S. Mandal, “Modeling and analysis of quadruple optical ring resonator performance as optical filter using Vernier principle,” Opt. Commun. 285, 439–446 (2012).
[CrossRef]

2011 (1)

P. P. Yupapin and B. Vanishkorn, “Mathematical simulation of light pulse propagating within a microring resonator system and applications,” Appl. Math. Model. 35, 1729–1738 (2011).
[CrossRef]

2010 (3)

S. Mitatha, K. Dejhan, P. P. Yupapin, and N. Pornsuwancharoen, “Chaotic signal generation and coding using a nonlinear micro ring resonator,” Optik 121, 120–125 (2010).
[CrossRef]

P. A. Costanzo-Caso, Y. Jin, S. Granieri, and A. Siahmakoun, “Optical bistability in a nonlinear-SOA-based fiber ring resonator,” J. Nonlinear Opt. Phys. Mater. 20, 281–292 (2010).
[CrossRef]

R. Boeck, N. A. Jaeger, N. Rouger, and L. Chrostowski, “Series-coupled silicon racetrack resonators and the Vernier effect: theory and measurement,” Opt. Express 18, 25151–25157 (2010).
[CrossRef]

2009 (4)

P. P. Yupapin and S. Suchat, “Nonlinear penalties and benefits of light traveling in a fiber optic ring resonator,” Optik. 120, 216–221 (2009).
[CrossRef]

A. Bahrampour, S. Zakeri, S. M. A. Mirzaee, Z. Ghaderi, and F. Farman, “All-optical set–reset flip–flop based on frequency bistability in semiconductor microring lasers,” Opt. Commun. 282, 2451–2456 (2009).
[CrossRef]

B. Li, M. I. Memon, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “All-optical digital logic gates using bistable semiconductor ring lasers,” J. Opt. Commun. 30, 190–194 (2009).
[CrossRef]

P. P. Yupapin and N. Pornsuwancharoen, “Proposed nonlinear microring resonator arrangement for stopping and storing light,” IEEE Photon. Technol. Lett. 21, 404–406 (2009).
[CrossRef]

2008 (6)

P. P. Yupapin, N. Pornsuwancharoen, and S. Chaiyasoonthorn, “Attosecond pulse generation using the multistage nonlinear microring resonators,” Microw. Opt. Technol. Lett. 50, 3108–3111 (2008).
[CrossRef]

P. P. Yupapin, “Coupler-loss and coupling-coefficient-dependent bistability and instability in a fiber ring resonator,” Optik 119, 492–494 (2008).
[CrossRef]

P. Saeung and P. P. Yupapin, “Generalized analysis of multiple ring resonator filters: modeling by using graphical approach,” Optik 119, 465–472 (2008).
[CrossRef]

A. Bahrampour, M. Karimi, M. Qamsari, H. R. Nejad, and S. Keyvaninia, “All-optical set–reset flip–flop based on the passive microring-resonator bistability,” Opt. Commun. 281, 5104–5113 (2008).
[CrossRef]

N. Dou and C. Li, “Optical bistability in fiber ring resonator containing an EDFA,” Opt. Commun. 281, 2238–2242 (2008).
[CrossRef]

A. Malacarne, J. Wang, Y. Zhang, A. D. Barman, G. Berrettini, L. Poti, and A. Bogoni, “20 ps transition time all-optical SOA-based flip-flop used for photonic 10  Gb/s switching operation without any bit loss,” IEEE J. Sel. Top. Quantum Electron. 14, 808–815 (2008).
[CrossRef]

2007 (3)

P. P. Yupapin, P. Saeung, and C. Li, “Characteristics of complementary ring-resonator add/drop filters modeling by using graphical approach,” Opt. Commun. 272, 81–86 (2007).
[CrossRef]

P. P. Yupapin and W. Suwancharoen, “Chaotic signal generation and cancellation using a micro ring resonator incorporating an optical add/drop multiplexer,” Opt. Commun. 280, 343–350 (2007).
[CrossRef]

P. Saeung and P. P. Yupapin, “Vernier effect of multiple-ring resonator filters modeling by a graphical approach,” Opt. Eng. 46, 075005 (2007).
[CrossRef]

2006 (1)

S. Mandal, K. Dasgupta, T. Basak, and S. Ghosh, “A generalized approach for modeling and analysis of ring-resonator performance as optical filter,” Opt. Commun. 264, 97–104 (2006).
[CrossRef]

2005 (1)

S. Zhang, D. Owens, Y. Liu, M. Hill, D. Lenstra, A. Tzanakaki, G. D. Khoe, and H. Dorren, “Multistate optical memory based on serially interconnected lasers,” IEEE Photon. Technol. Lett. 17, 1962–1964 (2005).
[CrossRef]

2003 (1)

M. Soljacic, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in nonlinear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
[CrossRef]

2002 (2)

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002).
[CrossRef]

J. Harbold, F. Ö. Ilday, F. Wise, J. Sanghera, V. Nguyen, L. Shaw, and I. Aggarwal, “Highly nonlinear As-S-Se glasses for all-optical switching,” Opt. Lett. 27, 119–121 (2002).
[CrossRef]

1999 (1)

1997 (1)

F. Sanchez, “Optical bistability in a 2×2 coupler fiber ring resonator: parametric formulation,” Opt. Commun. 142, 211–214 (1997).
[CrossRef]

1987 (1)

C. L. Tang, A. Schremer, and T. Fujita, “Bistability in two-mode semiconductor lasers via gain saturation,” Appl. Phys. Lett. 51, 1392–1394 (1987).
[CrossRef]

1984 (1)

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, “Fiber optic signal lattice processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

1983 (1)

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

1981 (1)

D. Miller, “Refractive Fabry–Perot bistability with linear absorption: theory of operation and cavity optimization,” IEEE J. Quantum Electron. 17, 306–311 (1981).
[CrossRef]

1980 (1)

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[CrossRef]

1978 (1)

J. H. Marburger and F. S. Felber, “Theory of a lossless nonlinear Fabry–Perot interferometer,” Phys. Rev. A 17, 335–342 (1978).
[CrossRef]

1976 (1)

F. S. Felber and J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28, 731–733 (1976).
[CrossRef]

1956 (1)

S. J. Mason, “Feedback theory-further properties of signal flow graphs,” Proc. IRE 44, 920–926 (1956).
[CrossRef]

1953 (1)

S. J. Mason, “Feedback theory-some properties of signal flow graphs,” Proc. IRE 41, 1144–1156 (1953).
[CrossRef]

Absil, P.

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002).
[CrossRef]

Afroozeh, A.

M. Bahadoran, A. Afroozeh, J. Ali, and P. P. Yupapin, “Slow light generation using microring resonators for optical buffer application,” Opt. Eng. 51, 044601 (2012).
[CrossRef]

Aggarwal, I.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).

Akimoto, O.

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[CrossRef]

Ali, J.

M. Bahadoran, A. Afroozeh, J. Ali, and P. P. Yupapin, “Slow light generation using microring resonators for optical buffer application,” Opt. Eng. 51, 044601 (2012).
[CrossRef]

C. Sirawattananon, M. Bahadoran, J. Ali, S. Mitatha, and P. P. Yupapin, “Analytical Vernier effects of a PANDA ring resonator for micro force sensing application,” IEEE Trans. Nanotech. 11, 707–712 (2012).
[CrossRef]

M. S. Aziz, S. Daud, M. Bahadoran, J. Ali, and P. P. Yupapin, “Light pulse in a modified add–drop optical filter for optical tweezers generation,” J. Nonlinear Opt. Phys. Mater. 21, 1250047 (2012).
[CrossRef]

Asaka, S.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

Aziz, M. S.

M. S. Aziz, S. Daud, M. Bahadoran, J. Ali, and P. P. Yupapin, “Light pulse in a modified add–drop optical filter for optical tweezers generation,” J. Nonlinear Opt. Phys. Mater. 21, 1250047 (2012).
[CrossRef]

Bahadoran, M.

C. Sirawattananon, M. Bahadoran, J. Ali, S. Mitatha, and P. P. Yupapin, “Analytical Vernier effects of a PANDA ring resonator for micro force sensing application,” IEEE Trans. Nanotech. 11, 707–712 (2012).
[CrossRef]

M. S. Aziz, S. Daud, M. Bahadoran, J. Ali, and P. P. Yupapin, “Light pulse in a modified add–drop optical filter for optical tweezers generation,” J. Nonlinear Opt. Phys. Mater. 21, 1250047 (2012).
[CrossRef]

M. Bahadoran, A. Afroozeh, J. Ali, and P. P. Yupapin, “Slow light generation using microring resonators for optical buffer application,” Opt. Eng. 51, 044601 (2012).
[CrossRef]

Bahrampour, A.

A. Bahrampour, S. Zakeri, S. M. A. Mirzaee, Z. Ghaderi, and F. Farman, “All-optical set–reset flip–flop based on frequency bistability in semiconductor microring lasers,” Opt. Commun. 282, 2451–2456 (2009).
[CrossRef]

A. Bahrampour, M. Karimi, M. Qamsari, H. R. Nejad, and S. Keyvaninia, “All-optical set–reset flip–flop based on the passive microring-resonator bistability,” Opt. Commun. 281, 5104–5113 (2008).
[CrossRef]

Barman, A. D.

A. Malacarne, J. Wang, Y. Zhang, A. D. Barman, G. Berrettini, L. Poti, and A. Bogoni, “20 ps transition time all-optical SOA-based flip-flop used for photonic 10  Gb/s switching operation without any bit loss,” IEEE J. Sel. Top. Quantum Electron. 14, 808–815 (2008).
[CrossRef]

Basak, T.

S. Mandal, K. Dasgupta, T. Basak, and S. Ghosh, “A generalized approach for modeling and analysis of ring-resonator performance as optical filter,” Opt. Commun. 264, 97–104 (2006).
[CrossRef]

Berrettini, G.

A. Malacarne, J. Wang, Y. Zhang, A. D. Barman, G. Berrettini, L. Poti, and A. Bogoni, “20 ps transition time all-optical SOA-based flip-flop used for photonic 10  Gb/s switching operation without any bit loss,” IEEE J. Sel. Top. Quantum Electron. 14, 808–815 (2008).
[CrossRef]

Boeck, R.

Bogoni, A.

A. Malacarne, J. Wang, Y. Zhang, A. D. Barman, G. Berrettini, L. Poti, and A. Bogoni, “20 ps transition time all-optical SOA-based flip-flop used for photonic 10  Gb/s switching operation without any bit loss,” IEEE J. Sel. Top. Quantum Electron. 14, 808–815 (2008).
[CrossRef]

Boyd, R. W.

Chaiyasoonthorn, S.

P. P. Yupapin, N. Pornsuwancharoen, and S. Chaiyasoonthorn, “Attosecond pulse generation using the multistage nonlinear microring resonators,” Microw. Opt. Technol. Lett. 50, 3108–3111 (2008).
[CrossRef]

Chrostowski, L.

Costanzo-Caso, P. A.

P. A. Costanzo-Caso, Y. Jin, S. Granieri, and A. Siahmakoun, “Optical bistability in a nonlinear-SOA-based fiber ring resonator,” J. Nonlinear Opt. Phys. Mater. 20, 281–292 (2010).
[CrossRef]

Daido, H.

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[CrossRef]

Dasgupta, K.

S. Mandal, K. Dasgupta, T. Basak, and S. Ghosh, “A generalized approach for modeling and analysis of ring-resonator performance as optical filter,” Opt. Commun. 264, 97–104 (2006).
[CrossRef]

Daud, S.

M. S. Aziz, S. Daud, M. Bahadoran, J. Ali, and P. P. Yupapin, “Light pulse in a modified add–drop optical filter for optical tweezers generation,” J. Nonlinear Opt. Phys. Mater. 21, 1250047 (2012).
[CrossRef]

Dejhan, K.

S. Mitatha, K. Dejhan, P. P. Yupapin, and N. Pornsuwancharoen, “Chaotic signal generation and coding using a nonlinear micro ring resonator,” Optik 121, 120–125 (2010).
[CrossRef]

Dey, S.

S. Dey and S. Mandal, “Modeling and analysis of quadruple optical ring resonator performance as optical filter using Vernier principle,” Opt. Commun. 285, 439–446 (2012).
[CrossRef]

Dorren, H.

S. Zhang, D. Owens, Y. Liu, M. Hill, D. Lenstra, A. Tzanakaki, G. D. Khoe, and H. Dorren, “Multistate optical memory based on serially interconnected lasers,” IEEE Photon. Technol. Lett. 17, 1962–1964 (2005).
[CrossRef]

Dou, N.

N. Dou and C. Li, “Optical bistability in fiber ring resonator containing an EDFA,” Opt. Commun. 281, 2238–2242 (2008).
[CrossRef]

Fan, S.

M. Soljacic, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in nonlinear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
[CrossRef]

Farman, F.

A. Bahrampour, S. Zakeri, S. M. A. Mirzaee, Z. Ghaderi, and F. Farman, “All-optical set–reset flip–flop based on frequency bistability in semiconductor microring lasers,” Opt. Commun. 282, 2451–2456 (2009).
[CrossRef]

Felber, F. S.

J. H. Marburger and F. S. Felber, “Theory of a lossless nonlinear Fabry–Perot interferometer,” Phys. Rev. A 17, 335–342 (1978).
[CrossRef]

F. S. Felber and J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28, 731–733 (1976).
[CrossRef]

Fink, Y.

M. Soljacic, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in nonlinear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
[CrossRef]

França, F. H. R.

A. Maurente, F. H. R. França, K. Miki, and J. R. Howell, “Application of approximations for joint cumulative k-distributions for mixtures to FSK radiation heat transfer in multi-component high temperature non-LTE plasmas,” J. Quant. Spectrosc. Radiat. Transfer 113, 1521–1535 (2012).
[CrossRef]

Fujita, T.

C. L. Tang, A. Schremer, and T. Fujita, “Bistability in two-mode semiconductor lasers via gain saturation,” Appl. Phys. Lett. 51, 1392–1394 (1987).
[CrossRef]

Ghaderi, Z.

A. Bahrampour, S. Zakeri, S. M. A. Mirzaee, Z. Ghaderi, and F. Farman, “All-optical set–reset flip–flop based on frequency bistability in semiconductor microring lasers,” Opt. Commun. 282, 2451–2456 (2009).
[CrossRef]

Ghosh, S.

S. Mandal, K. Dasgupta, T. Basak, and S. Ghosh, “A generalized approach for modeling and analysis of ring-resonator performance as optical filter,” Opt. Commun. 264, 97–104 (2006).
[CrossRef]

Goodman, J. W.

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, “Fiber optic signal lattice processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Granieri, S.

P. A. Costanzo-Caso, Y. Jin, S. Granieri, and A. Siahmakoun, “Optical bistability in a nonlinear-SOA-based fiber ring resonator,” J. Nonlinear Opt. Phys. Mater. 20, 281–292 (2010).
[CrossRef]

Grover, R.

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002).
[CrossRef]

Harbold, J.

Heebner, J. E.

Hill, M.

S. Zhang, D. Owens, Y. Liu, M. Hill, D. Lenstra, A. Tzanakaki, G. D. Khoe, and H. Dorren, “Multistate optical memory based on serially interconnected lasers,” IEEE Photon. Technol. Lett. 17, 1962–1964 (2005).
[CrossRef]

Ho, P. T.

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002).
[CrossRef]

Howell, J. R.

A. Maurente, F. H. R. França, K. Miki, and J. R. Howell, “Application of approximations for joint cumulative k-distributions for mixtures to FSK radiation heat transfer in multi-component high temperature non-LTE plasmas,” J. Quant. Spectrosc. Radiat. Transfer 113, 1521–1535 (2012).
[CrossRef]

Huang, B.

N. Zou, W. Li, B. Huang, Z. Xu, S. Xu, and C. Yang, “An optical continuous phase FSK modulation scheme with an arbitrary modulation index over long-haul transmission fiber link,” Opt. Commun. 285, 2591–2595 (2012).
[CrossRef]

Ibanescu, M.

M. Soljacic, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in nonlinear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
[CrossRef]

Ibrahim, T.

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002).
[CrossRef]

Ikeda, K.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[CrossRef]

Ilday, F. Ö.

Itoh, H.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

Jaeger, N. A.

Jin, Y.

P. A. Costanzo-Caso, Y. Jin, S. Granieri, and A. Siahmakoun, “Optical bistability in a nonlinear-SOA-based fiber ring resonator,” J. Nonlinear Opt. Phys. Mater. 20, 281–292 (2010).
[CrossRef]

Joannopoulos, J. D.

M. Soljacic, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in nonlinear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
[CrossRef]

Johnson, F.

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002).
[CrossRef]

Johnson, S. G.

M. Soljacic, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in nonlinear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
[CrossRef]

Karimi, M.

A. Bahrampour, M. Karimi, M. Qamsari, H. R. Nejad, and S. Keyvaninia, “All-optical set–reset flip–flop based on the passive microring-resonator bistability,” Opt. Commun. 281, 5104–5113 (2008).
[CrossRef]

Keyvaninia, S.

A. Bahrampour, M. Karimi, M. Qamsari, H. R. Nejad, and S. Keyvaninia, “All-optical set–reset flip–flop based on the passive microring-resonator bistability,” Opt. Commun. 281, 5104–5113 (2008).
[CrossRef]

Khoe, G. D.

S. Zhang, D. Owens, Y. Liu, M. Hill, D. Lenstra, A. Tzanakaki, G. D. Khoe, and H. Dorren, “Multistate optical memory based on serially interconnected lasers,” IEEE Photon. Technol. Lett. 17, 1962–1964 (2005).
[CrossRef]

Lenstra, D.

S. Zhang, D. Owens, Y. Liu, M. Hill, D. Lenstra, A. Tzanakaki, G. D. Khoe, and H. Dorren, “Multistate optical memory based on serially interconnected lasers,” IEEE Photon. Technol. Lett. 17, 1962–1964 (2005).
[CrossRef]

Li, B.

B. Li, M. I. Memon, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “All-optical digital logic gates using bistable semiconductor ring lasers,” J. Opt. Commun. 30, 190–194 (2009).
[CrossRef]

Li, C.

N. Dou and C. Li, “Optical bistability in fiber ring resonator containing an EDFA,” Opt. Commun. 281, 2238–2242 (2008).
[CrossRef]

P. P. Yupapin, P. Saeung, and C. Li, “Characteristics of complementary ring-resonator add/drop filters modeling by using graphical approach,” Opt. Commun. 272, 81–86 (2007).
[CrossRef]

Li, W.

N. Zou, W. Li, B. Huang, Z. Xu, S. Xu, and C. Yang, “An optical continuous phase FSK modulation scheme with an arbitrary modulation index over long-haul transmission fiber link,” Opt. Commun. 285, 2591–2595 (2012).
[CrossRef]

Liu, Y.

S. Zhang, D. Owens, Y. Liu, M. Hill, D. Lenstra, A. Tzanakaki, G. D. Khoe, and H. Dorren, “Multistate optical memory based on serially interconnected lasers,” IEEE Photon. Technol. Lett. 17, 1962–1964 (2005).
[CrossRef]

Luo, C.

M. Soljacic, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in nonlinear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
[CrossRef]

Malacarne, A.

A. Malacarne, J. Wang, Y. Zhang, A. D. Barman, G. Berrettini, L. Poti, and A. Bogoni, “20 ps transition time all-optical SOA-based flip-flop used for photonic 10  Gb/s switching operation without any bit loss,” IEEE J. Sel. Top. Quantum Electron. 14, 808–815 (2008).
[CrossRef]

Mandal, S.

S. Dey and S. Mandal, “Modeling and analysis of quadruple optical ring resonator performance as optical filter using Vernier principle,” Opt. Commun. 285, 439–446 (2012).
[CrossRef]

S. Mandal, K. Dasgupta, T. Basak, and S. Ghosh, “A generalized approach for modeling and analysis of ring-resonator performance as optical filter,” Opt. Commun. 264, 97–104 (2006).
[CrossRef]

Marburger, J. H.

J. H. Marburger and F. S. Felber, “Theory of a lossless nonlinear Fabry–Perot interferometer,” Phys. Rev. A 17, 335–342 (1978).
[CrossRef]

F. S. Felber and J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28, 731–733 (1976).
[CrossRef]

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S. J. Mason, “Feedback theory-further properties of signal flow graphs,” Proc. IRE 44, 920–926 (1956).
[CrossRef]

S. J. Mason, “Feedback theory-some properties of signal flow graphs,” Proc. IRE 41, 1144–1156 (1953).
[CrossRef]

Matsuoka, M.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

Maurente, A.

A. Maurente, F. H. R. França, K. Miki, and J. R. Howell, “Application of approximations for joint cumulative k-distributions for mixtures to FSK radiation heat transfer in multi-component high temperature non-LTE plasmas,” J. Quant. Spectrosc. Radiat. Transfer 113, 1521–1535 (2012).
[CrossRef]

Memon, M. I.

B. Li, M. I. Memon, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “All-optical digital logic gates using bistable semiconductor ring lasers,” J. Opt. Commun. 30, 190–194 (2009).
[CrossRef]

Mezosi, G.

B. Li, M. I. Memon, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “All-optical digital logic gates using bistable semiconductor ring lasers,” J. Opt. Commun. 30, 190–194 (2009).
[CrossRef]

Miki, K.

A. Maurente, F. H. R. França, K. Miki, and J. R. Howell, “Application of approximations for joint cumulative k-distributions for mixtures to FSK radiation heat transfer in multi-component high temperature non-LTE plasmas,” J. Quant. Spectrosc. Radiat. Transfer 113, 1521–1535 (2012).
[CrossRef]

Miller, D.

D. Miller, “Refractive Fabry–Perot bistability with linear absorption: theory of operation and cavity optimization,” IEEE J. Quantum Electron. 17, 306–311 (1981).
[CrossRef]

Mirzaee, S. M. A.

A. Bahrampour, S. Zakeri, S. M. A. Mirzaee, Z. Ghaderi, and F. Farman, “All-optical set–reset flip–flop based on frequency bistability in semiconductor microring lasers,” Opt. Commun. 282, 2451–2456 (2009).
[CrossRef]

Mitatha, S.

C. Sirawattananon, M. Bahadoran, J. Ali, S. Mitatha, and P. P. Yupapin, “Analytical Vernier effects of a PANDA ring resonator for micro force sensing application,” IEEE Trans. Nanotech. 11, 707–712 (2012).
[CrossRef]

S. Mitatha, K. Dejhan, P. P. Yupapin, and N. Pornsuwancharoen, “Chaotic signal generation and coding using a nonlinear micro ring resonator,” Optik 121, 120–125 (2010).
[CrossRef]

Moslehi, B.

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, “Fiber optic signal lattice processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Nakatsuka, H.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

Nejad, H. R.

A. Bahrampour, M. Karimi, M. Qamsari, H. R. Nejad, and S. Keyvaninia, “All-optical set–reset flip–flop based on the passive microring-resonator bistability,” Opt. Commun. 281, 5104–5113 (2008).
[CrossRef]

Nguyen, V.

Owens, D.

S. Zhang, D. Owens, Y. Liu, M. Hill, D. Lenstra, A. Tzanakaki, G. D. Khoe, and H. Dorren, “Multistate optical memory based on serially interconnected lasers,” IEEE Photon. Technol. Lett. 17, 1962–1964 (2005).
[CrossRef]

Pornsuwancharoen, N.

S. Mitatha, K. Dejhan, P. P. Yupapin, and N. Pornsuwancharoen, “Chaotic signal generation and coding using a nonlinear micro ring resonator,” Optik 121, 120–125 (2010).
[CrossRef]

P. P. Yupapin and N. Pornsuwancharoen, “Proposed nonlinear microring resonator arrangement for stopping and storing light,” IEEE Photon. Technol. Lett. 21, 404–406 (2009).
[CrossRef]

P. P. Yupapin, N. Pornsuwancharoen, and S. Chaiyasoonthorn, “Attosecond pulse generation using the multistage nonlinear microring resonators,” Microw. Opt. Technol. Lett. 50, 3108–3111 (2008).
[CrossRef]

Poti, L.

A. Malacarne, J. Wang, Y. Zhang, A. D. Barman, G. Berrettini, L. Poti, and A. Bogoni, “20 ps transition time all-optical SOA-based flip-flop used for photonic 10  Gb/s switching operation without any bit loss,” IEEE J. Sel. Top. Quantum Electron. 14, 808–815 (2008).
[CrossRef]

Qamsari, M.

A. Bahrampour, M. Karimi, M. Qamsari, H. R. Nejad, and S. Keyvaninia, “All-optical set–reset flip–flop based on the passive microring-resonator bistability,” Opt. Commun. 281, 5104–5113 (2008).
[CrossRef]

Rouger, N.

Saeung, P.

P. Saeung and P. P. Yupapin, “Generalized analysis of multiple ring resonator filters: modeling by using graphical approach,” Optik 119, 465–472 (2008).
[CrossRef]

P. P. Yupapin, P. Saeung, and C. Li, “Characteristics of complementary ring-resonator add/drop filters modeling by using graphical approach,” Opt. Commun. 272, 81–86 (2007).
[CrossRef]

P. Saeung and P. P. Yupapin, “Vernier effect of multiple-ring resonator filters modeling by a graphical approach,” Opt. Eng. 46, 075005 (2007).
[CrossRef]

Sanchez, F.

F. Sanchez, “Optical bistability in a 2×2 coupler fiber ring resonator: parametric formulation,” Opt. Commun. 142, 211–214 (1997).
[CrossRef]

Sanghera, J.

Schremer, A.

C. L. Tang, A. Schremer, and T. Fujita, “Bistability in two-mode semiconductor lasers via gain saturation,” Appl. Phys. Lett. 51, 1392–1394 (1987).
[CrossRef]

Shaw, H. J.

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, “Fiber optic signal lattice processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Shaw, L.

Siahmakoun, A.

P. A. Costanzo-Caso, Y. Jin, S. Granieri, and A. Siahmakoun, “Optical bistability in a nonlinear-SOA-based fiber ring resonator,” J. Nonlinear Opt. Phys. Mater. 20, 281–292 (2010).
[CrossRef]

Sirawattananon, C.

C. Sirawattananon, M. Bahadoran, J. Ali, S. Mitatha, and P. P. Yupapin, “Analytical Vernier effects of a PANDA ring resonator for micro force sensing application,” IEEE Trans. Nanotech. 11, 707–712 (2012).
[CrossRef]

Soljacic, M.

M. Soljacic, M. Ibanescu, C. Luo, S. G. Johnson, S. Fan, Y. Fink, and J. D. Joannopoulos, “All-optical switching using optical bistability in nonlinear photonic crystals,” Proc. SPIE 5000, 200–214 (2003).
[CrossRef]

Sorel, M.

B. Li, M. I. Memon, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “All-optical digital logic gates using bistable semiconductor ring lasers,” J. Opt. Commun. 30, 190–194 (2009).
[CrossRef]

Suchat, S.

P. P. Yupapin and S. Suchat, “Nonlinear penalties and benefits of light traveling in a fiber optic ring resonator,” Optik. 120, 216–221 (2009).
[CrossRef]

Suwancharoen, W.

P. P. Yupapin and W. Suwancharoen, “Chaotic signal generation and cancellation using a micro ring resonator incorporating an optical add/drop multiplexer,” Opt. Commun. 280, 343–350 (2007).
[CrossRef]

Tang, C. L.

C. L. Tang, A. Schremer, and T. Fujita, “Bistability in two-mode semiconductor lasers via gain saturation,” Appl. Phys. Lett. 51, 1392–1394 (1987).
[CrossRef]

Tur, M.

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, “Fiber optic signal lattice processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Tzanakaki, A.

S. Zhang, D. Owens, Y. Liu, M. Hill, D. Lenstra, A. Tzanakaki, G. D. Khoe, and H. Dorren, “Multistate optical memory based on serially interconnected lasers,” IEEE Photon. Technol. Lett. 17, 1962–1964 (2005).
[CrossRef]

Van, V.

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002).
[CrossRef]

Vanishkorn, B.

P. P. Yupapin and B. Vanishkorn, “Mathematical simulation of light pulse propagating within a microring resonator system and applications,” Appl. Math. Model. 35, 1729–1738 (2011).
[CrossRef]

Wang, J.

A. Malacarne, J. Wang, Y. Zhang, A. D. Barman, G. Berrettini, L. Poti, and A. Bogoni, “20 ps transition time all-optical SOA-based flip-flop used for photonic 10  Gb/s switching operation without any bit loss,” IEEE J. Sel. Top. Quantum Electron. 14, 808–815 (2008).
[CrossRef]

Wang, Z.

B. Li, M. I. Memon, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “All-optical digital logic gates using bistable semiconductor ring lasers,” J. Opt. Commun. 30, 190–194 (2009).
[CrossRef]

Wise, F.

Xu, S.

N. Zou, W. Li, B. Huang, Z. Xu, S. Xu, and C. Yang, “An optical continuous phase FSK modulation scheme with an arbitrary modulation index over long-haul transmission fiber link,” Opt. Commun. 285, 2591–2595 (2012).
[CrossRef]

Xu, Z.

N. Zou, W. Li, B. Huang, Z. Xu, S. Xu, and C. Yang, “An optical continuous phase FSK modulation scheme with an arbitrary modulation index over long-haul transmission fiber link,” Opt. Commun. 285, 2591–2595 (2012).
[CrossRef]

Yang, C.

N. Zou, W. Li, B. Huang, Z. Xu, S. Xu, and C. Yang, “An optical continuous phase FSK modulation scheme with an arbitrary modulation index over long-haul transmission fiber link,” Opt. Commun. 285, 2591–2595 (2012).
[CrossRef]

Yu, S.

B. Li, M. I. Memon, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “All-optical digital logic gates using bistable semiconductor ring lasers,” J. Opt. Commun. 30, 190–194 (2009).
[CrossRef]

Yupapin, P. P.

M. Bahadoran, A. Afroozeh, J. Ali, and P. P. Yupapin, “Slow light generation using microring resonators for optical buffer application,” Opt. Eng. 51, 044601 (2012).
[CrossRef]

C. Sirawattananon, M. Bahadoran, J. Ali, S. Mitatha, and P. P. Yupapin, “Analytical Vernier effects of a PANDA ring resonator for micro force sensing application,” IEEE Trans. Nanotech. 11, 707–712 (2012).
[CrossRef]

M. S. Aziz, S. Daud, M. Bahadoran, J. Ali, and P. P. Yupapin, “Light pulse in a modified add–drop optical filter for optical tweezers generation,” J. Nonlinear Opt. Phys. Mater. 21, 1250047 (2012).
[CrossRef]

P. P. Yupapin and B. Vanishkorn, “Mathematical simulation of light pulse propagating within a microring resonator system and applications,” Appl. Math. Model. 35, 1729–1738 (2011).
[CrossRef]

S. Mitatha, K. Dejhan, P. P. Yupapin, and N. Pornsuwancharoen, “Chaotic signal generation and coding using a nonlinear micro ring resonator,” Optik 121, 120–125 (2010).
[CrossRef]

P. P. Yupapin and N. Pornsuwancharoen, “Proposed nonlinear microring resonator arrangement for stopping and storing light,” IEEE Photon. Technol. Lett. 21, 404–406 (2009).
[CrossRef]

P. P. Yupapin and S. Suchat, “Nonlinear penalties and benefits of light traveling in a fiber optic ring resonator,” Optik. 120, 216–221 (2009).
[CrossRef]

P. P. Yupapin, N. Pornsuwancharoen, and S. Chaiyasoonthorn, “Attosecond pulse generation using the multistage nonlinear microring resonators,” Microw. Opt. Technol. Lett. 50, 3108–3111 (2008).
[CrossRef]

P. P. Yupapin, “Coupler-loss and coupling-coefficient-dependent bistability and instability in a fiber ring resonator,” Optik 119, 492–494 (2008).
[CrossRef]

P. Saeung and P. P. Yupapin, “Generalized analysis of multiple ring resonator filters: modeling by using graphical approach,” Optik 119, 465–472 (2008).
[CrossRef]

P. Saeung and P. P. Yupapin, “Vernier effect of multiple-ring resonator filters modeling by a graphical approach,” Opt. Eng. 46, 075005 (2007).
[CrossRef]

P. P. Yupapin, P. Saeung, and C. Li, “Characteristics of complementary ring-resonator add/drop filters modeling by using graphical approach,” Opt. Commun. 272, 81–86 (2007).
[CrossRef]

P. P. Yupapin and W. Suwancharoen, “Chaotic signal generation and cancellation using a micro ring resonator incorporating an optical add/drop multiplexer,” Opt. Commun. 280, 343–350 (2007).
[CrossRef]

Zakeri, S.

A. Bahrampour, S. Zakeri, S. M. A. Mirzaee, Z. Ghaderi, and F. Farman, “All-optical set–reset flip–flop based on frequency bistability in semiconductor microring lasers,” Opt. Commun. 282, 2451–2456 (2009).
[CrossRef]

Zhang, S.

S. Zhang, D. Owens, Y. Liu, M. Hill, D. Lenstra, A. Tzanakaki, G. D. Khoe, and H. Dorren, “Multistate optical memory based on serially interconnected lasers,” IEEE Photon. Technol. Lett. 17, 1962–1964 (2005).
[CrossRef]

Zhang, Y.

A. Malacarne, J. Wang, Y. Zhang, A. D. Barman, G. Berrettini, L. Poti, and A. Bogoni, “20 ps transition time all-optical SOA-based flip-flop used for photonic 10  Gb/s switching operation without any bit loss,” IEEE J. Sel. Top. Quantum Electron. 14, 808–815 (2008).
[CrossRef]

Zou, N.

N. Zou, W. Li, B. Huang, Z. Xu, S. Xu, and C. Yang, “An optical continuous phase FSK modulation scheme with an arbitrary modulation index over long-haul transmission fiber link,” Opt. Commun. 285, 2591–2595 (2012).
[CrossRef]

Appl. Math. Model. (1)

P. P. Yupapin and B. Vanishkorn, “Mathematical simulation of light pulse propagating within a microring resonator system and applications,” Appl. Math. Model. 35, 1729–1738 (2011).
[CrossRef]

Appl. Phys. Lett. (2)

C. L. Tang, A. Schremer, and T. Fujita, “Bistability in two-mode semiconductor lasers via gain saturation,” Appl. Phys. Lett. 51, 1392–1394 (1987).
[CrossRef]

F. S. Felber and J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28, 731–733 (1976).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Miller, “Refractive Fabry–Perot bistability with linear absorption: theory of operation and cavity optimization,” IEEE J. Quantum Electron. 17, 306–311 (1981).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

A. Malacarne, J. Wang, Y. Zhang, A. D. Barman, G. Berrettini, L. Poti, and A. Bogoni, “20 ps transition time all-optical SOA-based flip-flop used for photonic 10  Gb/s switching operation without any bit loss,” IEEE J. Sel. Top. Quantum Electron. 14, 808–815 (2008).
[CrossRef]

V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

S. Zhang, D. Owens, Y. Liu, M. Hill, D. Lenstra, A. Tzanakaki, G. D. Khoe, and H. Dorren, “Multistate optical memory based on serially interconnected lasers,” IEEE Photon. Technol. Lett. 17, 1962–1964 (2005).
[CrossRef]

P. P. Yupapin and N. Pornsuwancharoen, “Proposed nonlinear microring resonator arrangement for stopping and storing light,” IEEE Photon. Technol. Lett. 21, 404–406 (2009).
[CrossRef]

IEEE Trans. Nanotech. (1)

C. Sirawattananon, M. Bahadoran, J. Ali, S. Mitatha, and P. P. Yupapin, “Analytical Vernier effects of a PANDA ring resonator for micro force sensing application,” IEEE Trans. Nanotech. 11, 707–712 (2012).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (2)

M. S. Aziz, S. Daud, M. Bahadoran, J. Ali, and P. P. Yupapin, “Light pulse in a modified add–drop optical filter for optical tweezers generation,” J. Nonlinear Opt. Phys. Mater. 21, 1250047 (2012).
[CrossRef]

P. A. Costanzo-Caso, Y. Jin, S. Granieri, and A. Siahmakoun, “Optical bistability in a nonlinear-SOA-based fiber ring resonator,” J. Nonlinear Opt. Phys. Mater. 20, 281–292 (2010).
[CrossRef]

J. Opt. Commun. (1)

B. Li, M. I. Memon, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “All-optical digital logic gates using bistable semiconductor ring lasers,” J. Opt. Commun. 30, 190–194 (2009).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

A. Maurente, F. H. R. França, K. Miki, and J. R. Howell, “Application of approximations for joint cumulative k-distributions for mixtures to FSK radiation heat transfer in multi-component high temperature non-LTE plasmas,” J. Quant. Spectrosc. Radiat. Transfer 113, 1521–1535 (2012).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

P. P. Yupapin, N. Pornsuwancharoen, and S. Chaiyasoonthorn, “Attosecond pulse generation using the multistage nonlinear microring resonators,” Microw. Opt. Technol. Lett. 50, 3108–3111 (2008).
[CrossRef]

Opt. Commun. (9)

P. P. Yupapin and W. Suwancharoen, “Chaotic signal generation and cancellation using a micro ring resonator incorporating an optical add/drop multiplexer,” Opt. Commun. 280, 343–350 (2007).
[CrossRef]

S. Dey and S. Mandal, “Modeling and analysis of quadruple optical ring resonator performance as optical filter using Vernier principle,” Opt. Commun. 285, 439–446 (2012).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Architecture of a PANDA Vernier filter: (a) waveguide layout and (b) Z-transform diagram SFG.

Fig. 2.
Fig. 2.

Influence of variation of middle coupling coefficients on insertion loss of the PANDA Vernier resonator with resonant mode number of (N;NR;NL)=(5;4;4), Lc=15μm, R1=6.545μm, RR=RL=4225μm, k1=k2=0.5, γ=0.01, α=20dB/cm: (a) through-port response and (b) drop-port response.

Fig. 3.
Fig. 3.

Effect of changing middle coupling coefficients on insertion loss of the PANDA Vernier resonator with resonant mode number of (N;NR;NL)=(9;7;7), with racetrack circumference of L1=127.91μm, LR=LL=99.487μm, k1=k2=0.5, γ=0.01, α=3dB/cm: (a) through-port response and (b) drop-port response.

Fig. 4.
Fig. 4.

Optical bistability diagram of PANDA Vernier resonator for resonant mode number of 5;4;4.

Fig. 5.
Fig. 5.

Optical bistability diagram of PANDA Vernier resonator for resonant mode number of 9;7;7.

Fig. 6.
Fig. 6.

Optical transfer function of PANDA Vernier resonator with resonant mode number of 5;4;4 for (a) through port and (b) drop port.

Fig. 7.
Fig. 7.

Optical transfer function of PANDA Vernier resonator with resonant mode number of 9;7;7 for (a) through port and (b) drop port.

Equations (23)

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FSRtot=N·FSR1=NR·FSRR=NL·FSRL
H=EnEi=i=1nTiΔiΔ,
L1=C1C2CRCLξN,
L2=LR=CRξRNR,
L3=LL=CLξLNL,
L4=L1R=C1C2CLSR2ξRNRξN,
L5=L1L=C1C2CRSL2ξLNLξN,
L6=C1C2SR2SL2ξLNLξRNRξN.
L12=L1·LR=C1C2CR2CLξRNRξN,
L13=L1·LL=C1C2CL2CRξLNLξN,
L23=LR·LL=CLCRξLNLξRNR,
L25=LR·L5=C1C2CR2SL2ξLNLξRNRξN,
L34=LL·L4=C1C2CL2SR2ξLNLξRNRξN,
L123=L1·L2·L3=C1C2CL2CR2ξLNLξRNRξN.
T1thr=C1,Δ1=1i=16Li+L12+L13+L23+L25+L34L123.
T2thr=C2CRCLS12ξN,Δ2=1L2L3+L23.
T3thr=C2CLS12SR2ξRRξN,Δ3=1L3.
T4thr=C2S12SR2SL2ξRRξLLξN,Δ4=1.
T5thr=C2CRS12SL2ξLLξN,Δ5=1L2.
H31=EThroughEin={C1(1CRξRNRCLξLNL+CRCLξRNRξLNL)+C2ξN(CRξLNL+CLξRNRCLCRξLNLξRNR)}{1C1C2CRCLξNCRξRNRCLξLL+C1C2CLSR2ξRNRξN+C1C2CRSL2ξLNLξNC1C2SR2SL2ξLNLξRNRξN+C1C2CR2CL×ξRNRξN+C1C2CL2CRξLNLξN+CLCR×ξLNLξRNRC1C2CR2SL2ξLNLξRNRξN+C1C2CL2SR2ξLNLξRNRξN+C1C2CL2CR2ξLNLξRNRξN}.
T1drp=CRS1S2ξN,Δ1=1L2L3+L23.
T2drp=S1S2SR2ξRNRξN,Δ2=1L3.
H81=EDropEin={S1S2ξN(CRCLξLNLCLξLNLξRNR+ξRNRCR)}{1C1C2CRCLξNCRξRNRCLξLL+C1C2CLSR2ξRNRξN+C1C2CRSL2ξLNLξNC1C2SR2SL2ξLNLξRNRξN+C1C2CR2CL×ξRNRξN+C1C2CL2CRξLNLξN+CLCR×ξLNLξRNRC1C2CR2SL2ξLNLξRNRξN+C1C2CL2SR2ξLNLξRNRξN+C1C2CL2CR2ξLNLξRNRξN}.

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