## Nonlinearity enhancement in finite coupled-resonator slow-light waveguides

Optics Express, Vol. 12, Issue 15, pp. 3353-3366 (2004)

http://dx.doi.org/10.1364/OPEX.12.003353

Acrobat PDF (924 KB)

### 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

## 1. Introduction

1. 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]

2. 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]

3. A. Melloni, R. Costa, P. Monguzzi, and M. Martinelli“Ring-resonator filters in silicon oxynitride technology for dense wavelength-division multiplexing systems,” Opt. Lett. **28**, 1567–1569 (2003). [CrossRef] [PubMed]

*v*

_{g}is much less than the free-space speed of light

*c*. The group velocity is determined by the derivative of the radial frequency

*ω*with respect to the propagation constant

*k*, given the optical dispersion relation

*k*(

*ω*). A dispersion relation has been established for the coupled resonator optical waveguide (CROW) in the tight-binding approximation and simplified with the assumption of nearest neighbor coupling [4

4. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef]

5. S. Mookherjea, D. S. Cohen, and A. Yariv “Nonlinear dispersion in a coupled-resonator optical waveguide,” Opt. Lett. **27**, 933–935 (2002). [CrossRef]

6. 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]

*c/v*

_{g})

^{2}[6

6. 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. M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B **19**, 2052–2059 (2002). [CrossRef]

4. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef]

6. 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]

7. 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]

1. 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]

1. 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]

9. S. Olivier, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterlé “Miniband transmission in a photonic crystal coupled-resonator optical waveguide,” Opt. Lett. **26**, 1019–1021 (2001). [CrossRef]

10. Y. Chen, G. Pasrija, B. Farhang-Boroujeny, and S. Blair “Engineering the nonlinear phase shift,” Opt. Lett. **28**, 1945–1947 (2003). [CrossRef] [PubMed]

## 2. Coupled ring-resonator waveguide

**M̂**can be expressed in a general form as [12

12. 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]

13. 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]

14. A. Melloni and M. Martinelli “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. **20**, 296–303 (2002). [CrossRef]

*d*=

*L*/2 is the sum of two

*L*/4 arcs for the ring-resonator, and

*k*=

*nω/c*is the material wave number where

*n*is the effective refractive index,

*ω*=2

*πc*=λ is the radial frequency, and λ is the wavelength in free space.

*T*

_{1}=|

*t*

_{1}|

^{2}=

*t*

^{2}, which is of the same value as the coupler transmittance.

### 2.1. Infinite waveguide

**35**, 365–379 (2003). [CrossRef]

12. 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]

*β*is the propagation constant of the Bloch wave. This expression is consistent with Yariv’s derivation [4

4. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef]

5. S. Mookherjea, D. S. Cohen, and A. Yariv “Nonlinear dispersion in a coupled-resonator optical waveguide,” Opt. Lett. **27**, 933–935 (2002). [CrossRef]

*ω*with respect to the propagation constant returns the group velocity [6

**35**, 365–379 (2003). [CrossRef]

*kd*=2

*mπ*with integer

*m*),

*n*

_{g}=

*n/t*, meaning that the group velocity is 1/

*t*times slower than the phase velocity

*ν*=

*c/n*in the underlying material. The weaker the coupling between cavities, the more strongly the light is slowed; however, there is a tradeoff between the light slowing ratio and the range of frequencies that can propagate through the waveguide. The bandwidth Δ

*ω*depends on the coupling coefficient as [6

**35**, 365–379 (2003). [CrossRef]

*c*/(

*nL*) is the free spectral range. The bandwidth increases with increasing coupling efficiency, which in turn weakens the light slowing ability.

### 2.2. Finite waveguide

*t*

_{N}and reflection coefficient

*r*

_{N}with relation |

*t*

_{N}|

^{2}+|

*r*

_{N}|

^{2}=1, the transfer matrix [12

12. 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]

*t*

_{N}can be described by its amplitude and phase as

*T*

_{N}is the waveguide transmittance and

*ϕ*

_{N}is the total phase shift determined by

*k*

_{N}is the propagation constant in the finite waveguide, being described in the dispersion relation of Eq. (15), and

*L*

_{N}is the total waveguide length. For an

*N*-cavity waveguide (i.e.

*N*+1 unit cells),

*L*

_{N}=(

*N*+1)

*d*. Assuming that all the unit cells are identical, the overall transfer matrix of the

*N*-cavity waveguide can also be computed from the unit cell transfer matrix

**M̂**using [12

**53**, 4107–4121 (1996). [CrossRef]

**Î**is the unit matrix. Inserting Eq. (2) into Eq. (12) and comparing the resulting matrix elements with those in Eq. (9),

*t*

_{N}is expressed in terms of the unit cell parameters as [12

**53**, 4107–4121 (1996). [CrossRef]

*t*

_{N}in the form of Eq. (10), with sin(

*kd*)=

*t*cos(

*βd*) from Eq. (5) and sin(

*Nβd*)=sin(

*βL*

_{N}-

*βd*)=sin(

*βL*

_{N}) cos(

*βd*)-cos(

*βL*

_{N}) sin(

*βd*), the exact dispersion relation of the finite

*N*-cavity coupled ring-resonator waveguide is

*β*is determined from Eq. (5).

*k*

_{N}with respect to radial frequency as

*n*

_{g}

*N*=

*c/v*

_{g}

*N*is the group index and

*v*

_{g}

*N*is the group velocity in the finite structure. With some mathematical efforts, the group index is derived from Eqs. (15) and (16) as

*n*

_{g}is the slowing ratio of the infinitely long waveguide given in Eq. (7). The total group delay is then calculated as

#### 2.2.1. Linear response

*L*=50

*µm*(FSR=4 THz) and miniband Δ

*ν*=FSR/3; the coupling coefficient is

*t*=0.5 (the lower the

*t*, the sharper and deeper each resonance). In the top figure, observing the transmittance

*I*

_{out}=

*I*

_{in}at the central frequency, the 11-cavity waveguide produces full transmission, but the 10-cavity waveguide is at a local minimum; they are out-of-phase. Moving towards the band edge, the two transmittance responses gradually become in-phase. The total linear phase in the miniband is determined by the number of cavities, where each ring contributes one

*π*phase shift. Much of the phase, however, is accumulated at the edge of the band, manifested by the band-edge spikes in the group delay responses shown in Fig. 2 (c). This is characteristic of auto-regressive (AR) optical filters. (In the transmission transfer function

*t*

_{N}given by Eq. (14),

*t*

_{N}possesses

*N*poles and (

*N*+1)=2 zeros at the origin, and is thus characterized as an AR filter [15].) The dispersion relations

*ω*(

*k*) are plotted in Fig. 3 (a). Also drawn for comparison are the dispersion relation of an infinite waveguide (thick solid line), which is a smooth curve, and the dispersion relation of the underlying material (thick dashed line), which is a straight line. The plots of normalized group velocity

*v*

_{g}

*/c*(Fig. 3(b)) and group index (Fig. 3(c)) show clearly the difference resulting from the ripples in the dispersion relation of finite waveguides. While approaching the band edge, the group index at some frequencies increases significantly, theoretically reaching infinity in an infinite system, resulting in zero group velocity [4

**24**, 711–713 (1999). [CrossRef]

7. 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]

*n*

_{g}

*N*at

*ν*

_{m}, where sin(

*kd*)=0, and hence cos(

*βd*)=0 from Eq. (5). The expression for

*n*

_{g}

*N*is reduced to

*βL*

_{N})=0 and cos(

*βL*

_{N})=1, so that

*t*times larger than the value for that of an infinite structure, meaning that the propagation of light on resonance is slower in the finite waveguide even though the propagation distance (waveguide length) is shorter. This phenomenon results from the miniband resonances at specific wavelengths. The resonance bandwidth

*δν*at

*ν*

_{m}, i.e. the full-width at half-magnitude (FWHM), is much narrower than the width of the entire miniband Δ

*ν*, which is also the resonance bandwidth for infinite structures. Otherwise, if

*N*is an even number, we have cos(

*βL*

_{N})=0 and sin(

*βL*

_{N})=1 instead, thus

*ν*

_{m}is chosen, such as at neighboring resonance peaks, which forces cos(

*βL*

_{N})=1 and sin(

*βL*

_{N})=0, light slowing does occur for even-

*N*cavity waveguide, as shown in Fig. 3(b)(c). To create effective slow-light periodic structures operating at the center of the miniband, the waveguide needs to contain odd number of cavities.

#### 2.2.2. Nonlinear response

*n*

_{2}

*I*

_{in}, where

*n*

_{2}is the nonlinear coefficient of the underlying material and

*I*

_{in}is the incident intensity. All the couplers are considered to be linear and intensity independent. With increasing

*n*

_{2}

*I*

_{in}, the transmittance goes down and the nonlinear phase change ΔΦ accumulates gradually. When the incident intensity is high enough, both the transmittance and phase shift responses experience rapid transitions (emphasized by arrows), and switch to a higher stable branch. Beyond this point, if the intensity is reduced, the transmittance and phase shift do not switch back immediately to the lower branch but stay on the higher branch until the input intensity is lower than the point of low-to-high transition, as depicted in the figure. This phenomenon, known as optical bistability (or multistability in this case), can occur when an optical system possesses both nonlinearity and feedback, as here. The region within the hysteresis loop, illustrated by the dotted lines, represents the unstable region. Optical bistability is a useful phenomenon for making optical switches and optical logic gates [16].

*d*ΔΦ=

*dI*

_{in}is an important metric of the performance of nonlinear devices. Prediction of the nonlinear phase sensitivity at low intensity input can be obtained from the linear filter response, which is fully controlled by two parameters: the coupling coefficient t and ring length

*L*. Figure 5 shows the normalized nonlinear phase sensitivity

*d*ΔΦ=

*d*(

*n*

_{2}

*I*

_{in}) (proportional to

*n*

_{2}where

_{m}=500

*nm*. Figure 5(a) indicates that the nonlinear phase sensitivity is proportional to 1=

*t*

^{4}, hence proportional to the group index squared according to Eq. (9) or inversely proportional to group delay squared with Eq. (18) as

*ν*

_{m}). According to Eq. (8), the finesse

*F*=FSR=Δ

*ν*∝1=sin

^{-1}(t); which is therefore constant with

*L*and

*N*; however,

*δν*scales inversely with

*L*and

*N*. In situations where

*t*is constant, then the nonlinear sensitivity is proportional to 1/

*δν*. Obviously, the most efficient way to enhance the nonlinear phase sensitivity is to control the coupling coefficient, but at the loss of overall bandwidth (i.e. Δ

*ν*). Vertical couplers with precise coupling efficiency have been demonstrated using a thermal wafer bonding technique [17

17. 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]

## 3. Optimizing coupled ring-resonator waveguide as a nonlinear phase shifter

*π*phase shift is achieved. The lower the intensity needed to obtain the

*π*phase shift, the better the nonlinear sensitivity. To avoid optical bistability and get better nonlinear phase response, an optical structure should be optimized to produce a flat-top magnitude response with a steep linear phase response within the passband, as shown in Fig. 6; a purely periodic structure produces strong ripples unless the finesse is extremely low (i.e.

*t*≈1 in Eq. (8)). Light incident on the optimized waveguide will be transmitted with efficiency given by the magnitude response, but will also experience a phase change due to the phase response. As the light intensity increases, the overall filter response will redshift due to intensity-induced changes in the structure components, which are themselves constructed from (weakly) nonlinear materials. Ideally, under weak detuning, the transmitted intensity fraction will not change given a flat-topped magnitude response, but the phase at the output will change due to the steep linear phase response within the passband. The slope of the phase determines the group delay. In effect, what this approach does is to amplify the intrinsic nonlinearity of a material, where the efficiency of the process improves with increasing filter group delay. Strong detuning in multi-resonator systems can result in distortions of the filter response, however.

### 3.1. Optimized waveguide response

14. A. Melloni and M. Martinelli “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. **20**, 296–303 (2002). [CrossRef]

*µm*) and ~100 GHz bandwidth with 3 to 11 rings are designed, and the optimized coupling coefficients are listed in Table 1, showing that these aperiodic systems are symmetric. The linear spectral responses are shown in Fig. 7. With increasing filter order, the passband gradually approaches the ideal square shape. In the passband, the transmission ripples have been minimized and the phase is approximately linear. The total phase change as well as the inband phase increase with filter order. Some fraction of the total phase change locates outside the band to reduce the inband transmission ripples. Since there are no independent zeros in AR filters, much of the inband phase is built up at the edge of the passband and hence results in group delay spikes. By adding independent zeros into the transfer function to create auto-regressive moving-average (ARMA) optical filters, better phase and group delay responses can be produced [10

10. Y. Chen, G. Pasrija, B. Farhang-Boroujeny, and S. Blair “Engineering the nonlinear phase shift,” Opt. Lett. **28**, 1945–1947 (2003). [CrossRef] [PubMed]

*N*≥3.

*µm*producing about 21.6 ps group delay; to produce the same group delay, a material length of 4.3

*mm*is required. There’s about 17 times size reduction. For structures of much higher orders, an asymptotic size reduction of 25 is estimated by dividing the slopes of the lines in Fig. 8(b).

### 3.2. Optimized nonlinear phase shift

*n*

_{2}

*I*

_{in}. The incident frequency is at the center of the passband. Because of the flattened linear response, optical bistability has been reduced. The nonlinear phase sensitivity is almost constant at low incident intensity, as shown in Fig. 9(c), and scales proportionally with filter order (

*N*≥3) as shown in Fig. 10, where plotted for comparison is the nonlinear phase sensitivity of the underlying material of equal group delay. In the case of the 9th order filter, there’s about 10 times improvement. For much higher order filters, the asymptotic enhancement is about 12 obtained from the line slopes. If we define a figure-of-merit as

*π*phase shift, i.e. lower

*n*

_{2}

*Iπ*, and also produce higher transmittance. For example, for the 9th order waveguide, a

*π*phase shift is obtained at

*n*

_{2}

*Iπ*=5×10

^{-6}with almost 100% transmittance, while

*n*

_{2}

*Iπ*=1.3×10

^{-5}with transmittance less than 70% is obtained for a 5th order waveguide.

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

10. Y. Chen, G. Pasrija, B. Farhang-Boroujeny, and S. Blair “Engineering the nonlinear phase shift,” Opt. Lett. **28**, 1945–1947 (2003). [CrossRef] [PubMed]

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

*n*

_{2}

*I*

_{π/4})

^{-1}and (

*n*

_{2}

*I*

_{π})

^{-1}produced by a 7-cavity waveguide are studied as a function of FSR. Note that this case is different to that described by Eq. (23) where the finesse was constant.

*n*

_{2}

*I*

_{π/4}and

*n*

_{2}

*I*

_{π}is plotted in Fig. 11 (a) and (b), respectively. Figure 11 (a) shows that the nonlinear sensitivity (roughly proportional to (

*n*

_{2}

*I*

_{π/4})

^{-1}since

*π*/4 is a relatively small phase change) scales linearly with filter order. In Fig. 11 (b), although there’s large nonlinear change with increasing filter order from 3 to 7, (

*n*

_{2}

*I*

_{π})

^{-1}also approximates a linear relation to filter order if

*N*≥7. In both figures, nonlinear sensitivity of a bulk material is plotted for comparison. For the 9-cavity waveguide, the nonlinear sensitivity enhancement factor is about 10.3 and 11.7 at

*n*

_{2}

*I*

_{π/4}and

*n*

_{2}

*I*

_{π}, respectively; considering the size reduction, the FOM is about 177 and 202, respectively. FOMs of higher order waveguides can be approximated from the slope of the lines in Fig. 11, which is around 300. Figure 12 (a) shows the nonlinear response at frequencies of

*ν*=

*ν*

_{m}±Δ

*ν*/4 for the 9-cavity structure. Because of the flattened linear filter response, transmittance above 90% can be obtained at the normalized incident intensity

*n*

_{2}

*I*

_{π}where a

*π*phase change is produced, demonstrating a broadband nonlinearity. The value of

*n*

_{2}

*I*

_{π}decreases with increasing frequency across the passband. Compared with the unstructured material at

*ν*

_{m}(thick-solid line in Fig. 12) (a), the coupled resonator structure requires much less intensity to produce a

*π*phase shift, and at a given moderate incident intensity, the nonlinear phase shift produced by the coupled resonator structure is much greater, although they allow the same group delay.

## 4. Discussion and conclusions

*π*nonlinear phase shift, compared to the underlying material producing the same group delay.

*α*=1cm

^{-1}) at low incident intensity, while the amplitude response maintains flattened transmission characteristics up to a

*π*phase change under moderate TPA (

*K*=0.03 [18

18. S. Blair, J. Heebner, and R. Boyd “Beyond the absorption-limited nonlinear phase shift with microring resonators,” Opt. Lett. **27**, 357–359 (2002). [CrossRef]

*T*~0.75). In the presence of loss, a slightly greater value of

*n*

_{2}

*I*

_{π}is obtained, resulting in slightly reduced nonlinear sensitivity. For large TPA (

*K*=0.1 or

*T*~2:5), the transmission at incident intensity

*n*

_{2}

*I*

_{π}is further decreased, and the nonlinear sensitivity is reduced as well.

## References and links

1. | M. Bayindir, B. Temelkuran, and E. Ozbay “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett. |

2. | 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. |

3. | A. Melloni, R. Costa, P. Monguzzi, and M. Martinelli“Ring-resonator filters in silicon oxynitride technology for dense wavelength-division multiplexing systems,” Opt. Lett. |

4. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

5. | S. Mookherjea, D. S. Cohen, and A. Yariv “Nonlinear dispersion in a coupled-resonator optical waveguide,” Opt. Lett. |

6. | A. Melloni, F. Morichetti, and M. Martinelli “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. |

7. | 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. |

8. | M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B |

9. | S. Olivier, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterlé “Miniband transmission in a photonic crystal coupled-resonator optical waveguide,” Opt. Lett. |

10. | Y. Chen, G. Pasrija, B. Farhang-Boroujeny, and S. Blair “Engineering the nonlinear phase shift,” Opt. Lett. |

11. | 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. | 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 |

13. | 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. |

14. | A. Melloni and M. Martinelli “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. |

15. | C. K. Madsen and J. H. Zhao |

16. | H. M. Gibbs |

17. | 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. |

18. | S. Blair, J. Heebner, and R. Boyd “Beyond the absorption-limited nonlinear phase shift with microring resonators,” Opt. Lett. |

19. | Y. Chen and S. Blair “Nonlinear phase shift of cascaded microring resonators,” J. Opt. Soc. Am. B |

20. | V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco “Two- photon absorption as a limitation to all-optical switching,” Opt. Lett. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.3270) Nonlinear optics : Kerr effect

(230.5750) Optical devices : Resonators

(230.7370) Optical devices : Waveguides

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 18, 2004

Revised Manuscript: July 9, 2004

Published: July 26, 2004

**Citation**

Yan Chen and Steve Blair, "Nonlinearity enhancement in finite coupled-resonator slow-light waveguides," Opt. Express **12**, 3353-3366 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-15-3353

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### References

- 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]
- A. Melloni, R. Costa, P. Monguzzi, and M. Martinelli �??Ring-resonator filters in silicon oxynitride technology for dense wavelength-division multiplexing systems,�?? Opt. Lett. 28, 1567-1569 (2003). [CrossRef] [PubMed]
- A. Yariv, Y. Xu, R. K. Lee, and A. Scherer �??Coupled-resonator optical waveguide: a proposal and analysis,�?? Opt. Lett. 24, 711-713 (1999). [CrossRef]
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