Abstract

We present an analytical study on the effects of coupler-induced localized backscattering (CILB) on the nature of superluminal and subliminal propagations in a traveling wave microresonator (TWMR). It is found that in the weak CILB regime, the flexibility to control the light propagation velocity is improved, and fast and slow light with high output power can be generated for a TWMR with a small net optical gain.

© 2010 Optical Society of America

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2010

2009

2008

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[CrossRef]

S. Fürst, A. Perez-Serrano, A. Scire, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: Experiment and theory,” Appl. Phys. Lett. 93, 251109 (2008).
[CrossRef]

2007

J. Yao, D. Leuenberger, M. C. M. Lee, and M. C. Wu, “Silicon microtoroidal resonators with integrated MEMS tunable coupler,” IEEE J. Sel. Top. Quantum Electron. 13, 202–208 (2007).
[CrossRef]

J. K. S. Poon, L. Zhu, J. M. Choi, G. A. DeRose, A. Scherer, and A. Yariv, “Active coupled-resonator optical waveguides. II Current injection InP-InGaAsP Fabry–Perot resonator arrays,” J. Opt. Soc. Am. B 24, 2389–2393 (2007).
[CrossRef]

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nature 3, 406–408 (2007), and references therein.

2006

J. Čtyroký, I. Richter, and M. Šiňor, “Dual resonance in a waveguide-coupled ring microresonator,” Opt. Quantum Electron. 38, 781 (2006).
[CrossRef]

2005

2002

J. E. Heebner and R. W. Boyd, “Slow and fast light in resonator-coupled waveguides,” J. Mod. Opt. 49, 2629–2636 (2002).
[CrossRef]

2001

2000

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

1999

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis (Wiley, 1999).
[CrossRef]

1997

1995

1992

M. L. Gorodetsky and V. S. Ilchenko, “Thermal nonlinear effects in optical whispering-gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[CrossRef]

Balle, S.

S. Fürst, A. Perez-Serrano, A. Scire, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: Experiment and theory,” Appl. Phys. Lett. 93, 251109 (2008).
[CrossRef]

Benech, P.

Blaize, S.

Borselli, M.

Boyd, R.

R. Boyd and D. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef] [PubMed]

Boyd, R. W.

J. E. Heebner and R. W. Boyd, “Slow and fast light in resonator-coupled waveguides,” J. Mod. Opt. 49, 2629–2636 (2002).
[CrossRef]

Bruyant, A.

Choi, J. M.

Chu, S. T.

Ctyroký, J.

J. Čtyroký, I. Richter, and M. Šiňor, “Dual resonance in a waveguide-coupled ring microresonator,” Opt. Quantum Electron. 38, 781 (2006).
[CrossRef]

DeRose, G. A.

Dogariu, A.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

Dong, P.

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nature 3, 406–408 (2007), and references therein.

Fedeli, J. -M.

Fürst, S.

S. Fürst, A. Perez-Serrano, A. Scire, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: Experiment and theory,” Appl. Phys. Lett. 93, 251109 (2008).
[CrossRef]

Gauthier, D.

R. Boyd and D. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef] [PubMed]

Gesuele, F.

Gorodetsky, M. L.

M. L. Gorodetsky and V. S. Ilchenko, “Thermal nonlinear effects in optical whispering-gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

Hare, J.

Haroche, S.

Heebner, J. E.

J. E. Heebner and R. W. Boyd, “Slow and fast light in resonator-coupled waveguides,” J. Mod. Opt. 49, 2629–2636 (2002).
[CrossRef]

Ilchenko, V. S.

M. L. Gorodetsky and V. S. Ilchenko, “Thermal nonlinear effects in optical whispering-gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

Johnson, T.

Kuzmich, A.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

Laine, J. P.

Lee, M. C. M.

J. Yao, D. Leuenberger, M. C. M. Lee, and M. C. Wu, “Silicon microtoroidal resonators with integrated MEMS tunable coupler,” IEEE J. Sel. Top. Quantum Electron. 13, 202–208 (2007).
[CrossRef]

Lee, R. K.

Lefèvre-Seguin, V.

Lérondel, G.

Leuenberger, D.

J. Yao, D. Leuenberger, M. C. M. Lee, and M. C. Wu, “Silicon microtoroidal resonators with integrated MEMS tunable coupler,” IEEE J. Sel. Top. Quantum Electron. 13, 202–208 (2007).
[CrossRef]

Li, Q.

Lipson, M.

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nature 3, 406–408 (2007), and references therein.

Little, B. E.

Madsen, C. K.

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis (Wiley, 1999).
[CrossRef]

Martin, B.

Morand, A.

Painter, O.

Perez-Serrano, A.

S. Fürst, A. Perez-Serrano, A. Scire, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: Experiment and theory,” Appl. Phys. Lett. 93, 251109 (2008).
[CrossRef]

Poon, J. K. S.

Qiu, M.

Raimond, J. -M.

Richter, I.

J. Čtyroký, I. Richter, and M. Šiňor, “Dual resonance in a waveguide-coupled ring microresonator,” Opt. Quantum Electron. 38, 781 (2006).
[CrossRef]

Royer, P.

Sandoghdar, V.

Scherer, A.

Scire, A.

S. Fürst, A. Perez-Serrano, A. Scire, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: Experiment and theory,” Appl. Phys. Lett. 93, 251109 (2008).
[CrossRef]

Šinor, M.

J. Čtyroký, I. Richter, and M. Šiňor, “Dual resonance in a waveguide-coupled ring microresonator,” Opt. Quantum Electron. 38, 781 (2006).
[CrossRef]

Sorel, M.

S. Fürst, A. Perez-Serrano, A. Scire, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: Experiment and theory,” Appl. Phys. Lett. 93, 251109 (2008).
[CrossRef]

Stefanon, I.

Su, Y.

Wang, J.

Wang, L. J.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

Weiss, D. S.

Wu, M. C.

J. Yao, D. Leuenberger, M. C. M. Lee, and M. C. Wu, “Silicon microtoroidal resonators with integrated MEMS tunable coupler,” IEEE J. Sel. Top. Quantum Electron. 13, 202–208 (2007).
[CrossRef]

Xu, Q.

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nature 3, 406–408 (2007), and references therein.

Yao, J.

J. Yao, D. Leuenberger, M. C. M. Lee, and M. C. Wu, “Silicon microtoroidal resonators with integrated MEMS tunable coupler,” IEEE J. Sel. Top. Quantum Electron. 13, 202–208 (2007).
[CrossRef]

Yariv, A.

Zhang, Z.

Zhao, J. H.

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis (Wiley, 1999).
[CrossRef]

Zhu, L.

Appl. Phys. Lett.

S. Fürst, A. Perez-Serrano, A. Scire, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: Experiment and theory,” Appl. Phys. Lett. 93, 251109 (2008).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

J. Yao, D. Leuenberger, M. C. M. Lee, and M. C. Wu, “Silicon microtoroidal resonators with integrated MEMS tunable coupler,” IEEE J. Sel. Top. Quantum Electron. 13, 202–208 (2007).
[CrossRef]

J. Mod. Opt.

J. E. Heebner and R. W. Boyd, “Slow and fast light in resonator-coupled waveguides,” J. Mod. Opt. 49, 2629–2636 (2002).
[CrossRef]

J. Opt. Soc. Am. B

Laser Phys.

M. L. Gorodetsky and V. S. Ilchenko, “Thermal nonlinear effects in optical whispering-gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

Nat. Photonics

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[CrossRef]

Nature

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nature 3, 406–408 (2007), and references therein.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

J. Čtyroký, I. Richter, and M. Šiňor, “Dual resonance in a waveguide-coupled ring microresonator,” Opt. Quantum Electron. 38, 781 (2006).
[CrossRef]

Science

R. Boyd and D. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef] [PubMed]

Other

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis (Wiley, 1999).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic of a TWMR with an add–drop configuration.

Fig. 2
Fig. 2

(a) Group delay of the transmitted light at the degenerate resonance frequency ω 0 for the through port of a TWMR with k 1 = k 2 = 0.2 . Critical coupling (oscillation) occurs at τ = 1 ( τ = 1.0417 ) . (b) The pole-zero dynamics on the unit circle as τ is varied. The cross (circle) denotes the pole (zero) of the transfer function.

Fig. 3
Fig. 3

(a) Group delay of the transmitted light at the degenerate resonance frequency ω o for the drop port of a TWMR with k 1 = k 2 = 0.2 . Oscillation occurs at τ = 1.0417 . (b) The pole-zero dynamics on the unit circle as τ is varied. The cross (circle) denotes the pole (zero) of the transfer function.

Fig. 4
Fig. 4

Relation between the transmission and the group delay at the degenerate resonance frequency ω o for the output light at the (a) through and (b) drop ports of a conventional TWMR with k 1 = k 2 = 0.2 as τ is varied.

Fig. 5
Fig. 5

Group delay at the degenerate resonance frequency ω o for the through port of a TWMR with k 1 = k 2 = 0.2 as a function of τ at different χ. (a) In the inset, the group delay curve changes from an increasing function to a deceasing function when χ is tuned from 0 to 0.0006 or 0.0009. (b) Further increment in χ shifts the asymptotes in the direction of the arrows. (c) If χ becomes too large, the group delay enhancement is degraded.

Fig. 6
Fig. 6

Group delay at the degenerate resonance frequency ω o for the drop port of a TWMR with k 1 = k 2 = 0.2 as a function of τ at different χ. (a) As shown in the inset, the group delay curve changes from an increasing function to a deceasing function when χ is tuned from 0 to 0.0006 or 0.0009. Further increment in χ smoothens the group delay curves. (c) At large χ, the group delay curves shift in the direction of the arrows, without degrading the group delay enhancement, unlike the case at the through port in Fig. 5c.

Fig. 7
Fig. 7

Evolution in the transmission and group delay spectra around the degenerate resonance frequency ω o for a TWMR with k 1 = k 2 = 0.2 for (a) the net loss regime ( τ = 0.99 ) and (b) net gain regime ( τ = 1.035 ) as χ is varied.

Fig. 8
Fig. 8

Relation between the transmission and the group delay at the degenerate resonance frequency ω o for the output light at the (a) through and (b) drop ports of a non-degenerate TWMR with k 1 = k 2 = 0.2 as τ is varied for χ = 0.006 .

Tables (2)

Tables Icon

Table 1 Summary of FL and SL Performances at the Degenerate Resonance Frequency ω 0 in the Stable Operation Regime a of a TWMR

Tables Icon

Table 2 Number of Poles and Zeros That Correspond to the Degenerate Resonance Frequency ω 0 in the Transfer Functions of the Output Ports of a TWMR a

Equations (30)

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H T = b 1 a 1 = | H T | Φ T = r 2 τ   exp ( j θ ) r 1 r 1 r 2 τ   exp ( j θ ) 1 ,
H D = b 2 a 1 = | H D | Φ D = k 1 k 2 τ   exp ( j θ / 2 ) r 1 r 2 τ   exp ( j θ ) 1 ,
z z , T = τ r 2 / r 1 ,     z p , T = τ r 1 r 2 ,
z z , D = 0 ,     z p , D = τ r 1 r 2 .
t g , T = τ r 2 ( r 1 2 1 ) ( τ 2 r 1 r 2 2   cos   θ + r 1   cos   θ τ r 1 2 r 2 τ r 2 ) ( r 1 2 2 τ r 1 r 2   cos   θ + τ 2 r 2 2 ) ( 1 2 τ r 1 r 2   cos   θ + τ 2 r 1 2 r 2 2 ) T ,
t g , D = 1 τ 2 r 1 2 r 2 2 2 ( 1 2 τ r 1 r 2   cos   θ + τ 2 r 1 2 r 2 2 ) T ,
t g , T ( ω o ) = τ r 2 ( 1 r 1 2 ) ( τ r 2 r 1 ) ( 1 τ r 1 r 2 ) T = ( z p , T z z , T ) ( z z , T 1 ) ( z p , T 1 ) T .
τ < r 1 / r 2 t g , T < 0 ,     τ > r 1 / r 2 t g , T > 0.
t g , D ( ω 0 ) = 1 + τ r 1 r 2 2 ( 1 τ r 1 r 2 ) T = 1 + z p , D 2 ( 1 z p , D ) T .
τ < 1 / ( r 1 r 2 ) t g , T > 0.
( b i b i d i d i ) T = ( [ P i ] [ U i ] [ U i ] [ P i ] ) ( a i a i c i c i ) T = N i ( a i a i c i c i ) T ,
[ P i ] = ( j r i r i r i j r i ) ,     [ U i ] = ( k i j k i j k i k i ) ,     i = 1   or   2.
| r i | 2 + | r i | 2 + | k i | 2 + | k i | 2 = χ i 2 + | r i | 2 + | k i | 2 = 1 ,
k i r i = k i r i ,
χ i 2 = | r i | 2 + | k i | 2 ,
r i = 1 χ i 2 | k i | 2 ,
k i = 1 χ i 2 | r i | 2 ,
r i = χ i r i / 1 χ i 2 ,
k i = χ i 1 χ i 2 | r i | 2 / 1 χ i 2 ,
c 1 = τ d 2   exp ( j θ / 2 ) ,     c 2 = τ d 1   exp ( j θ / 2 ) ,
c 1 = τ d 2   exp ( j θ / 2 ) ,     c 2 = τ d 1   exp ( j θ / 2 ) .
H T = b 1 a 1 = τ 2 A   exp ( j 2 θ ) τ B   exp ( j θ ) + r 1 τ 2 C   exp ( j 2 θ ) + 2 τ D   exp ( j θ ) + 1 ,
H D = b 2 a 1 = τ 3 / 2 E   exp ( j 3 θ / 2 ) + τ F   exp ( j θ / 2 ) τ 2 C   exp ( j 2 θ ) + 2 τ D   exp ( j θ ) + 1 ,
A = r 1 r 2 2 / ε 2 ,     B = χ 1 r 2 ( χ 1 r 1 2 ε 1 1 / 2 2 χ 2 ε 1 1 / 2 ) / ε 2 + r 2 ( r 1 2 2 χ 1 2 + 1 ) ,
C ( r 1 r 2 ) 2 / ( ε 1 ε 2 ) ,     D = r 1 ( χ 1 χ 2 r 2 / ε 1 ε 2 r 2 ) ,
E = r 1 r 2 ( ε 1 r 1 2 ) ( ε 2 r 2 2 ) / ( ε 1 ε 2 ) ,     F = ( χ 2 χ 1 ε 1 ε 2 ) ( ε 1 r 1 2 ) ( ε 2 r 2 2 ) / ε 1 ε 2 ,
z z , T = ( B ± B 2 4 A r 1 ) / ( 2 r 1 τ 1 ) ,     z p , T = τ [ ( r 1 r 2 r 1 r 2 ) ± j ( r 1 r 2 + r 1 r 2 ) ] ,
z z , D = τ E / F ,     z p , D = τ [ ( r 1 r 2 r 1 r 2 ) ± j ( r 1 r 2 + r 1 r 2 ) ] .
t g , T ( ω 0 ) = [ 2 τ 2 A ( | r 2 | 2 + | r 2 | 2 ) τ B τ 2 A ( | r 2 | 2 + | r 2 | 2 ) τ B + r 1 2 τ 2 ( | r 1 | 2 + | r 1 | 2 ) ( | r 2 | 2 + | r 2 | 2 ) τ ( 2 r 1 r 2 2 r 1 r 2 ) τ 2 ( | r 1 | 2 + | r 1 | 2 ) ( | r 2 | 2 + | r 2 | 2 ) + τ ( 2 r 1 r 2 2 r 1 r 2 ) + 1 ] T ,
t g , D ( ω 0 ) = 2 [ τ ( 3 τ C + k 1 k 2 k 1 k 2 ) 4 ( τ C + k 1 k 2 k 1 k 2 ) - r 1 r 2 ( 1 τ r 1 r 2 ) r 1 r 2 ( 1 + τ r 1 r 2 ) τ ( | r 1 | 2 | r 2 | 2 + | r 1 | 2 | r 2 | 2 ) τ 1 ( 1 + τ r 1 r 2 ) 2 + τ | r 1 | 2 ( | r 2 | 2 + | r 2 | ) + r 2 ( τ r 2 | r 1 | 2 2 r 1 ) ] T ,

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