## Modulation instability in nonlinear coupled resonator optical waveguides and photonic crystal waveguides

Optics Express, Vol. 17, Issue 3, pp. 1299-1307 (2009)

http://dx.doi.org/10.1364/OE.17.001299

Acrobat PDF (783 KB)

### Abstract

Modulation instability (MI) in a coupled resonator optical waveguide (CROW) and photonic-crystal waveguide (PCW) with nonlinear Kerr media was studied by using the tight-binding theory. By considering the coupling between the defects, we obtained a discrete nonlinear evolution equation and termed it the extended discrete nonlinear Schrödinger (EDNLS) equation. By solving this equation for CROWs and PCWs, we obtained the MI region and the MI gains, G(*p,q*), for different wavevectors of the incident plane wave (*p*) and perturbation (*q*) analytically. In CROWs, the MI region, in which solitons can be formed, can only occur for *pa* being located either before or after π/2, where *a* is the separation of the cavities. The location of the MI region is determined by the number of the separation rods between defects and the sign of the Kerr coefficient. However, in the PCWs, *pa* in the MI region can exceed the π/2. For those wavevectors close to π/2, the MI profile, G(*q*), can possess two gain maxima at fixed *pa*. It is quite different from the results of the nonlinear CROWs and optical fibers. By numerically solving the EDNLS equation using the 4^{th} order Runge-Kutta method to observe exponential growth of small perturbation in the MI region, we found it is consistent with our analytic solutions.

© 2009 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Photonic band-gap crystals,” J. Phys. Condens. Matter **5**, 2443–2460 (1993). [CrossRef]

2. E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B **10**, 283–295 (1993). [CrossRef]

3. D. W. Prather, S. Y. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. H. Chen, and B. L. Miao, “Photonic crystal structures and applications: Perspective, overview, and development,“ IEEE J. Sel. Top. Quantum Electron. **12**, 1416–1437 (2006). [CrossRef]

6. A. Imhof, W. L. Vos, R. Sprik, and A. Lagendijk, “Large dispersive effects near the band edges of photonic crystals,” Phys. Rev. Lett. **83**, 2942–2945 (1999). [CrossRef]

7. W. J. Kim, W. Kuang, and J. D. O’Brien, ”Dispersion characteristics of photonic crystal coupled resonator optical waveguides,“ Opt. Express **11**, 3431–3437 (2003). [CrossRef] [PubMed]

8. S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, “Long-range interaction and nonlinear localized modes in photonic crystal waveguides,” Phys. Rev. E **62**, 5777–5782 (2000). [CrossRef]

12. A. G. Shagalov, “Modulational instability of nonlinear waves in the range of zero dispersion,” Physics Lett. A **239**, 41–45 (1998). [CrossRef]

14. F. K. Abdullaev, A. Bouketir, A. Messikh, and B. A. Umarov, “Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrodinger equation,” Physica D-Nonlinear Phenomena **232**, 54–61 (2007). [CrossRef]

15. F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. **11**, 659–661 (1986). [CrossRef] [PubMed]

13. L. Hadzievski, M. Stepic, and M. M. Skoric, “Modulation instability in two-dimensional nonlinear Schrodinger lattice models with dispersion and long-range interactions,” Phys. Rev. B **68**, 014305 (2003). [CrossRef]

10. D. N. Christodoulides and N. K. Efremidis, “Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals,” Opt. Lett. **27**, 568–570 (2002). [CrossRef]

16. T. Kamalakis and T. Sphicopoulos, “Analytical expressions for the resonant frequencies and modal fields of finite coupled optical cavity chains,” IEEE J. Quantum Electron. **41**, 1419–1425 (2005). [CrossRef]

17. S. Mookherjea, “Dispersion characteristics of coupled-resonator optical waveguides,” Opt. Lett. **30**, 2406–2408 (2005). [CrossRef] [PubMed]

10. D. N. Christodoulides and N. K. Efremidis, “Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals,” Opt. Lett. **27**, 568–570 (2002). [CrossRef]

18. F. S. S. Chien, J. B. Tu, W. F. Hsieh, and S. C. Cheng, “Tight-binding theory for coupled photonic crystal waveguides,” Phys. Rev. B **75**, 125113 (2007). [CrossRef]

## 2. Theory

*a*. The distance between successive defect points or cavities is

_{L}*a*, and the Kerr media, in which the refraction index is proportional to the intensity of the incident wave, is put in the defect region, shown in Fig. 1. Assuming the isolated point defect is a single mode with eigenfrequency of ω

_{0}, we can express the mode fields of each point defect as

**E**(

**r**,

*t*) =

**E**

_{0}(

**r**)

*exp(-iω*and

_{0}t)**H**(

**r**,

*t*) =

**H**

_{0}(

**r**)

*exp(-iω*. The electric field

_{0}t)**E**′

_{0}(

**r**,

*t*) and magnetic field

**H**′

_{0}(

**r**,

*t*) of the waveguide can be expressed as a superposition of the bound states, i.e.,

**E**′

_{0}(

**r**,

*t*) = ∑

*b*(

_{m}*t*)

**E**

_{0m}and

**H**′

_{0}(

**r**,

*t*) = ∑

*b*(

_{m}*t*)

**H**

_{0m}, where

**E**

_{0m}=

**E**

_{0}(

**r**-

*ma*) and

**H**

_{0m}=

**H**

_{0}(

**r**-

*ma*).

18. F. S. S. Chien, J. B. Tu, W. F. Hsieh, and S. C. Cheng, “Tight-binding theory for coupled photonic crystal waveguides,” Phys. Rev. B **75**, 125113 (2007). [CrossRef]

*c*is defined as [10

_{m}10. D. N. Christodoulides and N. K. Efremidis, “Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals,” Opt. Lett. **27**, 568–570 (2002). [CrossRef]

*ε*(

**r**) =

*ε*′(

**r**) −

*ε*(

**r**) being the difference of dielectric constants of the waveguide (

*ε*′(

**r**)) and the point-defected PC (

*ε*(

**r**)) and

*c*

_{0}representing a small shift in the eigenfrequency

*ω*that arises from present of the neighbor defects or cavities. The self-phase modulation strength γ is given by

_{0}_{2}being the Kerr coefficient. Let the plane wave with amplitude

*ϕ*, propagation wavevector

_{0}*p*, and frequency

*ω*in site

*n*as b

_{n}= ϕ

_{0}

*exp*(

*i*n

*pa*-

*iωt*) be the solution of Eq. (1). The dispersion relation of the nonlinear PCW can be derived as

*v*

_{n}(t) superimposed on a plane wave, shown as [14

14. F. K. Abdullaev, A. Bouketir, A. Messikh, and B. A. Umarov, “Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrodinger equation,” Physica D-Nonlinear Phenomena **232**, 54–61 (2007). [CrossRef]

*v*

_{n}(t) as this form [14

14. F. K. Abdullaev, A. Bouketir, A. Messikh, and B. A. Umarov, “Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrodinger equation,” Physica D-Nonlinear Phenomena **232**, 54–61 (2007). [CrossRef]

*q*and Ω are the wavevector and frequency of the modulation perturbation.

*V*and

_{1}*V*

_{2}^{*}represent small perturbation with perturbation wavevectors of

*q*and -

*q*. Substituting

*v*into Eq.(6), we osbtioned the dispersion relation of the perturbation:

_{n}(t)## 3. Analyses and discussion

_{z}) of a single point defect, simulated by the plane wave expansion method in the square lattice with the dielectric constant, radii of dielectric rods and the radius (r

_{d}) of the defect rods being 12, 0.2a

_{L}and 0.05a

_{L}for frequency f = 0.364

*c*/

*a*is shown in Fig. 2. And the field profile along the red dash line in Fig. 2(a) is plotted in Fig. 2(b), it has the opposite sign when it extends to the nearest-neighbor defects for the PCW (

_{L}**E**

_{0}(0,0)

^{*}

**E**

_{0}(a

_{L},0) < 0) and the CROW (

**E**

_{0}(0,0)

^{*}

**E**

_{0}(3a

_{L},0) < 0) with even (2) separation rods. To maintain a single mode propagating in the waveguides, the radii or the refraction index of the rods in the waveguides is reduced therefore Δε is negative in the following discussion. Since the electric field is mainly localized around the dielectric rods of the waveguides, we can use the maximum values to replace the integral values for a simple estimation in Eq. (2). Therefore, c

_{1}is positive in even-separated-rod CROWs [16

16. T. Kamalakis and T. Sphicopoulos, “Analytical expressions for the resonant frequencies and modal fields of finite coupled optical cavity chains,” IEEE J. Quantum Electron. **41**, 1419–1425 (2005). [CrossRef]

_{2}would be two orders of magnitude smaller than c

_{1}so that we considered only the nearest-neighbor coupling in the CROWs and let c

_{2}≈ 0 [10

**27**, 568–570 (2002). [CrossRef]

18. F. S. S. Chien, J. B. Tu, W. F. Hsieh, and S. C. Cheng, “Tight-binding theory for coupled photonic crystal waveguides,” Phys. Rev. B **75**, 125113 (2007). [CrossRef]

**E**(0,0)

^{*}

**E**(2a

_{L},0) is positive in the odd-separation-rod (1) CROWs so c

_{1}would be negative and c

_{2}≈ 0.

_{2}≈ 0 for the CROWs, the coefficient

*A*can be rewritten as

*A*= 4

*c*

_{1}cos(

*pa*)sin

^{2}(

*qa*/2), in which the sign of

*A*is determined only by

*pa*and it changes sign at

*pa*= π/2. Here the region of

*pa*(or

*qa*) is defined between 0 and π. For positive (negative)

*A*, γ must also be positive (negative) and γ|

*ϕ*|

_{0}^{2}>

*A*> 0(

*γ*|

*ϕ*|

_{0}^{2}<

*A*< 0) to support MI, which can be easily derived by Eq. (9); in other words, c

_{1}cos(

*pa*)γ must be positive in MI region. Therefore, the boundary of MI must be located at

*pa*= π/2. In odd-separation-rod CROWs, c

_{1}is negative, therefore

*A*and

*γ*must be both negative when 0 <

*pa*<

*π*/2 and positive as

*pa*>

*π*/2. However, in even-separation-rod CROWs, c

_{1}is positive, therefore

*A*and

*γ*must be both positive when 0 <

*pa*< π/2 and negative as

*pa*> π/2, shown in Table 1. When the structure of the waveguide (c

_{1}) has been chosen, |

*A*| increases if

*q*increases at constant c

_{1}and

*p*. When we plot the gain profile as the graph of G vs.

*q*at a given

*p*and defined the gain maximum as the maximal values in the graph, from Eq. (9), the gain maximum would be located at

*A*= 0.5

*γ*|

*ϕ*|

_{0}^{2}and cut off at

*A*=

*γ*|

*ϕ*|

_{0}^{2}when 4|

*c*

_{1}cos(

*pa*)|> 0.5 |

*γ*||

*ϕ*

_{0}|

^{2}; otherwise, the gain maximum would be located at

*qa*= π.

_{1}for an odd-separation-rod (even-separation-rod) case, the dispersion relation slop is negative (positive) [19

19. K. Hosomi and T. Katsuyama, “A dispersion compensator using coupled defects in a photonic crystal,” IEEE J. Quantum Electron. **38**, 825–829 (2002). [CrossRef]

^{2}ω/dk

^{2}is negative (positive) when

*pa*< π/2 and positive (negative) for

*pa*> π/2 from Eq. (4). Therefore, for negative D (

*pa*< π/2 for the odd-separation-rod case and

*pa*> π/2 for the even-separation-rod case), the negative

*γ*is needed to support MI and positive

*γ*is needed to support MI for positive D. In other words, the MI regions of the CROWs in

*pa*can also be decided by simply considering the parameters of D and

*γ*.

**E**

_{n}

**E**

_{n+1}or

**E**

_{0}(0,0)

^{*}

**E**

_{0}(a

_{L},0) < 0 and

**E**

_{n}

**E**

_{n+2}> 0 in PCWs with

*a*=

_{L}*a*, therefore, c

_{1}is positive and c

_{2}, which cannot be neglected, is negative. First, we consider the positive Kerr media having positive n

_{2}(or

*γ*) so the criterion of the MI is

*γ*|

*ϕ*|

_{0}^{2}>

*A*> 0. From the criterion of

*A*= 4

*c*

_{1}cos(

*pa*)sin

^{2}(

*qa*/2) + 4

*c*

_{2}cos(2

*pa*)sin

^{2}(

*qa*) > 0, since c

_{2}is an order of magnitude smaller than c

_{1}, this criterion can be further reduced to cos(

*pa*) > -4|c

_{2}/c

_{1}|cos

^{2}(

*qa*/2). Under this circumstance, the MI region is determined not only by

*pa*but also by

*qa*, and

*pa*in the MI region can exceed π/2, unlike in CROWs that the MI boundary for

*pa*is located at π/2 and is independent of

*qa*. From the other criterion:

*γ*|

*ϕ*|

_{0}^{2}>

*A*, we found A is dominated by the c

_{1}term as

*pa*is located away from π/2, in this case the MI gain is similar to that in the CROWs with even separation rods. Contrarily, when

*pa*approaches to π/2, the c

_{1}term is almost zero and

*A*becomes dominated by the c

_{2}term. In this case,

*A*would not increase as increasing

*qa*. From Eq. (9), we knew that the maximum of the gain profile, G(q), is located at

*A*= 0.5

*γ*|

*ϕ*|

_{0}^{2}or

*dA/dq*= 0. For the latter case, the peak gain would be smaller than that of the former condition. When 4c

_{2}cos(2

*pa*) < 0.5

*γ*|

*ϕ*|

_{0}^{2}, there would be two gain maxima at a fixed

*pa*and the gain maxima is located at

*A*= 0.5

*γ*|

*ϕ*|

_{0}^{2}, but there would be only one gain maximum located at

*dA/dq*= 0 as 4c

_{2}cos(2

*pa*) < 0.5

*γ*|

*ϕ*|

_{0}^{2}.

*γ*, the first criterion is cos(

*pa*) < -4|c

_{2}/c

_{1}|• cos

^{2}(

*qa*/2). We found the MI would happen only when

*pa*> π/a. However, when 0 > cos(

*pa*) > -4|c

_{2}/c

_{1}|, the MI region is located at the higher

*q*rather than the general case in which the perturbation would have gain at

*qa*= 0

^{+}. The cutoff gain is also decided by

*A*=

*γ*|

*ϕ*|

_{0}^{2}.

## 4. Simulation results

_{L}, where a

_{L}is the lattice constant of the PCs. The radii (r

_{d}) of the defect rods are reduced to be 0.05a

_{L}and the Kerr media are introduced around the defects between one separation rod to create the CROW and sequentially to create the PCW. The structures and dispersion relations of the CROW and PCW in TM polarization (the electric field parallels the rod axis) without Kerr media are shown in Fig. 3, which are simulated by the plane wave expansion method.

_{1}is -0.00841 (2πc/

*a*), where c is the speed of light in the vacuum. Because c

_{L}_{1}is negative, the eigenfrequencies will decrease as increasing k. Figure 4(a) shows

*A*vs.

*qa*with different

*p*. Let

*A*’ be

*γ*|

*ϕ*|

_{0}*-*

^{2}*A*so that

*A*lies between 0 and

*γ*|

*ϕ*|

_{0}*and the maximum of G appears when*

^{2}*A*equals (or is the closest) to 0.5

*γ*|

*ϕ*|

_{0}*. Figure 4(b) shows G(*

^{2}*p,a*) with

*γ*|

*ϕ*|

_{0}*=0.01 (2πc/*

^{2}*a*). It can be seen that there is no MI gain when

_{L}*pa*≤ 0.5π and only a single gain maximum at given

*pa*in the condition of

*pa*> 0.576π.

_{1}and c

_{2}are 0.039 and -0.0047(2πc/a), and ω

_{0}-Δω is 0.3632 (2πc/

*a*). The values of

*A*at a given

*pa*were shown in Fig. 5(a). When

*pa*is small, i.e., in [0, 0.4π],

*A*is dominated by c

_{1}term and

*A*increases as

*qa*increases. Due to c

_{1}is positive, the properties of MI would be similar to the CROWs with even separation rods that possesses a single gain maximum as the solid curve in Fig. 6(a) for

*pa*= 0.4π. However, as

*pa*is in (0.4π, 0.6π],

*A*is not simple increasing or decreasing function of

*qa*, shown in Fig. 5(b). At a given

*pa*with positive Kerr media (γ > 0), when the values of A(q) is always smaller than 0.5

*γ*|

*ϕ*|

_{0}*, e.g.,*

^{2}*γ*|

*ϕ*|

_{0}*= 0.01 (2πc/*

^{2}*a*) and

*pa*= 0.6π, there would be a maximal gain as the solid curve in Fig. 6(d). However, when A(q) is larger than 0.5

*γ*|

*ϕ*|

_{0}*, e.g.,*

^{2}*γ*|

*ϕ*|

_{0}*=0.01 (2πc/*

^{2}*a*) and

*pa*= 0.49π and 0.55π, there would have 2 gain maxima, solid curves shown in Fig. 6(b) and (c). And the MI region with positive γ can extend to

*pa*= 0.6π, as shown in Fig. 5(c). On the other hand, the MI region with negative Kerr media is shown in Fig. 5(d) which is located within π/2 <

*pa*< π but having the MI region located at high

*qa*as

*pa*close to π/2.

^{th}order Runge-Kutta method to simulate the evolution of the perturbation. A plane wave with 10% initial sinusoidal perturbation is used as the input source in a square-array PCW with

*γ*|

*ϕ*|

_{0}*= 0.01 (2πc/*

^{2}*a*). The perturbation will grow exponentially in the MI region to become a discrete soliton before it splits, as shown in Fig. 7(a), but the perturbation would never grow outside the MI region in Fig. 7(b). We plot the gain coefficients with square dots in Fig. 6 by evaluating the growing rate by the Runge-Kutta method and then compare with gain profiles (solid curves) calculated by using Eq. (9). The results show a quite good agreement.

## 5. Conclusion

_{1}) and the next nearest-neighbor coefficient (c

_{2}) can be neglected because it is more than 2 orders of magnitude smaller than c

_{1}. This leads to positive dispersion for positive coupling coefficient and vice versa. Although the signs of the coupling coefficient could be different, the criteria: c

_{1}cos(

*pa*)γ > 0 for obtaining modulation instability is the same for incident plane wave of wavevector

*p*. Therefore, the MI region can only be located in either

*pa*< π/2 or

*pa*> π/2 with only one gain maximum. In the air-defect PCWs, c

_{1}is positive and c

_{2}, which is no longer negligible, is negative. It makes the MI gain of positive Kerr media located at low wavevectors in the first Brilluoin zone and vice versus. The boundary of gain region of

*pa*is not exactly at π/2 due to the MI is mainly dominated by c

_{2}term as

*pa*approaches π/2 and there could exist two gain maxima. Furthermore, the numerical simulation using the 4

^{th}order Runge-Kutta method reveals exponentially growing perturbation intensity as it propagates and the growing rate matches with the gain coefficient of MI in the analytic solution.

## Acknowledgments

## References and links

1. | E. Yablonovitch, “Photonic band-gap crystals,” J. Phys. Condens. Matter |

2. | E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B |

3. | D. W. Prather, S. Y. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. H. Chen, and B. L. Miao, “Photonic crystal structures and applications: Perspective, overview, and development,“ IEEE J. Sel. Top. Quantum Electron. |

4. | N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,“ Phys. Rev. B |

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

6. | A. Imhof, W. L. Vos, R. Sprik, and A. Lagendijk, “Large dispersive effects near the band edges of photonic crystals,” Phys. Rev. Lett. |

7. | W. J. Kim, W. Kuang, and J. D. O’Brien, ”Dispersion characteristics of photonic crystal coupled resonator optical waveguides,“ Opt. Express |

8. | S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, “Long-range interaction and nonlinear localized modes in photonic crystal waveguides,” Phys. Rev. E |

9. | S. F. Mingaleev and Y. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. |

10. | D. N. Christodoulides and N. K. Efremidis, “Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals,” Opt. Lett. |

11. | S. F. Mingaleev, A. E. Miroshnichenko, Y. S. Kivshar, and K. Busch, ”All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures,“ Phys. Rev. E |

12. | A. G. Shagalov, “Modulational instability of nonlinear waves in the range of zero dispersion,” Physics Lett. A |

13. | L. Hadzievski, M. Stepic, and M. M. Skoric, “Modulation instability in two-dimensional nonlinear Schrodinger lattice models with dispersion and long-range interactions,” Phys. Rev. B |

14. | F. K. Abdullaev, A. Bouketir, A. Messikh, and B. A. Umarov, “Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrodinger equation,” Physica D-Nonlinear Phenomena |

15. | F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. |

16. | T. Kamalakis and T. Sphicopoulos, “Analytical expressions for the resonant frequencies and modal fields of finite coupled optical cavity chains,” IEEE J. Quantum Electron. |

17. | S. Mookherjea, “Dispersion characteristics of coupled-resonator optical waveguides,” Opt. Lett. |

18. | F. S. S. Chien, J. B. Tu, W. F. Hsieh, and S. C. Cheng, “Tight-binding theory for coupled photonic crystal waveguides,” Phys. Rev. B |

19. | K. Hosomi and T. Katsuyama, “A dispersion compensator using coupled defects in a photonic crystal,” IEEE J. Quantum Electron. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(190.0190) Nonlinear optics : Nonlinear optics

(230.7370) Optical devices : Waveguides

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: November 17, 2008

Revised Manuscript: January 12, 2009

Manuscript Accepted: January 12, 2009

Published: January 22, 2009

**Citation**

Chih-Hsien Huang, Ying-Hsiuan Lai, Szu-Cheng Cheng, and Wen-Feng Hsieh, "Modulation instability in nonlinear coupled resonator optical waveguides and photonic crystal waveguides," Opt. Express **17**, 1299-1307 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-3-1299

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### References

- E. Yablonovitch, "Photonic band-gap crystals," J. Phys. Condens. Matter 5, 2443-2460 (1993). [CrossRef]
- E. Yablonovitch, "Photonic band-gap structures," J. Opt. Soc. Am. B 10, 283-295 (1993). [CrossRef]
- D. W. Prather, S. Y. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. H. Chen, and B. L. Miao, "Photonic crystal structures and applications: Perspective, overview, and development," IEEE J. Sel. Top. Quantum Electron. 12, 1416-1437 (2006). [CrossRef]
- N. Stefanou and A. Modinos, "Impurity bands in photonic insulators," Phys. Rev. B 57, 12127-12133 (1998). [CrossRef]
- 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]
- A. Imhof, W. L. Vos, R. Sprik, and A. Lagendijk, "Large dispersive effects near the band edges of photonic crystals," Phys. Rev. Lett. 83, 2942-2945 (1999). [CrossRef]
- W. J. Kim, W. Kuang, and J. D. O'Brien, "Dispersion characteristics of photonic crystal coupled resonator optical waveguides," Opt. Express 11, 3431-3437 (2003). [CrossRef] [PubMed]
- S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, "Long-range interaction and nonlinear localized modes in photonic crystal waveguides," Phys. Rev. E 62, 5777-5782 (2000). [CrossRef]
- S. F. Mingaleev, and Y. S. Kivshar, "Self-trapping and stable localized modes in nonlinear photonic crystals," Phys. Rev. Lett. 86, 5474-5477 (2001). [CrossRef] [PubMed]
- D. N. Christodoulides and N. K. Efremidis, "Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals," Opt. Lett. 27, 568-570 (2002). [CrossRef]
- S. F. Mingaleev, A. E. Miroshnichenko, Y. S. Kivshar, and K. Busch, "All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures," Phys. Rev. E 74, 046603 (2006). [CrossRef]
- A. G. Shagalov, "Modulational instability of nonlinear waves in the range of zero dispersion," Physics Lett. A 239, 41-45 (1998). [CrossRef]
- L. Hadzievski, M. Stepic, and M. M. Skoric, "Modulation instability in two-dimensional nonlinear Schrodinger lattice models with dispersion and long-range interactions," Phys. Rev. B 68, 014305 (2003). [CrossRef]
- F. K. Abdullaev, A. Bouketir, A. Messikh, and B. A. Umarov, "Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrodinger equation," Physica D 232, 54-61 (2007). [CrossRef]
- F. M. Mitschke and L. F. Mollenauer, "Discovery of the soliton self-frequency shift," Opt. Lett. 11, 659-661 (1986). [CrossRef] [PubMed]
- T. Kamalakis and T. Sphicopoulos, "Analytical expressions for the resonant frequencies and modal fields of finite coupled optical cavity chains," IEEE J. Quantum Electron. 41, 1419-1425 (2005). [CrossRef]
- S. Mookherjea, "Dispersion characteristics of coupled-resonator optical waveguides," Opt. Lett. 30, 2406-2408 (2005). [CrossRef] [PubMed]
- F. S. S. Chien, J. B. Tu, W. F. Hsieh, and S. C. Cheng, "Tight-binding theory for coupled photonic crystal waveguides," Phys. Rev. B 75, 125113 (2007). [CrossRef]
- K. Hosomi and T. Katsuyama, "A dispersion compensator using coupled defects in a photonic crystal," IEEE J. Quantum Electron. 38, 825-829 (2002). [CrossRef]

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