Abstract

In this paper, we study analytically and numerically the light propagation in microring resonator chains that exhibit distributed loss or gain. We derive the stability conditions for the latter and demonstrate the feasibility of the group index control within a certain amplification range. Possible applications of the discussed effects are proposed.

©2007 Optical Society of America

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References

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  1. J. Capmany and M.A. Muriel, “A new transfer matrix for the analysis of fiber ring resonators: Compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 90,1904–1919 (1990).
    [Crossref]
  2. S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11,691–693 (1999).
    [Crossref]
  3. C.K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10,994–996 (1998).
    [Crossref]
  4. P.P. Absil, J.V. Hryniewicz, B.E. Little, P.S. Cho, R.A. Wilson, L.G. Joneckis, and P.T. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25,554–556 (2000).
    [Crossref]
  5. 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]
  6. D. Biallo, A. D’ Orazio, M. De Sario, V. Petruzzelli, and F. Prudenzano, “Time domain analysis of optical amplification in Er3+ doped SiO2-TiO2 planar waveguide,” Opt. Express 13,4683–4692 (2005).
    [Crossref] [PubMed]
  7. A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35,365–379 (2003).
    [Crossref]
  8. Y. Chen and S. Blair, “Nonlinearity enhancement in finite coupled-resonator slow-light waveguides,” Opt. Express 12,3353–3366 (2004).
    [Crossref] [PubMed]
  9. A. Agarwal, P. Toliver, R. Menendez, S. Etemad, J. Jackel, J. Young, T. Banwell, B.E. Little, S.T. Chu, W. Chen, W. Chen, J. Hryniewicz, F. Johnson, D. Gill, O. King, R. Davidson, K. Donovan, and P. J. Delfyett, “Fully programmable ring-resonator-based integrated photonic circuit for phase coherent applications,” J. Lightwave Technol. 24,77–87 (2006).
    [Crossref]
  10. J. Capmany and J. Cascon, “Discrete time fiber-optic signal processors using optical amplifiers,” J. Lightwave Technol. 12,106–117 (1994).
    [Crossref]
  11. J. Capmany, B. Ortega, D. Pastor, and S. Sales, “Discrete-time optical processing of microwave signals,” J. Lightwave Technol. 23,702–723 (2005).
    [Crossref]
  12. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12,90–103 (2004).
    [Crossref] [PubMed]
  13. Y.M. Landobasa and M.K. Chin, “Defect modes in micro-ring resonator arrays,” Opt. Express 13,7800–7815 (2005).
    [Crossref] [PubMed]
  14. A.A. Tovar and L.W. Casperson, “Generalized Sylvester theorems for periodic applications in matrix optics,” J. Opt. Soc. Am. A 12,578–590 (1995).
    [Crossref]
  15. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53,4107–4121 (1996).
    [Crossref]
  16. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3,233–245 (1970).
    [Crossref]
  17. S. Mookherjea, “Using gain to tune the dispersion relation of coupled-resonator optical waveguides,” IEEE Photon. Technol. Lett. 18,715–717 (2006).
    [Crossref]
  18. C.G.B. Garret and D.E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev A 1,305–313 (1970).
    [Crossref]
  19. G. Lenz, B.J. Eggleton, C.R. Giles, C.K Madsen, and R.E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34,1390–1402 (1998).
    [Crossref]
  20. A. V. Oppenheim, Willsky A.S., and S. H. Nawab, Signals and Systems (Prentice Hall, 1997).
  21. Y. Chen and S. Blair, “Nonlinear phase-shift of cascaded microring resonators,” J. Opt. Soc. Am. B 20,2125–2132 (2003).
    [Crossref]
  22. J.B. Khurgin, “Expanding the bandwidth of slow-light photonic devices based on coupled resonators,” Opt. Lett. 30,513–515 (2005).
    [Crossref] [PubMed]

2006 (2)

2005 (4)

2004 (3)

2003 (2)

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35,365–379 (2003).
[Crossref]

Y. Chen and S. Blair, “Nonlinear phase-shift of cascaded microring resonators,” J. Opt. Soc. Am. B 20,2125–2132 (2003).
[Crossref]

2000 (1)

1999 (1)

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11,691–693 (1999).
[Crossref]

1998 (2)

C.K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10,994–996 (1998).
[Crossref]

G. Lenz, B.J. Eggleton, C.R. Giles, C.K Madsen, and R.E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34,1390–1402 (1998).
[Crossref]

1996 (1)

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53,4107–4121 (1996).
[Crossref]

1995 (1)

1994 (1)

J. Capmany and J. Cascon, “Discrete time fiber-optic signal processors using optical amplifiers,” J. Lightwave Technol. 12,106–117 (1994).
[Crossref]

1990 (1)

J. Capmany and M.A. Muriel, “A new transfer matrix for the analysis of fiber ring resonators: Compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 90,1904–1919 (1990).
[Crossref]

1970 (2)

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3,233–245 (1970).
[Crossref]

C.G.B. Garret and D.E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev A 1,305–313 (1970).
[Crossref]

A.S., Willsky

A. V. Oppenheim, Willsky A.S., and S. H. Nawab, Signals and Systems (Prentice Hall, 1997).

Absil, P.P.

Agarwal, A.

Banwell, T.

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53,4107–4121 (1996).
[Crossref]

Biallo, D.

Blair, S.

Capmany, J.

J. Capmany, B. Ortega, D. Pastor, and S. Sales, “Discrete-time optical processing of microwave signals,” J. Lightwave Technol. 23,702–723 (2005).
[Crossref]

J. Capmany and J. Cascon, “Discrete time fiber-optic signal processors using optical amplifiers,” J. Lightwave Technol. 12,106–117 (1994).
[Crossref]

J. Capmany and M.A. Muriel, “A new transfer matrix for the analysis of fiber ring resonators: Compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 90,1904–1919 (1990).
[Crossref]

Cascon, J.

J. Capmany and J. Cascon, “Discrete time fiber-optic signal processors using optical amplifiers,” J. Lightwave Technol. 12,106–117 (1994).
[Crossref]

Casperson, L.W.

Chen, W.

Chen, Y.

Chin, M.K.

Cho, P.S.

Chu, S. T.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11,691–693 (1999).
[Crossref]

Chu, S.T.

Davidson, R.

De Sario, M.

Delfyett, P. J.

Donovan, K.

Dowling, J. P.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53,4107–4121 (1996).
[Crossref]

Eggleton, B.J.

G. Lenz, B.J. Eggleton, C.R. Giles, C.K Madsen, and R.E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34,1390–1402 (1998).
[Crossref]

Etemad, S.

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]

Garret, C.G.B.

C.G.B. Garret and D.E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev A 1,305–313 (1970).
[Crossref]

Giles, C.R.

G. Lenz, B.J. Eggleton, C.R. Giles, C.K Madsen, and R.E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34,1390–1402 (1998).
[Crossref]

Gill, D.

Ho, P.T.

Hryniewicz, J.

Hryniewicz, J.V.

Huang, Y.

Jackel, J.

Johnson, F.

Joneckis, L.G.

Kaneko, T.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11,691–693 (1999).
[Crossref]

Khurgin, J.B.

King, O.

Kokubun, Y.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11,691–693 (1999).
[Crossref]

Landobasa, Y.M.

Lenz, G.

C.K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10,994–996 (1998).
[Crossref]

G. Lenz, B.J. Eggleton, C.R. Giles, C.K Madsen, and R.E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34,1390–1402 (1998).
[Crossref]

Little, B. E.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11,691–693 (1999).
[Crossref]

Little, B.E.

Loudon, R.

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3,233–245 (1970).
[Crossref]

Madsen, C.K

G. Lenz, B.J. Eggleton, C.R. Giles, C.K Madsen, and R.E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34,1390–1402 (1998).
[Crossref]

Madsen, C.K.

C.K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10,994–996 (1998).
[Crossref]

Martinelli, M.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35,365–379 (2003).
[Crossref]

McCumber, D.E.

C.G.B. Garret and D.E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev A 1,305–313 (1970).
[Crossref]

Melloni, A.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35,365–379 (2003).
[Crossref]

Menendez, R.

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]

Mookherjea, S.

S. Mookherjea, “Using gain to tune the dispersion relation of coupled-resonator optical waveguides,” IEEE Photon. Technol. Lett. 18,715–717 (2006).
[Crossref]

J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12,90–103 (2004).
[Crossref] [PubMed]

Morichetti, F.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35,365–379 (2003).
[Crossref]

Muriel, M.A.

J. Capmany and M.A. Muriel, “A new transfer matrix for the analysis of fiber ring resonators: Compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 90,1904–1919 (1990).
[Crossref]

Nawab, S. H.

A. V. Oppenheim, Willsky A.S., and S. H. Nawab, Signals and Systems (Prentice Hall, 1997).

Oppenheim, A. V.

A. V. Oppenheim, Willsky A.S., and S. H. Nawab, Signals and Systems (Prentice Hall, 1997).

Orazio, A. D’

Ortega, B.

Paloczi, G. T.

Pan, W.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11,691–693 (1999).
[Crossref]

Pastor, D.

Petruzzelli, V.

Poon, J. K. S.

Prudenzano, F.

Sales, S.

Sato, S.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11,691–693 (1999).
[Crossref]

Scalora, M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53,4107–4121 (1996).
[Crossref]

Scheuer, J.

Slusher, R.E.

G. Lenz, B.J. Eggleton, C.R. Giles, C.K Madsen, and R.E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34,1390–1402 (1998).
[Crossref]

Toliver, P.

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]

Tovar, A.A.

Wilson, R.A.

Yariv, A.

Young, J.

IEEE J. Quantum Electron. (1)

G. Lenz, B.J. Eggleton, C.R. Giles, C.K Madsen, and R.E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34,1390–1402 (1998).
[Crossref]

IEEE Photon. Technol. Lett. (4)

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett. 11,691–693 (1999).
[Crossref]

C.K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10,994–996 (1998).
[Crossref]

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]

S. Mookherjea, “Using gain to tune the dispersion relation of coupled-resonator optical waveguides,” IEEE Photon. Technol. Lett. 18,715–717 (2006).
[Crossref]

J. Lightwave Technol. (4)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Phys. A (1)

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3,233–245 (1970).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35,365–379 (2003).
[Crossref]

Phys. Rev A (1)

C.G.B. Garret and D.E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev A 1,305–313 (1970).
[Crossref]

Phys. Rev. E (1)

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53,4107–4121 (1996).
[Crossref]

Other (1)

A. V. Oppenheim, Willsky A.S., and S. H. Nawab, Signals and Systems (Prentice Hall, 1997).

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Figures (9)

Fig. 1.
Fig. 1. Scheme of a simple microring chain resonator. A chain with N rings has N + 1 periodic cells.
Fig. 2.
Fig. 2. Transmisivity T 3 for a lossless pasive (αd = 0), lossy (αd = 0.015) and amplifying (αd < 0) 3-ring resonator. In this example, (k - k 0)d/π = ∓0.165 is equivalent to λ = 1.5513 μm and 1.5498 μm, respectively. Note the graphic is semilogarithmic to allow for a proper representation of the large dynamic margin. Intentionally, no gain saturation has been considered. The inset shows a closer, non-truncated view of the left side peak.
Fig. 3.
Fig. 3. The phase Ψ3(k) corresponding to the case of Fig. 2.
Fig. 4.
Fig. 4. Group index of the 3-ring structure for several values of the normalized gain -αd within the stable and unstable zones. The spectral range depicted in the figure includes the central and left side peak (the right side peak behaving similarly to the latter). As -αd increases, n g,3 grows and eventually reaches +∞ at the two side peaks when -αd = -(αd)(-) c , immediately switching to -∞ for -αd = -(αd)(+) c (not shown here).
Fig. 5.
Fig. 5. Stability of the 3-ring resonator. The position of the 6 poles of (10), with respect to the unit circle, for a lossy and an amplifying resonator with several gain values. For the latter, only the - αd ≃ 0.005 case (beyond the first threshold) yields a stable structure.
Fig. 6.
Fig. 6. Group index for an active 5-ring structure, all other parameters being the same as in the N = 3 case. Several values of the normalized gain αd < 0 are considered, the first two corresponding to the stable operation zone.
Fig. 7.
Fig. 7. Output field envelope for the input signal as given in Eq. (20) tuned to one of the side resonances of the three peaks miniband for various values of αd < 0 (dashed-dotted) and αd > 0 both below and above the (αd) c threshold (solid and dashed, respectively).
Fig. 8.
Fig. 8. Group velocity (in logarithmic scale) around the left side peak of the 3-ring structure for several values of the normalized gain parameter αd < 0, and αd = 0 for comparison. To illustrate the order of the spectral widths, the abscissa shows the optical frequency, with νref =1.934×1014 Hz.
Fig. 9.
Fig. 9. Normalized output optical field envelope when the input signal (20) is tuned to one of the side peaks for values of ad in the stable region: -αd < -(αd) c .

Equations (20)

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A n 1 + A n 1 = M A n + A n i t ( e ik̃d r r e ik̃d ) A n + A n .
= k ,
M N + 1 = sin ( [ N + 1 ] βd ) sin ( βd ) M sin [ Nβd ] sin ( βd ) I ,
cos ( βd ) = 1 t sin ( k̅d ) .
t N A N + 1 + A 0 + = 1 ( M N + 1 ) 11 .
t N ( k ) = T N ( k ) e i Φ N ( k ) ,
v g ( N ) = ( N + 1 ) L ( N ( k ) ) 1 = c ( N + 1 ) L n g ( N ( k ) d k ) 1 .
t 3 ( k ) = t 4 2 r 2 + r 4 + exp [ 4 d ( ik + α ) ] r 2 exp [ 2 d ( ik + α ) ] 3 r 2 exp [ 2 d ( ik + α ) ] .
z = e σd e ikd ρ e ikd ( k real )
Q 3 ( z ) = Bz 2 1 + b 2 z 2 + b 4 z 4 + b 6 z 6
b 2 = ( 2 + r 2 ) exp ( 2 αd ) , b 4 = 3 exp ( 4 αd ) , b 6 = r 2 exp ( 6 αd ) .
R 3 ( z ) α = 0 = C j = 1 6 [ z r j ( 0 ) ] .
b j ( α ) = a j exp ( jαd ) ,
R 3 ( z ) = 1 + a 2 e 2 αd z 2 + a 4 e 4 αd z 4 + a 6 e 6 αd z 6
1 + a 2 u 2 + a 4 u 4 + a 6 u 6 .
u e αd z .
R 3 ( z ) = C j = 1 6 [ u r j ( 0 ) ] = C e αd j = 1 6 [ z r j ( 0 ) e αd ] .
r j ( α ) = r j ( 0 ) e αd .
( αd ) c = log ( max j r j ( 0 ) ) ,
E i ( t ) = exp ( t 2 2 T 0 2 ) exp ( i ω 0 t )

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