Abstract

In this paper, we derive the exact dispersion relation of one dimensional periodic coupled-resonator optical waveguides of finite length, from which the reduced group velocity of light is obtained. We show that the group index strongly depends on the number of cavities in the system, especially for operation at the center frequency. The nonlinear phase sensitivity shows an enhancement proportional to the square of the group index (or light slowing ratio). Aperiodic coupled ring-resonator optical waveguides with optimized linear properties are then synthesized to give an almost ideal nonlinear phase shift response. For a given application and bandwidth requirement, the nonlinear sensitivity can be increased by either decreasing resonator length or by using higher-order structures. The impact of optical loss, including linear and two-photon absorption is discussed in post-analysis.

© 2004 Optical Society of America

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References

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IEEE J. Quantum Electron.

E. Ozbay, M. Bayindir, I. Bulu, and E. Cubuku �??Investigation of localized coupled-cavity modes in two-dimensional photonic bandgap structures,�?? IEEE J. Quantum Electron. 38, 837-843 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

T. J. Karle, D. H. Brown, R. Wilson, M. Steer, and T. F. Krauss �??Planar photonic crystal coupled cavity waveguides,�?? IEEE J. Sel. Top. Quantum Electron. 8, 909-918 (2002).
[CrossRef]

IEEE Photon. Technol. Lett.

D. V. Tishinin, P. D. Dapkus, A. E. Bond, I. Kim, C. K. Chin, and J. O'Brien �??Vertical resonant couplers with precise coupling efficiency control fabricated by wafer bonding,�?? IEEE Photon. Technol. Lett. 11, 1003-1005 (1999).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

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. E

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]

Phys. Rev. Lett.

M. Bayindir, B. Temelkuran, and E. Ozbay �??Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,�?? Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama �??Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,�?? Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Other

C. K. Madsen and J. H. Zhao Optical Filter Design and Analysis: A Signal Processing Approach. Wiley 1999.

H. M. Gibbs Optical Bistability : Controlling Light with Light. Academic Press, Inc. 1985.

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Figures (12)

Fig. 1.
Fig. 1.

Coupled resonator waveguides: (a) coupled Fabry-Perot type cavities in thin film and (b) coupled ring-resonators, where the shaded portions in dotted-boxes represent the unit cells.

Fig. 2.
Fig. 2.

The filter responses (a) transmission, (b) phase and (c) group delay of periodic coupled ring-resonator waveguides with central frequency νm and miniband width Δν=33.33 GHz. The solid line represents the response of an 11th order filter and dotted line for a 10th order filter. An effective refractive index of 1.5 and central wavelength of 500 nm are used.

Fig. 3.
Fig. 3.

Dispersion relations (a), normalized group velocity (b) and group index (c) of infinite, 10th, and 11th order periodic coupled ring-resonator waveguides, and the underlying material, respectively. ω 0=2πνm , and X=kd,βd and kNd for the material, infinite waveguide and finite waveguide respectively. The insets show details in the central band.

Fig. 4.
Fig. 4.

Nonlinear response produced by a periodic 11-cavity coupled ring-resonator waveguide where optical bistability occurs at high intensity input. There’s also a small region where the two bistable cycles overlap, producing a region where there are three stable states.

Fig. 5.
Fig. 5.

Nonlinear phase sensitivity of periodic coupled ring-resonator waveguides: (a) varying with coupling strength (The numbers shown beside the line are the coupling coefficients) but constant length; (b) varying with ring length but constant coupling.

Fig. 6.
Fig. 6.

Ideal linear filter response for nonlinear phase shift

Fig. 7.
Fig. 7.

Linear filter responses (a) transmission, (b) phase and (c) group delay of optimized aperiodic coupled ring-resonator waveguides having 4 THz FSR, ~100 GHz bandwidth, and finesse of 40. The arrow indicates filter order increasing from 3 to 11.

Fig. 8.
Fig. 8.

(a) Group delay vs. filter order for optimized aperiodic coupled ring-resonator waveguides. For N≥3, group delay is proportional to filter order. (b) Physical length of coupled ring-resonator waveguides and underlying material of equal group delay vs. filter order.

Fig. 9.
Fig. 9.

Nonlinear response of optimized aperiodic coupled ring-resonator waveguides. The arrow indicates filter order increasing from 3 to 11.

Fig. 10.
Fig. 10.

(a) Nonlinear sensitivity at low intensity as a function of filter order, for the optimized aperiodic coupled ring-resonator waveguide. (b) Nonlinear sensitivity of a 7th order optimized aperiodic coupled ring-resonator waveguide vs. FSR, given a bandwidth ~100 GHz.

Fig. 11.
Fig. 11.

Nonlinear sensitivity of optimized aperiodic coupled ring-resonator waveguides having constant bandwidth but increased filter order, studied at the normalized intensity to produce: (a) π/4 phase shift and (b) π phase shift, respectively.

Fig. 12.
Fig. 12.

Nonlinear responses of a 9th order optimized aperiodic coupled ring-resonator waveguide: (a) at different frequencies; (b) to optical loss, where the linear attenuation is α=1 cm-1 [19]; the normalized two photon absorption (TPA) coefficients are K=0,0.03,0.1, corresponding to zero, moderate and large TPA, respectively. The TPA figure of merit [20] is given as T=8πK, where T<1 is desired.

Tables (1)

Tables Icon

Table 1. Optimized coupling coefficients t of aperiodic coupled ring-resonator waveguides (N=3 to 11)

Equations (25)

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[ A n + A n ] = M ̂ [ A n + 1 + A n + 1 ]
M ̂ = [ 1 t 1 r 1 * t 1 * r 1 t 1 1 t 1 * ]
M ̂ = j t [ e jkd r r e jkd ]
t 1 = j t e j k d
cos ( β d ) = Re { 1 t 1 } = sin ( k d ) t
v g = c n t 2 sin 2 ( k d ) cos ( k d ) ,
n g = c v g = n cos ( k d ) t 2 sin 2 ( k d ) .
Δ ω = 2 π Δ ν = 4 FSR sin 1 ( t ) ,
M ̂ N = [ 1 t N r N * t N * r N t N 1 t N * ]
t N = T N e j ϕ N ,
ϕ N = k N L N .
M ̂ N = M ̂ sin ( β L N ) sin ( βd ) I ̂ sin ( N β d ) sin ( β d ) ,
1 t N = 1 t 1 sin ( β L N ) sin ( βd ) sin ( N β d ) sin ( β d ) .
t N = j t sin ( β d ) e jkd sin ( β L N ) j t sin ( N β d ) .
tan ( k N L N ) = cos ( k d ) t tan ( β L N ) sin ( β d ) ,
d k N d ω = n g N c = 1 L N d d ω [ tan 1 ( RHS ) ]
n g N = n g t t 2 sin 2 ( β d ) cos 2 ( β L N ) + cos 2 ( k d ) sin 2 ( β L N ) [ cos ( k d ) sin ( β d )
n d sin ( 2 β L N ) 2 L N ( sin ( k d ) sin ( β d ) n g + cos ( k d ) cos ( β d ) n ) ]
g . d . = n g N L N c .
n g N = n t 2 cos 2 ( β L N ) + sin 2 ( β L N ) ,
n g N ( odd ) = n t 2 ,
n g N ( even ) = n ,
d Δ Φ d ( n 2 I in ) n g N 2 = ( n t 2 ) 2 ,
d Δ Φ d ( n 2 I in ) L
FOM = L m L w { d Δ Φ d I } w { d Δ Φ d I } m ,

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