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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 25 — Dec. 6, 2010
  • pp: 25509–25518
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Low-threshold bistability in nonlinear microring tower resonator

Mehdi Shafiei and Mohammad Khanzadeh  »View Author Affiliations


Optics Express, Vol. 18, Issue 25, pp. 25509-25518 (2010)
http://dx.doi.org/10.1364/OE.18.025509


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Abstract

Microring tower resonators, which are a chain of microring resonators stacked on top of each other, are of great interest for nonlinear optics due to their unique features such as very high compactness, coupling efficiency and quality factor. In this research, we investigate the optical bistability in microring tower (MRT) with Kerr nonlinearity by using the coupled mode theory, and demonstrate how a proper defect into the structure can lead to low threshold bistability. In particular, we observed optical bistability in nonlinear defect modes with switching power as low as 165 μ W through numerical calculations in a structure with a overall loss on the order of 0.01 m m 1 . In addition, we also develop an analytical model that excellently gives the position of defect modes in linear regime.

© 2010 OSA

1. Introduction

Recently, optical bistability has been attracted increasing attention due to several applications in fast all-optical devices; including switches, logic gates, transistors, flip-flops, and optical memories [1

1. H. Gibbs, Optical Bistability: Controlling Light with Light (Academic Press, Orlando, 1985).

5

5. M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601–055604 (2002). [CrossRef]

]. Microring arrays, among a large number of devices that provide bistable behavior, have more interesting features because these structures can simultaneously take the unique advantages of the microring resonators and photonic crystal structures [6

6. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997). [CrossRef]

, 7

7. K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

]. In this paper, we focus on vertically stacked multi-ring resonator [8

8. M. Sumetsky, “Vertically-stacked multi-ring resonator,” Opt. Express 13(17), 6354–6375 (2005). [CrossRef] [PubMed]

], illustrated in Fig. 1(a)
Fig. 1 a) Sketch of MRT with N rings. b) Schematic of an infinite periodic VMR structure with a defect.
, and demonstrate wavelength and power bistability in these 3D arrays. In this paper, this configuration will be called microring tower (MRT) resonator.

2. Linear response

2.1. Simple model for fast calculation of cavity- modes

Inspiring from [10

10. D. N. Christodoulides and E. D. Eugenieva, “Minimizing bending losses in two-dimensional discrete soliton networks,” Opt. Lett. 26(23), 1876–1878 (2001). [CrossRef]

], we first suggest a theoretical model for defect modes in an infinite array of stacked and coupled microrings. This model is applicable to linear regime of a MRT which we abbreviate it as LMRT. Thus, let us consider an infinite array of vertically coupled microrings with the same radii, R. Making use of coupled mode theory, the fields can be described by following equations for the amplitudes An(s) in the nth ring [9

9. M. Shafiei, M. Khanzadeh, M. Agha-Bolorizadeh, and R. F. Moghaddam, “Linear transmission properties of a vertically stacked multiring resonator with a defect,” Appl. Opt. 48(31), G148–G155 (2009). [CrossRef] [PubMed]

]:
idAn/ds+κn1,nAn1ei(βn1βn)s+κn,n+1An+1ei(βn+1βn)s=0,n=0,±1,±2,
(1)
where βn is the propagation constant of the mode in the nth ring, κn1,n is the coupling coefficient between adjoining rings, and s is the coordinate along each ring. Furthermore, the amplitude An(s) at nth ring satisfies the condition of closure of rings as follows:
An(0)=An(L)exp(iβnL),
(2)
where L=2πR. Now, consider a uniform structure with the same coupling coefficients and propagation constants, (κn1,n=κ,βn=β), apart from a single defect located at n=0 (as shown in Fig. 1(b)), which its propagation constant differs from the rest by δβdef. Moreover, we assume that the coupling coefficient between the defect site and its nearest neighbors is κ. Making use of variable changes a0(s)=A0(s)exp(iδβdefs) and an(s)=An(s) for n0, Eq. (1) for the middle three sites can be rewritten as follows:
{ida1/ds+κa0+κa2=0ida0/ds+δβdefa0+κ(a1+a1)=0ida1/ds+κa0+κa2=0.
(3)
The field propagation in a uniform infinite array is described by exp(iμs+iξzn) where ξ is effective propagation constant along the vertical direction, μ=2κcos(ξd), zn=nd, and d is the period of the structure [9

9. M. Shafiei, M. Khanzadeh, M. Agha-Bolorizadeh, and R. F. Moghaddam, “Linear transmission properties of a vertically stacked multiring resonator with a defect,” Appl. Opt. 48(31), G148–G155 (2009). [CrossRef] [PubMed]

]. Therefore the following solution may be proposed:
an={eiμs(eiξzn+reiξzn)   ​,  n<0qeiμs                                  ,  n=0teiμseiξzn                         ​,n>0
(4)
where t and r, to be determined, are transmission and reflection coefficients associated with a forward and reflected wave, respectively. Substituting Eq. (4) into Eqs. (3) and applying some straightforward manipulations, one arrives at the following equations for the coefficients r and q:
r=t1,q=κt/κ,
(5)
where
t=2iκ2sinξd2(κ2κ2)cosξd+2iκ2sinξd+κδβdef.
(6)
In steady state, we can also expect that the defect modes assure the conditions of ring closure and therefore satisfy the following dispersion relation [9

9. M. Shafiei, M. Khanzadeh, M. Agha-Bolorizadeh, and R. F. Moghaddam, “Linear transmission properties of a vertically stacked multiring resonator with a defect,” Appl. Opt. 48(31), G148–G155 (2009). [CrossRef] [PubMed]

]:
B=2πl2Kcos(ξd),
(7)
where K=κL, B=βL and l is an integer number. Note that for defect modes placed in the bandgap, the propagation constantξ is imaginary. Inserting the expression for sin(ξd) from Eq. (7) into Eq. (6) one obtains
t=K2[(B2πl)24K2]1/2(K2K2)(2πlB)K2[(B2πl)24K2]1/2+K2δBdef,
(8)
where, K=κL. Defect modes are obtained by Eq. (8) when the denominator is set to zero. Accordingly, the defect modes are given by the following simple equation:
Bdef=2πlδBdef(K2K2)±K2δBdef24(K22K2)K22K2.
(9)
In the case of κ=κ this equation reduces to Bdef=2πl±δBdef2+4K2. The two solutions conform to the two possible defect modes in each bandgap; the positive (negative) sign represent the modes in the left (right) of the band with δβdef<0 (δβdef>0). For the defect modes the coefficients t and r are infinitely large, so Eq. (4) shows that the field is localized at the defect position and decays exponentially away with the rate of exp(|iξzn|). These situations are in agreement with results in ref [9

9. M. Shafiei, M. Khanzadeh, M. Agha-Bolorizadeh, and R. F. Moghaddam, “Linear transmission properties of a vertically stacked multiring resonator with a defect,” Appl. Opt. 48(31), G148–G155 (2009). [CrossRef] [PubMed]

]. In order to determine the appropriateness of the model we compare it with numerical calculations. The results are shown in Fig. 2
Fig. 2 Defect modes Bdeff versus δBdeffor two different defect modes: (right) positive and (left) negative defects. The Curves denote the model results assuming l=68; and K = 0.8. Filled circles denote the numerical results where the number of rings is 9 and the coupling parameter between the input waveguide and its adjacent ring, K0, is 0.8.
. Except for small defects, there is an excellent agreement between numerical calculations and the model. Note that, in numerical calculations the number of the rings is inevitably finite. However, for sufficiently high number of the rings its behaviors are very similar to the infinite one [9

9. M. Shafiei, M. Khanzadeh, M. Agha-Bolorizadeh, and R. F. Moghaddam, “Linear transmission properties of a vertically stacked multiring resonator with a defect,” Appl. Opt. 48(31), G148–G155 (2009). [CrossRef] [PubMed]

].

2.1. Influence of cavity parameters on the Q factor

Now let us discuss one important cavity property that enhances processes in nonlinear optics, namely very high quality factor. To study the dependence of the Q factor to the cavity geometry, we first study the influence of δβdefon the Q factor. The results are shown in Fig. 3(a)
Fig. 3 dependence of the Q factor a) on the δβdef, and b) on the K0 and K.
where we consider a uniform MRT with 9 rings and coupling parameters K0=K=0.6,and see that for larger defects, the Q factor is higher. This can readily explain by considering the position of the defect modes in the bandgap. By increasing the defect size, the position of the defect modes goes far from the band edges. As the defect mode approaches the band edges, the linewidth broadens so that the defect mode has a smaller Q. Next, we study the influence of the coupling parameters. Calculation show that if the coupling parameter between adjacent microrings is equal to the coupling parameter between input/output waveguides and adjacent microring, i. e. K0=K, the Q factor has a minimum value (see Fig. 3(b)) whilst increasing the coupling parameter difference, |K0K|, noticeably increases the quality factor.

Also, the Q factor strongly depends on the number of the microrings. For example, for the choice of K=K0=0.8, and δBdefect=1.6, while for N=9, Q=4.25×104,for N=11, calculations show that Q=3×105. we attribute this improvement mainly to the fact that the structure with more microring give better transmission characteristics and simulate ideal infinite structures more exactly.

3. Nonlinear response

3.1. Coupled wave equations

3. 2. Nonlinear transmission properties

Being equipped with a proper method, the properties of NMRT are investigated. In this section, we ignore the loss in the Eq. (10)-(12) to identify the most general physical effects. Also, we use material parameters of AlGaAs at λ=1.55   μmwith n2=1.5×1013cm2W1 [15

15. A. Villeneuve, C. Yang, G. Stegeman, C.-H. Lin, and H.-H. Lin, “Nonlinear refractive-index and two photon-absorption near half the band gap in AlGaAs,” Appl. Phys. Lett. 62(20), 2465–2467 (1993). [CrossRef]

]. For conventional microring size the mode effective area is about 1   μm2. And hence, γ60   m1W1. At first, we consider a uniform NMRT consisting of 9 rings with coupling parameters K0=0.2,and K=2. Fig. 4(a)
Fig. 4 a) Detailed spectrum of a resonant mode at different input powers for a lossless uniform MRT with N = 9, K0=0.2,and K=2. b) Output power (Pout)versus input power (Pin) for various detuning parameters.
shows the transmission around a resonant mode with Q1.6×104 for deferent input powers. As can be seen, increasing the input power results in asymmetric resonance transmission pattern. From the steep declines in the transmission, it can be seen that optical bistability is obtained for powers of approximately 90mW and above for this configuration. In Fig. 4(b) we plot the input power versus output power while fixed input wavelength for different detuning parameters δ=BinB0, where B0 and Bin are normalized propagation constant of linear resonance mode and input field respectively. Note that, the device in this mode shows counterclockwise hysteresis cycles. As the detuning parameter increase, the bistability threshold and the width of the hysteresis cycle reduce.

Since, it is desirable an all-optical device to operate at a very low power, let us consider how can lower bistability threshold in our device. In order to achieve very low bistability threshold, we use high Q-factor defect modes. An example is shown in Fig. 6
Fig. 6 a) A section of transmission spectrum of a lossless MRT (N=9,K=K0=0.8, and δBdefect=1.6) for low input powers, a sharp dip shows a defect mode near B=424.998. b) Nonlinear transmission for different input powers. c) - d) Input/Output characteristic for two different δ.
, where we consider an NMRT consisting of 9 rings with coupling parameters K=K0=0.8, in which the dimensionless propagation constant of the center ring at n=5 differs from those of others rings by δBdefect=1.6. A section of the through transmission spectrum in linear or a low power case is shown in Fig. 6(a). A sharp resonant mode with quality factor of Q=4.25×104 can be seen inside the gap nearB=424.998. Figure 6(b) shows the details of transmission around the defect mode for different input powers, Pin, in nonlinear case with γ=60. One can observe, as Pin increases, the defect mode shifts to the left or according to B=βL1/λ shifts to longer wavelengths. Also, for defect modes located at the left side of the gap; i.e. δBdefect<0, calculations show that the red-shift occurs again. These red-shifts can readily be explained by considering the Kerr effect and localization of the field in the defect site. Also, as shown in Fig. 6(b) the bistability threshold is about 10 mW. furthermore, the power transmission in Fig. (c) - (d) show that again, we have multiple hysteresis cycle for sufficiently high δ. This behavior is useful in some types of optical switches and memories.

4. Effects of the loss upon device performance

In the following, we briefly investigate the effects of loss on quality factor and bistability threshold. To avoid unnecessary complications, we assume that coupling between waveguides and adjacent rings are lossless. The filled circle in Fig. 8(a)
Fig. 8 a) Filled circles show the calculated quality factors with different losses for the microcavity of the Fig. 7(a). The solid line is the curve 8.51/(0.13α+1). b) The relationship between bistability threshold and loss. The solid line is the curve c×α2 with appropriate constant c.
shows the calculated relationship between the quality factor and the loss for the microcavity of the Fig. 7(a). These circles can be fitted very well with an inverse decay curve. As similar to the cases of a ring resonator and microcoil resonators [6

6. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997). [CrossRef]

, 21

21. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B 21(9), 1665–1673 (2004). [CrossRef]

23

23. F. Xu, P. Horak, and G. Brambilla, “Optical microfiber coil resonator refractometric sensor,” Opt. Express 15(12), 7888–7893 (2007). [CrossRef] [PubMed]

], this result approximately suggests a relation of the form Q1/α between the quality factor and the loss (especially for α1). The constant of the proportionality depend on the geometrical parameters (e. g. the number of the rings) and the chosen eigen mode.

Figure 8(b) depicts the variation of the bistability threshold versus the loss. The calculated values can be fitted well with a binomial curve with a second order term. Therefore, the relation between the bistability threshold and loss can be approximated in the form ofptrα2, which agrees well with the result ptrQ-2 in photonic crystal microcavities [24

24. L. D. Haret, T. Tanabe, E. Kuramochi, and M. Notomi, “Extremely low power optical bistability in silicon demonstrated using 1D photonic crystal nanocavity,” Opt. Express 17(23), 21108–21117 (2009). [CrossRef] [PubMed]

]. Assuming that the loss α in the example of the Fig. 7(a) is 0.01mm1 (which is considerably smaller than those of used in the Refs [22

22. M. Sumetsky, “Optical fiber microcoil resonators,” Opt. Express 12(10), 2303–2316 (2004). [CrossRef] [PubMed]

, 23

23. F. Xu, P. Horak, and G. Brambilla, “Optical microfiber coil resonator refractometric sensor,” Opt. Express 15(12), 7888–7893 (2007). [CrossRef] [PubMed]

].), we find that the bistability threshold is Ptr=165μW.

5. Conclusion

In this paper we present an alternative bistable structure based on the vertically stacked microring arrays. This compact configuration is of interest to nonlinear processes and slow light applications, which has often been accomplished in CROWs. The implementation of the microring resonators along with the defect modes with very high quality factors significantly reduces the bistable threshold in these configurations. We observed optical bistability at very low input power of 165μW. The similarity between the nonlinear coupled wave equation in this structure and the DNSE is an interesting feature that can lead to many useful phenomena.

Owing to the small spatial period of MRT in comparison with similar planar structures such as CROWs, this structure exhibit lower group velocity and subsequently can improve the efficiency of the nonlinear processes much better. The constant coupling coefficient along entire length of each resonator makes it easy to analyze. Furthermore, this removes the challenging short coupling length problems that usually appear in similar structures such as CROWs and SISCORs.

We also introduce a theoretical model in linear regime for finding defect modes in the transmission spectrum. This model is in very good agreement with numerical calculations. The algebraic formula that is presented makes it possible to find defect modes exactly, without any need to the numerical calculations. We also study the dependence of the quality factor on the cavity geometry. We find in our calculations that if the cavity parameters carefully be chosen the quality factor can noticeably increase. Specially, the quality factor strongly depends on the number of the microrings and the magnitude of the defect.

Finally, it should be noted that many features of this structure are not known at this stage. Therefore the results obtained in this work are at the beginning and we expect that its many features warrant further investigations.

Acknowledgment

The authors gratefully acknowledge Mohammad Agha-bolorizadeh and Raza Farrahi-Moghaddam for fruitful discussions and their help in the course of this work. This research was supported by the Vali-e-Asr University of Rafsanjan under grant No. P. 4561.

References and links

1.

H. Gibbs, Optical Bistability: Controlling Light with Light (Academic Press, Orlando, 1985).

2.

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13(7), 2678–2687 (2005). [CrossRef] [PubMed]

3.

A. Hurtado, A. Quirce, A. Valle, L. Pesquera, and M. J. Adams, “Power and wavelength polarization bistability with very wide hysteresis cycles in a 1550 nm-VCSEL subject to orthogonal optical injection,” Opt. Express 17(26), 23637–23642 (2009). [CrossRef]

4.

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “Fast bistable all-optical switch and memory on a silicon photonic crystal on-chip,” Opt. Lett. 30(19), 2575–2577 (2005). [CrossRef] [PubMed]

5.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601–055604 (2002). [CrossRef]

6.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997). [CrossRef]

7.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

8.

M. Sumetsky, “Vertically-stacked multi-ring resonator,” Opt. Express 13(17), 6354–6375 (2005). [CrossRef] [PubMed]

9.

M. Shafiei, M. Khanzadeh, M. Agha-Bolorizadeh, and R. F. Moghaddam, “Linear transmission properties of a vertically stacked multiring resonator with a defect,” Appl. Opt. 48(31), G148–G155 (2009). [CrossRef] [PubMed]

10.

D. N. Christodoulides and E. D. Eugenieva, “Minimizing bending losses in two-dimensional discrete soliton networks,” Opt. Lett. 26(23), 1876–1878 (2001). [CrossRef]

11.

K. Okamoto, Fundamentals of Optical Waveguides, (Elsevier, 2006), Chap. 4.

12.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. 18(10), 1580–1583 (1982). [CrossRef]

13.

D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13(9), 794–796 (1988). [CrossRef] [PubMed]

14.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998). [CrossRef]

15.

A. Villeneuve, C. Yang, G. Stegeman, C.-H. Lin, and H.-H. Lin, “Nonlinear refractive-index and two photon-absorption near half the band gap in AlGaAs,” Appl. Phys. Lett. 62(20), 2465–2467 (1993). [CrossRef]

16.

A. Shinya, S. Matsuo, T. Yosia, E. Tanabe, T. Kuramochi, T. Sato, Kakitsuka, and M. Notomi, “All-optical on-chip bit memory based on ultra high Q InGaAsP photonic crystal,” Opt. Express 16(23), 19382–19387 (2008). [CrossRef]

17.

H. Zhang, V. Gauss, P. Wen, and S. Esener, “Observation of wavelength and multiple bistabilities in 850nm Vertical-Cavity Semiconductor Optical Amplifiers (VCSOAs),” Opt. Express 15(18), 11723–11730 (2007). [CrossRef] [PubMed]

18.

E. Weidner, S. Combri’e, A. de Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101–118 (2007). [CrossRef]

19.

G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express 13(23), 9623–9628 (2005). [CrossRef] [PubMed]

20.

N. G. R. Broderick, “Optical snakes and ladders: dispersion and nonlinearity in microcoil resonators,” Opt. Express 16(20), 16247–16254 (2008). [CrossRef] [PubMed]

21.

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B 21(9), 1665–1673 (2004). [CrossRef]

22.

M. Sumetsky, “Optical fiber microcoil resonators,” Opt. Express 12(10), 2303–2316 (2004). [CrossRef] [PubMed]

23.

F. Xu, P. Horak, and G. Brambilla, “Optical microfiber coil resonator refractometric sensor,” Opt. Express 15(12), 7888–7893 (2007). [CrossRef] [PubMed]

24.

L. D. Haret, T. Tanabe, E. Kuramochi, and M. Notomi, “Extremely low power optical bistability in silicon demonstrated using 1D photonic crystal nanocavity,” Opt. Express 17(23), 21108–21117 (2009). [CrossRef] [PubMed]

OCIS Codes
(190.1450) Nonlinear optics : Bistability
(230.4320) Optical devices : Nonlinear optical devices

ToC Category:
Nonlinear Optics

History
Original Manuscript: August 23, 2010
Revised Manuscript: November 11, 2010
Manuscript Accepted: November 12, 2010
Published: November 22, 2010

Citation
Mehdi Shafiei and Mohammad Khanzadeh, "Low-threshold bistability in nonlinear microring tower resonator," Opt. Express 18, 25509-25518 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25509


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References

  1. H. Gibbs, Optical Bistability: Controlling Light with Light (Academic Press, Orlando, 1985).
  2. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13(7), 2678–2687 (2005). [CrossRef] [PubMed]
  3. A. Hurtado, A. Quirce, A. Valle, L. Pesquera, and M. J. Adams, “Power and wavelength polarization bistability with very wide hysteresis cycles in a 1550 nm-VCSEL subject to orthogonal optical injection,” Opt. Express 17(26), 23637–23642 (2009). [CrossRef]
  4. T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “Fast bistable all-optical switch and memory on a silicon photonic crystal on-chip,” Opt. Lett. 30(19), 2575–2577 (2005). [CrossRef] [PubMed]
  5. M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601–055604 (2002). [CrossRef]
  6. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997). [CrossRef]
  7. K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).
  8. M. Sumetsky, “Vertically-stacked multi-ring resonator,” Opt. Express 13(17), 6354–6375 (2005). [CrossRef] [PubMed]
  9. M. Shafiei, M. Khanzadeh, M. Agha-Bolorizadeh, and R. F. Moghaddam, “Linear transmission properties of a vertically stacked multiring resonator with a defect,” Appl. Opt. 48(31), G148–G155 (2009). [CrossRef] [PubMed]
  10. D. N. Christodoulides and E. D. Eugenieva, “Minimizing bending losses in two-dimensional discrete soliton networks,” Opt. Lett. 26(23), 1876–1878 (2001). [CrossRef]
  11. K. Okamoto, Fundamentals of Optical Waveguides, (Elsevier, 2006), Chap. 4.
  12. S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. 18(10), 1580–1583 (1982). [CrossRef]
  13. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13(9), 794–796 (1988). [CrossRef] [PubMed]
  14. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998). [CrossRef]
  15. A. Villeneuve, C. Yang, G. Stegeman, C.-H. Lin, and H.-H. Lin, “Nonlinear refractive-index and two photon-absorption near half the band gap in AlGaAs,” Appl. Phys. Lett. 62(20), 2465–2467 (1993). [CrossRef]
  16. A. Shinya, S. Matsuo, T. Yosia, E. Tanabe, T. Kuramochi, T. Sato, Kakitsuka, and M. Notomi, “All-optical on-chip bit memory based on ultra high Q InGaAsP photonic crystal,” Opt. Express 16(23), 19382–19387 (2008). [CrossRef]
  17. H. Zhang, V. Gauss, P. Wen, and S. Esener, “Observation of wavelength and multiple bistabilities in 850nm Vertical-Cavity Semiconductor Optical Amplifiers (VCSOAs),” Opt. Express 15(18), 11723–11730 (2007). [CrossRef] [PubMed]
  18. E. Weidner, S. Combri’e, A. de Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101–118 (2007). [CrossRef]
  19. G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express 13(23), 9623–9628 (2005). [CrossRef] [PubMed]
  20. N. G. R. Broderick, “Optical snakes and ladders: dispersion and nonlinearity in microcoil resonators,” Opt. Express 16(20), 16247–16254 (2008). [CrossRef] [PubMed]
  21. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B 21(9), 1665–1673 (2004). [CrossRef]
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