Abstract

The work addresses the issues of the ever-expanding demand of wide free spectral range (FSR) compatible integrated optic passive devices used in commercial communication systems like dense wavelength division multiplexing and a host of other applications. FSR expansion has been achieved by using the Vernier principle. The analysis has been carried out using Mason’s rule and the delay line signal processing approach. Performances of two resonator structures made of silicon-on-insulator-based waveguides with two different effective refractive indices, viz., 1.7 and 2.811, have been compared. The maximum FSR of 3527 THz with finesse of 4409 has been obtained from the first architecture.

© 2012 Optical Society of America

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  24. Q. Huang, Y. Yu, and J. Yu, “Experimental investigation on submicron rib waveguide microring/racetrack resonators in silicon-on-insulator,” Opt. Commun. 282, 22–26 (2009).
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  27. 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]

2012

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

S. Dey and S. Mandal, “Wide free-spectral-range triple ring resonator as optical filter,” Opt. Eng. 50, 084601 (2011).
[CrossRef]

2009

Q. Huang, Y. Yu, and J. Yu, “Experimental investigation on submicron rib waveguide microring/racetrack resonators in silicon-on-insulator,” Opt. Commun. 282, 22–26 (2009).
[CrossRef]

2007

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007).
[CrossRef]

O. Schwelb, “The nature of spurious mode suppression in extended FSR microring multiplexers,” Opt. Commun. 271, 424–429 (2007).
[CrossRef]

2006

B. Jalali, M. Paniccia, and G. Reed, “Silicon photonics,” IEEE Microw. Mag. 7, 58–68 (2006).
[CrossRef]

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
[CrossRef]

F. Morichetti, A. Melloni, and M. Martinelli, “Effects of polarization rotation in optical ring-resonator-based devices,” J. Lightwave Technol. 24, 573–585 (2006).
[CrossRef]

2005

2004

V. M. Menon, W. Tong, and S. R. Forrest, “Control of quality factor and critical coupling in microring resonators through integration of a semiconductor optical amplifier,” IEEE Photon. Technol. Lett 16, 1343–1345 (2004).
[CrossRef]

M. J. Kobrinsky, B. A. Block, J.-F. Zheng, B. C. Barnett, E. Mohammed, M. Reshotko, F. Robertson, S. List, I. Young, and K. Cadien, “On-chip optical interconnects,” Intel Technol. J. 8, 129–143 (2004).

2002

A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002).
[CrossRef]

1999

D. Kajfez, S. Chebolu, M. R. Abdul-Gaffoor, and A. A. Kishk, “Uncertainty analysis of the transmission-type measurement of Q-factor,” IEEE Trans. Microwave Theory Tech. 47, 367–371 (1999).
[CrossRef]

1995

G. Barbarossa, A. M. Matteo, and M. N. Armenise, “Theoretical analysis of triple-coupler ring-based optical guided-wave resonator,” IEEE J. Lightwave Technol. 13, 148–157 (1995).
[CrossRef]

S. Suzuki, K. Oda, and Y. Hibino, “Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz,” IEEE J. Lightwave Technol. 13, 1766–1771 (1995).
[CrossRef]

1984

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

1983

R. C. Youngquist, L. F. Stokes, and H. J. Shaw, “Effects of normal mode loss in dielectric waveguide directional couplers and interferometers,” IEEE J. Quantum Electron. 19, 1888–1896 (1983).
[CrossRef]

1972

A. Deczky, “Synthesis of recursive digital filters using the minimum p-error criterion,” IEEE Trans. Audio Electroacoust. 20, 257–263 (1972).
[CrossRef]

1971

D. Marcuse, “Bending losses of the asymmetric slab waveguide,” Bell System Tech. J. 50, 2551–2563 (1971).

1956

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

Abdul-Gaffoor, M. R.

D. Kajfez, S. Chebolu, M. R. Abdul-Gaffoor, and A. A. Kishk, “Uncertainty analysis of the transmission-type measurement of Q-factor,” IEEE Trans. Microwave Theory Tech. 47, 367–371 (1999).
[CrossRef]

Armenise, M. N.

G. Barbarossa, A. M. Matteo, and M. N. Armenise, “Theoretical analysis of triple-coupler ring-based optical guided-wave resonator,” IEEE J. Lightwave Technol. 13, 148–157 (1995).
[CrossRef]

Barbarossa, G.

G. Barbarossa, A. M. Matteo, and M. N. Armenise, “Theoretical analysis of triple-coupler ring-based optical guided-wave resonator,” IEEE J. Lightwave Technol. 13, 148–157 (1995).
[CrossRef]

Barnett, B. C.

M. J. Kobrinsky, B. A. Block, J.-F. Zheng, B. C. Barnett, E. Mohammed, M. Reshotko, F. Robertson, S. List, I. Young, and K. Cadien, “On-chip optical interconnects,” Intel Technol. J. 8, 129–143 (2004).

Block, B. A.

M. J. Kobrinsky, B. A. Block, J.-F. Zheng, B. C. Barnett, E. Mohammed, M. Reshotko, F. Robertson, S. List, I. Young, and K. Cadien, “On-chip optical interconnects,” Intel Technol. J. 8, 129–143 (2004).

Cadien, K.

M. J. Kobrinsky, B. A. Block, J.-F. Zheng, B. C. Barnett, E. Mohammed, M. Reshotko, F. Robertson, S. List, I. Young, and K. Cadien, “On-chip optical interconnects,” Intel Technol. J. 8, 129–143 (2004).

Chebolu, S.

D. Kajfez, S. Chebolu, M. R. Abdul-Gaffoor, and A. A. Kishk, “Uncertainty analysis of the transmission-type measurement of Q-factor,” IEEE Trans. Microwave Theory Tech. 47, 367–371 (1999).
[CrossRef]

Deczky, A.

A. Deczky, “Synthesis of recursive digital filters using the minimum p-error criterion,” IEEE Trans. Audio Electroacoust. 20, 257–263 (1972).
[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]

S. Dey and S. Mandal, “Wide free-spectral-range triple ring resonator as optical filter,” Opt. Eng. 50, 084601 (2011).
[CrossRef]

S. Dey, and S. Mandal, “Enhancement of free spectral range in optical triple ring resonator: a Vernier principle approach,” in IEEE Proceedings of International Conference on Recent Advances in Information Technology (RAIT-2012) (IEEE, 2012), Vol. 1, pp. 246–250.

Dumon, P.

L. Y. M. Tobing and P. Dumon, “Fundamental principles of operation andnotes on fabrication of photonic microresonators,” in Photonic Microring Research and Application, Vol. 156 of Springer Series in Optical Sciences (Springer, 2010), Chap. 1.

Forrest, S. R.

V. M. Menon, W. Tong, and S. R. Forrest, “Control of quality factor and critical coupling in microring resonators through integration of a semiconductor optical amplifier,” IEEE Photon. Technol. Lett 16, 1343–1345 (2004).
[CrossRef]

Goodman, J.

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

Hibino, Y.

S. Suzuki, K. Oda, and Y. Hibino, “Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz,” IEEE J. Lightwave Technol. 13, 1766–1771 (1995).
[CrossRef]

Huang, Q.

Q. Huang, Y. Yu, and J. Yu, “Experimental investigation on submicron rib waveguide microring/racetrack resonators in silicon-on-insulator,” Opt. Commun. 282, 22–26 (2009).
[CrossRef]

Jalali, B.

B. Jalali, M. Paniccia, and G. Reed, “Silicon photonics,” IEEE Microw. Mag. 7, 58–68 (2006).
[CrossRef]

Kajfez, D.

D. Kajfez, S. Chebolu, M. R. Abdul-Gaffoor, and A. A. Kishk, “Uncertainty analysis of the transmission-type measurement of Q-factor,” IEEE Trans. Microwave Theory Tech. 47, 367–371 (1999).
[CrossRef]

Kishk, A. A.

D. Kajfez, S. Chebolu, M. R. Abdul-Gaffoor, and A. A. Kishk, “Uncertainty analysis of the transmission-type measurement of Q-factor,” IEEE Trans. Microwave Theory Tech. 47, 367–371 (1999).
[CrossRef]

Kobrinsky, M. J.

M. J. Kobrinsky, B. A. Block, J.-F. Zheng, B. C. Barnett, E. Mohammed, M. Reshotko, F. Robertson, S. List, I. Young, and K. Cadien, “On-chip optical interconnects,” Intel Technol. J. 8, 129–143 (2004).

Lipson, M.

List, S.

M. J. Kobrinsky, B. A. Block, J.-F. Zheng, B. C. Barnett, E. Mohammed, M. Reshotko, F. Robertson, S. List, I. Young, and K. Cadien, “On-chip optical interconnects,” Intel Technol. J. 8, 129–143 (2004).

Madsen, C. K.

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis, A Signal Processing Approach (Wiley, 1999).S

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. Dey and S. Mandal, “Wide free-spectral-range triple ring resonator as optical filter,” Opt. Eng. 50, 084601 (2011).
[CrossRef]

S. Dey, and S. Mandal, “Enhancement of free spectral range in optical triple ring resonator: a Vernier principle approach,” in IEEE Proceedings of International Conference on Recent Advances in Information Technology (RAIT-2012) (IEEE, 2012), Vol. 1, pp. 246–250.

Marcuse, D.

D. Marcuse, “Bending losses of the asymmetric slab waveguide,” Bell System Tech. J. 50, 2551–2563 (1971).

Martinelli, M.

Mason, S. J.

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

Matteo, A. M.

G. Barbarossa, A. M. Matteo, and M. N. Armenise, “Theoretical analysis of triple-coupler ring-based optical guided-wave resonator,” IEEE J. Lightwave Technol. 13, 148–157 (1995).
[CrossRef]

Melloni, A.

Menon, V. M.

V. M. Menon, W. Tong, and S. R. Forrest, “Control of quality factor and critical coupling in microring resonators through integration of a semiconductor optical amplifier,” IEEE Photon. Technol. Lett 16, 1343–1345 (2004).
[CrossRef]

Mohammed, E.

M. J. Kobrinsky, B. A. Block, J.-F. Zheng, B. C. Barnett, E. Mohammed, M. Reshotko, F. Robertson, S. List, I. Young, and K. Cadien, “On-chip optical interconnects,” Intel Technol. J. 8, 129–143 (2004).

Morichetti, F.

F. Morichetti, A. Melloni, and M. Martinelli, “Effects of polarization rotation in optical ring-resonator-based devices,” J. Lightwave Technol. 24, 573–585 (2006).
[CrossRef]

F. Morichetti, “Polarization rotation in ring resonators,” in Photonic Microresonator Research and Application, Vol.  156 of Springer Series in Optical Sciences (Springer, 2010), Chap 3.

Moslehi, B.

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

Oda, K.

S. Suzuki, K. Oda, and Y. Hibino, “Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz,” IEEE J. Lightwave Technol. 13, 1766–1771 (1995).
[CrossRef]

Oppenheim, A.

A. Oppenheim and R. Schafer, Digital Signal Processing, 2nd ed. (Prentice-Hall, 1975).

Paniccia, M.

B. Jalali, M. Paniccia, and G. Reed, “Silicon photonics,” IEEE Microw. Mag. 7, 58–68 (2006).
[CrossRef]

Reed, G.

B. Jalali, M. Paniccia, and G. Reed, “Silicon photonics,” IEEE Microw. Mag. 7, 58–68 (2006).
[CrossRef]

Reshotko, M.

M. J. Kobrinsky, B. A. Block, J.-F. Zheng, B. C. Barnett, E. Mohammed, M. Reshotko, F. Robertson, S. List, I. Young, and K. Cadien, “On-chip optical interconnects,” Intel Technol. J. 8, 129–143 (2004).

Robertson, F.

M. J. Kobrinsky, B. A. Block, J.-F. Zheng, B. C. Barnett, E. Mohammed, M. Reshotko, F. Robertson, S. List, I. Young, and K. Cadien, “On-chip optical interconnects,” Intel Technol. J. 8, 129–143 (2004).

Schafer, R.

A. Oppenheim and R. Schafer, Digital Signal Processing, 2nd ed. (Prentice-Hall, 1975).

Schwelb, O.

O. Schwelb, “The nature of spurious mode suppression in extended FSR microring multiplexers,” Opt. Commun. 271, 424–429 (2007).
[CrossRef]

O. Schwelb, “Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters—a tutorial overview,” IEEE J. Lightwave Technol.22, 1380–1394 (2004).

Sekaric, L.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007).
[CrossRef]

Shaw, H.

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

Shaw, H. J.

R. C. Youngquist, L. F. Stokes, and H. J. Shaw, “Effects of normal mode loss in dielectric waveguide directional couplers and interferometers,” IEEE J. Quantum Electron. 19, 1888–1896 (1983).
[CrossRef]

Soref, R.

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
[CrossRef]

Stokes, L. F.

R. C. Youngquist, L. F. Stokes, and H. J. Shaw, “Effects of normal mode loss in dielectric waveguide directional couplers and interferometers,” IEEE J. Quantum Electron. 19, 1888–1896 (1983).
[CrossRef]

Suzuki, S.

S. Suzuki, K. Oda, and Y. Hibino, “Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz,” IEEE J. Lightwave Technol. 13, 1766–1771 (1995).
[CrossRef]

Tobing, L. Y. M.

L. Y. M. Tobing and P. Dumon, “Fundamental principles of operation andnotes on fabrication of photonic microresonators,” in Photonic Microring Research and Application, Vol. 156 of Springer Series in Optical Sciences (Springer, 2010), Chap. 1.

Tomoyuki, K.

K. Yasuo and K. Tomoyuki, “Series-coupled and parallel-coupled add/drop filters and FSR extension,” in Photonic Microring Research and Application (Springer, 2010), Chap. 4.

Tong, W.

V. M. Menon, W. Tong, and S. R. Forrest, “Control of quality factor and critical coupling in microring resonators through integration of a semiconductor optical amplifier,” IEEE Photon. Technol. Lett 16, 1343–1345 (2004).
[CrossRef]

Tur, M.

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

Vlasov, Y.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007).
[CrossRef]

Xia, F.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007).
[CrossRef]

Yariv, A.

A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002).
[CrossRef]

Yasuo, K.

K. Yasuo and K. Tomoyuki, “Series-coupled and parallel-coupled add/drop filters and FSR extension,” in Photonic Microring Research and Application (Springer, 2010), Chap. 4.

Young, I.

M. J. Kobrinsky, B. A. Block, J.-F. Zheng, B. C. Barnett, E. Mohammed, M. Reshotko, F. Robertson, S. List, I. Young, and K. Cadien, “On-chip optical interconnects,” Intel Technol. J. 8, 129–143 (2004).

Youngquist, R. C.

R. C. Youngquist, L. F. Stokes, and H. J. Shaw, “Effects of normal mode loss in dielectric waveguide directional couplers and interferometers,” IEEE J. Quantum Electron. 19, 1888–1896 (1983).
[CrossRef]

Yu, J.

Q. Huang, Y. Yu, and J. Yu, “Experimental investigation on submicron rib waveguide microring/racetrack resonators in silicon-on-insulator,” Opt. Commun. 282, 22–26 (2009).
[CrossRef]

Yu, Y.

Q. Huang, Y. Yu, and J. Yu, “Experimental investigation on submicron rib waveguide microring/racetrack resonators in silicon-on-insulator,” Opt. Commun. 282, 22–26 (2009).
[CrossRef]

Zhao, J. H.

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis, A Signal Processing Approach (Wiley, 1999).S

Zheng, J.-F.

M. J. Kobrinsky, B. A. Block, J.-F. Zheng, B. C. Barnett, E. Mohammed, M. Reshotko, F. Robertson, S. List, I. Young, and K. Cadien, “On-chip optical interconnects,” Intel Technol. J. 8, 129–143 (2004).

Bell System Tech. J.

D. Marcuse, “Bending losses of the asymmetric slab waveguide,” Bell System Tech. J. 50, 2551–2563 (1971).

IEEE J. Lightwave Technol.

S. Suzuki, K. Oda, and Y. Hibino, “Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz,” IEEE J. Lightwave Technol. 13, 1766–1771 (1995).
[CrossRef]

G. Barbarossa, A. M. Matteo, and M. N. Armenise, “Theoretical analysis of triple-coupler ring-based optical guided-wave resonator,” IEEE J. Lightwave Technol. 13, 148–157 (1995).
[CrossRef]

IEEE J. Quantum Electron.

R. C. Youngquist, L. F. Stokes, and H. J. Shaw, “Effects of normal mode loss in dielectric waveguide directional couplers and interferometers,” IEEE J. Quantum Electron. 19, 1888–1896 (1983).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
[CrossRef]

IEEE Microw. Mag.

B. Jalali, M. Paniccia, and G. Reed, “Silicon photonics,” IEEE Microw. Mag. 7, 58–68 (2006).
[CrossRef]

IEEE Photon. Technol. Lett

V. M. Menon, W. Tong, and S. R. Forrest, “Control of quality factor and critical coupling in microring resonators through integration of a semiconductor optical amplifier,” IEEE Photon. Technol. Lett 16, 1343–1345 (2004).
[CrossRef]

IEEE Photon. Technol. Lett.

A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002).
[CrossRef]

IEEE Trans. Audio Electroacoust.

A. Deczky, “Synthesis of recursive digital filters using the minimum p-error criterion,” IEEE Trans. Audio Electroacoust. 20, 257–263 (1972).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

D. Kajfez, S. Chebolu, M. R. Abdul-Gaffoor, and A. A. Kishk, “Uncertainty analysis of the transmission-type measurement of Q-factor,” IEEE Trans. Microwave Theory Tech. 47, 367–371 (1999).
[CrossRef]

Intel Technol. J.

M. J. Kobrinsky, B. A. Block, J.-F. Zheng, B. C. Barnett, E. Mohammed, M. Reshotko, F. Robertson, S. List, I. Young, and K. Cadien, “On-chip optical interconnects,” Intel Technol. J. 8, 129–143 (2004).

J. Lightwave Technol.

Nat. Photonics

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007).
[CrossRef]

Opt. Commun.

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]

Q. Huang, Y. Yu, and J. Yu, “Experimental investigation on submicron rib waveguide microring/racetrack resonators in silicon-on-insulator,” Opt. Commun. 282, 22–26 (2009).
[CrossRef]

O. Schwelb, “The nature of spurious mode suppression in extended FSR microring multiplexers,” Opt. Commun. 271, 424–429 (2007).
[CrossRef]

Opt. Eng.

S. Dey and S. Mandal, “Wide free-spectral-range triple ring resonator as optical filter,” Opt. Eng. 50, 084601 (2011).
[CrossRef]

Proc. IEEE

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

Proc. IRE

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

Other

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis, A Signal Processing Approach (Wiley, 1999).S

K. Yasuo and K. Tomoyuki, “Series-coupled and parallel-coupled add/drop filters and FSR extension,” in Photonic Microring Research and Application (Springer, 2010), Chap. 4.

L. Y. M. Tobing and P. Dumon, “Fundamental principles of operation andnotes on fabrication of photonic microresonators,” in Photonic Microring Research and Application, Vol. 156 of Springer Series in Optical Sciences (Springer, 2010), Chap. 1.

A. Oppenheim and R. Schafer, Digital Signal Processing, 2nd ed. (Prentice-Hall, 1975).

O. Schwelb, “Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters—a tutorial overview,” IEEE J. Lightwave Technol.22, 1380–1394 (2004).

S. Dey, and S. Mandal, “Enhancement of free spectral range in optical triple ring resonator: a Vernier principle approach,” in IEEE Proceedings of International Conference on Recent Advances in Information Technology (RAIT-2012) (IEEE, 2012), Vol. 1, pp. 246–250.

F. Morichetti, “Polarization rotation in ring resonators,” in Photonic Microresonator Research and Application, Vol.  156 of Springer Series in Optical Sciences (Springer, 2010), Chap 3.

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Figures (6)

Fig. 1.
Fig. 1.

(a) Schematic of a directional optical coupler. E 1 i , E 2 i represent coupler inputs; E 1 0 , E 2 0 represent coupler outputs. (b) Block diagram model of directional optical coupler.

Fig. 2.
Fig. 2.

Two-pair system. (a) Feed-forward configuration; (b) feedback configuration.

Fig. 3.
Fig. 3.

(a) Single ring resonator with one directional optical coupler and single bus line with coupling coefficient κ , (b)  Z -transforms schematics of the single ring resonator.

Fig. 4.
Fig. 4.

(a) Schematic diagram of PORR; (b)  Z -transform equivalent of the PORR.

Fig. 5.
Fig. 5.

(a) Frequency response characteristics of QORR made up of SOI with effective refractive index ( n ) = 1.7 . (b) Zoom-in view of the first resonant peak at 3527 THz. (c) Group delay and (d) dispersion characteristics.

Fig. 6.
Fig. 6.

(a) Frequency response characteristics of QORR made up of SOI with effective refractive index ( n ) = 2.811 . (b) Zoom-in view of the first resonant peak at 2133 THz. (c) Group delay and (d) dispersion characteristics.

Equations (120)

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F ( ν ) = N = f ( N T ) exp ( j 2 π ν N T ) ,
F ( ν ) = N = f ( N ) exp ( j 2 π ν N ) .
H ( z ) = N = h ( N ) z N .
T = n L U / C ,
FSR = 1 / T = C / n L U = C / 2 π n R ,
FSR = N FSR 1 = M FSR 2 = O FSR 3 = P FSR 4 = Q FSR 5 ,
m = [ FSR desired / FSR unextended ] int ,
R i + 1 = ( m + 1 ) R i / m ,
[ E 1 0 E 2 0 ] = [ C j S j S C ] [ E 1 i E 2 i ] .
C = ( 1 κ ) 1 / 2 ,
j S = j κ 1 / 2 .
S 2 + C 2 = 1 .
[ E 1 0 E 2 0 ] = q [ C j S j S C ] [ E 1 i E 2 i ] .
T f = ( T n Δ n ) / Δ ,
Δ = 1 L 1 + L 2 L 3 + ,
T 1 = ( j S 1 ) Z N ( j S 2 ) Z M ( j S 3 ) Z O ( j S 4 ) Z P ( j S 5 ) Z Q ( j s 6 ) = { Z ( N + M + O + P + Q ) } 1 / 2 ( κ 1 κ 2 κ 3 κ 4 κ 5 κ 6 ) 1 / 2 ,
where s i = ( κ i ) 1 / 2 , i = 1 , 2 , , 6 .
Δ = 1 ( L 1 + L 2 + L 3 + L 4 + L 5 + L 6 + L 7 + L 8 + L 9 + L 10 + L 11 + L 12 + L 13 + L 14 + L 15 ) + ( L 21 + L 31 + L 41 + L 51 + L 71 + L 81 + L 91 + L 11 1 + L 121 + L 141 + L 23 + L 24 + L 25 + L 28 + L 29 + L 2 12 + L 34 + L 35 + L 36 + L 39 + L 45 + L 46 + L 47 + L 4 10 + L 56 + L 57 + L 58 + L 5 10 + L 5 11 + L 5 13 + L 68 + L 69 + L 6 12 + L 79 + L 9 10 ) ( L 123 + L 124 + L 125 + L 128 + L 129 + L 1 2 12 + L 134 + L 135 + L 139 + L 145 + L 147 + L 157 + L 158 + L 15 11 + L 179 + L 234 + L 235 + L 239 + L 245 + L 258 + L 345 + L 346 + L 356 + L 369 + L 456 + L 457 + L 45 10 + L 568 ) + ( L 1234 + L 1235 + L 1239 + L 1245 + L 1258 + L 1345 + L 1457 + L 2345 + L 3456 ) L 12345 .
Δ = 1 C 1 C 2 Z N C 2 C 3 Z M C 3 C 4 Z O C 4 C 5 Z P C 5 C 6 Z Q + C 1 C 3 Z ( N + M ) + C 2 C 4 Z ( M + O ) + C 3 C 5 Z ( O + P ) + C 4 C 6 Z ( P + Q ) + C 1 C 2 C 3 C 4 Z ( N + O ) + C 1 C 2 C 4 C 5 Z ( N + P ) + C 1 C 2 C 5 C 6 Z ( N + Q ) + C 2 C 3 C 4 C 5 Z ( M + P ) + C 2 C 3 C 5 C 6 Z ( M + Q ) + C 3 C 4 C 5 C 6 Z ( O + Q ) C 1 C 4 Z ( N + M + O ) C 2 C 5 Z ( M + O + P ) C 3 C 6 Z ( O + P + Q ) C 1 C 2 C 3 C 5 Z ( N + O + P ) C 1 C 2 C 4 C 6 Z ( N + P + Q ) C 2 C 3 C 4 C 6 Z ( M + P + Q ) C 1 C 3 C 4 C 5 Z ( N + M + P ) C 1 C 3 C 5 C 6 Z ( N + M + Q ) C 2 C 4 C 5 C 6 Z ( M + O + Q ) C 1 C 2 C 3 C 4 C 5 C 6 Z ( N + O + Q ) + C 1 C 5 Z ( N + M + O + P ) + C 2 C 6 Z ( M + O + P + Q ) + C 1 C 4 C 5 C 6 Z ( N + M + O + Q ) + C 1 C 2 C 3 C 6 Z ( N + O + P + Q ) + C 1 C 3 C 4 C 6 Z ( N + M + P + Q ) C 1 C 6 Z ( N + M + O + P + Q ) ,
T f = T 1 Δ 1 / Δ = T 1 / Δ ,
T f = Z ( N + M + O + P + Q ) κ 1 κ 2 κ 3 κ 4 κ 5 κ 6 1 C 1 C 2 Z N C 2 C 3 Z M C 3 C 4 Z O C 4 C 5 Z P C 5 C 6 Z Q + C 1 C 3 Z ( N + M ) + C 2 C 4 Z ( M + O ) + C 3 C 5 Z ( O + P ) + C 4 C 6 Z ( P + Q ) + C 1 C 2 C 3 C 4 Z ( N + O ) + C 1 C 2 C 4 C 5 Z ( N + P ) + C 1 C 2 C 5 C 6 Z ( N + Q ) + C 2 C 3 C 4 C 5 Z ( M + P ) + C 2 C 3 C 5 C 6 Z ( M + Q ) + C 3 C 4 C 5 C 6 Z ( O + Q ) C 1 C 4 Z ( N + M + O ) C 2 C 5 Z ( M + O + P ) C 3 C 6 Z ( O + P + Q ) C 1 C 2 C 3 C 5 Z ( N + O + P ) C 1 C 2 C 4 C 6 Z ( N + P + Q ) C 2 C 3 C 4 C 6 Z ( M + P + Q ) C 1 C 3 C 4 C 5 Z ( N + M + P ) C 1 C 3 C 5 C 6 Z ( N + M + Q ) C 2 C 4 C 5 C 6 Z ( M + O + Q ) C 1 C 2 C 3 C 4 C 5 C 6 Z ( N + O + Q ) + C 1 C 5 Z ( N + M + O + P ) + C 2 C 6 Z ( M + O + P + Q ) + C 1 C 4 C 5 C 6 Z ( N + M + O + Q ) + C 1 C 2 C 3 C 6 Z ( N + O + P + Q ) + C 1 C 3 C 4 C 6 Z ( N + M + P + Q ) C 1 C 6 Z ( N + M + O + P + Q ) .
T f = q Z ( N + M + O + P + Q ) γ 1 γ 2 γ 3 γ 4 γ 5 κ 1 κ 2 κ 3 κ 4 κ 5 κ 6 1 C 1 C 2 γ 1 Z N C 2 C 3 γ 2 Z M C 3 C 4 γ 3 Z O C 4 C 5 γ 4 Z P C 5 C 6 γ 5 Z Q + C 1 C 3 γ 1 γ 2 Z ( N + M ) + C 2 C 4 γ 2 γ 3 Z ( M + O ) + C 3 C 5 γ 3 γ 4 Z ( O + P ) + C 4 C 6 γ 4 γ 5 Z ( P + Q ) + C 1 C 2 C 3 C 4 γ 1 γ 3 Z ( N + O ) + C 1 C 2 C 4 C 5 γ 1 γ 4 Z ( N + P ) + C 1 C 2 C 5 C 6 γ 1 γ 5 Z ( N + Q ) + C 2 C 3 C 4 C 5 γ 2 γ 4 Z ( M + P ) + C 2 C 3 C 5 C 6 γ 2 γ 5 Z ( M + Q ) + C 3 C 4 C 5 C 6 γ 3 γ 5 Z ( O + Q ) C 1 C 4 γ 1 γ 2 γ 3 Z ( N + M + O ) C 2 C 5 γ 2 γ 3 γ 4 Z ( M + O + P ) C 3 C 6 γ 3 γ 4 γ 5 Z ( O + P + Q ) C 1 C 2 C 3 C 5 γ 1 γ 3 γ 4 Z ( N + O + P ) C 1 C 2 C 4 C 6 γ 1 γ 4 γ 5 Z ( N + P + Q ) C 1 C 2 C 3 C 6 γ 2 γ 4 γ 5 Z ( M + P + Q ) C 1 C 3 C 4 C 5 γ 1 γ 2 γ 4 Z ( N + M + P ) C 1 C 3 C 5 C 6 γ 1 γ 2 γ 5 Z ( N + M + Q ) C 2 C 4 C 5 C 6 γ 2 γ 3 γ 5 Z ( M + O + Q ) C 1 C 2 C 3 C 4 C 5 C 6 γ 1 γ 3 γ 5 Z ( N + O + Q ) + C 1 C 5 γ 1 γ 2 γ 3 γ 4 Z ( N + M + O + P ) + C 2 C 6 γ 2 γ 3 γ 4 γ 5 Z ( M + O + P + Q ) + C 1 C 4 C 5 C 6 γ 1 γ 2 γ 3 γ 5 Z ( N + M + O + Q ) + C 1 C 2 C 3 C 6 γ 1 γ 3 γ 4 γ 5 Z ( N + O + P + Q ) + C 1 C 3 C 4 C 6 γ 1 γ 2 γ 4 γ 5 Z ( N + M + P + Q ) C 1 C 6 γ 1 γ 2 γ 3 γ 4 γ 5 Z ( N + M + O + P + Q ) ,
T f = q Z 1681 γ 1 γ 2 γ 3 γ 4 γ 5 κ 1 κ 2 κ 3 κ 4 κ 5 κ 6 1 C 1 C 2 γ 1 Z 280 C 2 C 3 γ 2 Z 320 C 3 C 4 γ 3 Z 340 C 4 C 5 γ 4 Z 360 C 5 C 6 γ 5 Z 381 + C 1 C 3 γ 1 γ 2 Z 600 + C 2 C 4 γ 2 γ 3 Z 660 + C 3 C 5 γ 3 γ 4 Z 700 + C 4 C 6 γ 4 γ 5 Z 741 + C 1 C 2 C 3 C 4 γ 1 γ 3 Z 620 + C 1 C 2 C 4 C 5 γ 1 γ 4 Z 640 + C 1 C 2 C 5 C 6 γ 1 γ 5 Z 661 + C 2 C 3 C 4 C 5 γ 2 γ 4 Z 680 + C 2 C 3 C 5 C 6 γ 2 γ 5 Z 701 + C 3 C 4 C 5 C 6 γ 3 γ 5 Z 721 C 1 C 4 γ 1 γ 2 γ 3 Z 940 C 2 C 5 γ 2 γ 3 γ 4 Z 1020 C 3 C 6 γ 3 γ 4 γ 5 Z 1081 C 1 C 2 C 3 C 5 γ 1 γ 3 γ 4 Z 980 C 1 C 2 C 4 C 6 γ 1 γ 4 γ 5 Z 1021 C 1 C 2 C 3 C 6 γ 2 γ 4 γ 5 Z 1061 C 1 C 3 C 4 C 5 γ 1 γ 2 γ 4 Z 960 C 1 C 3 C 5 C 6 γ 1 γ 2 γ 5 Z 981 C 2 C 4 C 5 C 6 γ 2 γ 3 γ 5 Z 1041 C 1 C 2 C 3 C 4 C 5 C 6 γ 1 γ 3 γ 5 Z 1001 + C 1 C 5 γ 1 γ 2 γ 3 γ 4 Z 1300 + C 2 C 6 γ 2 γ 3 γ 4 γ 5 Z 1401 + C 1 C 4 C 5 C 6 γ 1 γ 2 γ 3 γ 5 Z 1321 + C 1 C 2 C 3 C 6 γ 1 γ 3 γ 4 γ 5 Z 1361 + C 1 C 3 C 4 C 6 γ 1 γ 2 γ 4 γ 5 Z 1341 C 1 C 6 γ 1 γ 2 γ 3 γ 4 γ 5 Z 1681 .
τ g = d d ω tan 1 [ Im { H ( z ) } / Re { H ( z ) } ] z = exp ( j ω ) .
τ g ( ω ) = i = 1 n r p i { cos ( ω ϕ p i ) r p i } { 1 2 r p i cos ( ω ϕ p i ) + r p i 2 } + i = 1 m r z i { r z i cos ( ω ϕ z i ) } { 1 2 r z i cos ( ω ϕ z i ) + r z i 2 } ,
D = d ( τ g ) / d ω .
β = β / ω = 1 / v g = τ g / L ,
τ g = L β / ω
τ g = T Φ / ω .
F = FSR / FWHM .
FSR 1 / FWHM 1 = FSR 2 / FWHM 2 = FSR 3 / FWHM 3 = FSR 4 / FWHM 4 = FSR 5 / FWHM 5 = Finesse ,
Q = f o / FWHM .
L 1 = C 1 C 2 Z N ,
L 2 = C 2 C 3 Z M ,
L 3 = C 3 C 4 Z O ,
L 4 = C 4 C 5 Z P ,
L 5 = C 5 C 6 Z Q ,
L 6 = C 1 C 3 S 2 2 Z ( N + M ) ,
L 7 = C 2 C 4 S 3 2 Z ( M + O ) ,
L 8 = C 3 C 5 S 4 2 Z ( O + P ) ,
L 9 = C 4 C 6 S 5 2 Z ( P + Q ) ,
L 10 = C 1 C 4 S 2 2 S 3 2 Z ( N + M + O ) ,
L 11 = C 2 C 5 S 3 2 S 4 2 Z ( M + O + P ) ,
L 12 = C 3 C 6 S 4 2 S 5 2 Z ( O + P + Q ) ,
L 13 = C 1 C 5 S 2 2 S 3 2 S 4 2 Z ( N + M + O + P ) ,
L 14 = C 2 C 6 S 3 2 S 4 2 S 5 2 Z ( M + O + P + Q ) ,
L 15 = C 1 C 6 S 2 2 S 3 2 S 4 2 S 5 2 Z ( N + M + O + P + Q ) .
L 21 = C 1 C 2 2 C 3 Z ( N + M ) ,
L 31 = C 1 C 2 C 3 C 4 Z ( N + O ) ,
L 41 = C 1 C 2 C 4 C 5 Z ( N + P ) ,
L 51 = C 1 C 2 C 5 C 6 Z ( N + Q ) ,
L 71 = C 1 C 2 2 C 4 S 3 2 Z ( N + M + O ) ,
L 81 = C 1 C 2 C 3 C 5 S 4 2 Z ( N + O + P ) ,
L 91 = C 1 C 2 C 4 C 6 S 3 2 Z ( N + P + Q ) ,
L 111 = C 1 C 2 2 C 5 S 3 2 S 4 2 Z ( N + M + O + P ) ,
L 121 = C 1 C 2 C 3 C 6 S 4 2 S 5 6 Z ( N + O + P + Q ) ,
L 141 = C 1 C 2 2 C 6 S 3 2 S 4 2 S 5 2 Z ( N + M + O + P + Q ) ,
L 23 = C 2 C 3 2 C 4 Z ( M + O ) ,
L 24 = C 2 C 3 C 4 C 5 Z ( M + P ) ,
L 25 = C 2 C 3 C 5 C 6 Z ( M + Q ) ,
L 28 = C 2 C 3 2 C 5 S 4 2 Z ( M + O + P ) ,
L 29 = C 2 C 3 C 4 C 6 S 5 2 Z ( M + P + Q ) ,
L 212 = C 2 C 3 2 C 6 S 4 2 S 5 2 Z ( M + O + P + Q ) ,
L 34 = C 3 C 4 2 C 5 Z ( O + P ) ,
L 35 = C 3 C 4 C 5 C 6 Z ( O + Q ) ,
L 36 = C 1 C 3 2 C 4 S 2 2 Z ( N + M + O ) ,
L 39 = C 3 C 4 2 C 6 S 5 2 Z ( O + P + Q ) ,
L 45 = C 4 C 5 2 C 6 Z ( P + Q ) ,
L 46 = C 1 C 3 C 4 C 5 S 2 2 Z ( N + M + P ) ,
L 47 = C 2 C 4 2 C 5 S 3 2 Z ( M + O + P ) ,
L 410 = C 1 C 4 2 C 5 S 2 2 S 3 2 Z ( N + M + O + P ) ,
L 56 = C 1 C 3 C 5 C 6 S 2 2 Z ( N + M + Q ) ,
L 57 = C 2 C 4 C 5 C 6 S 3 2 Z ( M + O + Q ) ,
L 58 = C 3 C 5 2 C 6 S 4 2 Z ( O + P + Q ) ,
L 510 = C 1 C 4 C 5 C 6 S 2 2 S 3 2 Z ( N + M + O + Q ) ,
L 511 = C 2 C 5 2 C 6 S 3 2 S 4 2 Z ( M + O + P + Q ) ,
L 513 = C 1 C 5 2 C 6 S 2 2 S 3 2 S 4 2 Z ( N + M + O + P + Q ) ,
L 68 = C 1 C 3 2 C 5 S 2 2 S 4 2 Z ( N + M + O + P ) ,
L 69 = C 1 C 3 C 4 C 6 S 2 2 S 5 2 Z ( N + M + P + Q ) ,
L 612 = C 1 C 3 2 C 6 S 2 2 S 4 2 S 5 2 Z ( N + M + O + P + Q ) ,
L 79 = C 2 C 4 2 C 6 S 3 2 S 5 2 Z ( M + O + P + Q ) ,
L 910 = C 1 C 4 2 C 6 S 2 2 S 3 2 S 5 2 Z ( N + M + O + P + Q ) .
L 123 = C 1 C 2 2 C 3 2 C 4 Z ( N + M + O ) ,
L 124 = C 1 C 2 2 C 3 C 4 C 5 Z ( N + M + P ) ,
L 125 = C 1 C 2 2 C 3 C 5 C 6 Z ( N + M + Q ) ,
L 128 = C 1 C 2 2 C 3 2 C 5 S 4 2 Z ( N + M + O + P ) ,
L 129 = C 1 C 2 2 C 3 C 4 C 6 S 4 2 Z ( N + M + P + Q ) ,
L 1212 = C 1 C 2 2 C 3 2 C 6 S 4 2 S 5 2 Z ( N + M + O + P + Q ) ,
L 134 = C 1 C 2 C 3 C 4 2 C 5 Z ( N + O + P ) ,
L 135 = C 1 C 2 C 3 C 4 C 5 C 6 Z ( N + O + Q ) ,
L 139 = C 1 C 2 C 3 C 4 2 C 6 S 5 2 Z ( N + O + P + Q ) ,
L 145 = C 1 C 2 C 4 C 5 2 C 6 Z ( N + P + Q ) ,
L 147 = C 1 C 2 2 C 4 2 C 5 S 3 2 Z ( N + M + O + P ) ,
L 157 = C 1 C 2 2 C 4 C 5 C 6 S 3 2 Z ( N + M + O + Q ) ,
L 158 = C 1 C 2 C 3 C 5 2 C 6 S 4 2 Z ( N + O + P + Q ) ,
L 1511 = C 1 C 2 2 C 5 2 C 6 S 3 2 S 4 2 Z ( N + M + O + P + Q ) ,
L 179 = C 1 C 2 2 C 4 2 C 6 S 3 2 S 5 2 Z ( N + M + O + P + Q ) ,
L 234 = C 2 C 3 2 C 4 2 C 5 Z ( M + O + P ) ,
L 235 = C 2 C 3 2 C 4 C 5 C 6 Z ( M + O + Q ) ,
L 239 = C 2 C 3 2 C 4 2 C 6 S 5 2 Z ( M + O + P + Q ) ,
L 245 = C 2 C 3 C 4 C 5 2 C 6 Z ( M + P + Q ) ,
L 258 = C 2 C 3 2 C 5 2 C 6 S 4 2 Z ( M + O + P + Q ) ,
L 345 = C 3 C 4 2 C 5 2 C 6 Z ( O + P + Q ) ,
L 346 = C 1 C 3 2 C 4 2 C 5 S 2 2 Z ( N + M + O + P ) ,
L 356 = C 1 C 3 2 C 4 C 5 C 6 S 2 2 Z ( N + M + O + Q ) ,
L 369 = C 1 C 3 2 C 4 2 C 6 S 2 2 S 5 2 Z ( N + M + O + P + Q ) ,
L 456 = C 1 C 3 C 4 C 5 2 C 6 S 2 2 Z ( N + M + P + Q ) ,
L 457 = C 2 C 4 2 C 5 2 C 6 S 3 2 Z ( M + O + P + Q ) ,
L 4510 = C 1 C 4 2 C 5 2 C 6 S 2 2 S 3 2 Z ( N + M + O + P + Q ) ,
L 568 = C 1 C 3 2 C 5 2 C 6 S 2 2 S 4 2 Z ( N + M + O + P + Q ) .
L 1234 = C 1 C 2 2 C 3 2 C 4 2 C 5 Z ( N + M + O + P ) ,
L 1235 = C 1 C 2 2 C 3 2 C 4 C 5 C 6 Z ( N + M + O + Q ) ,
L 1239 = C 1 C 2 2 C 3 2 C 4 2 C 6 S 5 2 Z ( N + M + O + P + Q ) ,
L 1245 = C 1 C 2 2 C 3 C 4 C 5 2 C 6 Z ( N + M + P + Q ) ,
L 1258 = C 1 C 2 2 C 3 2 C 5 2 C 6 S 4 2 Z ( N + M + O + P + Q ) ,
L 1345 = C 1 C 2 C 3 C 4 2 C 5 2 C 6 Z ( N + O + P + Q ) ,
L 1457 = C 1 C 2 2 C 4 2 C 5 2 C 6 S 3 2 Z ( N + M + O + P + Q ) ,
L 2345 = C 2 C 3 2 C 4 2 C 5 2 C 6 Z ( M + O + P + Q ) ,
L 3456 = C 1 C 3 2 C 4 2 C 5 2 C 6 S 2 2 Z ( N + M + O + P + Q ) .
L 12345 = C 1 C 2 2 C 3 2 C 4 2 C 5 2 C 6 Z ( N + M + O + P + Q ) .

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