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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 5 — Mar. 3, 2008
  • pp: 3069–3076
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Symmetry breaking with coupled Fano resonances

Björn Maes, Peter Bienstman, and Roel Baets  »View Author Affiliations


Optics Express, Vol. 16, Issue 5, pp. 3069-3076 (2008)
http://dx.doi.org/10.1364/OE.16.003069


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Abstract

We describe the effect of symmetry breaking in a system with two coupled Fano resonators. A general criterion is derived and optimal parameter regions for switching are identified. By extending the single resonator behavior we show that one achieves the effect at very low input powers.

© 2008 Optical Society of America

1. Introduction

The concept of a Fano resonance is very general [1

1. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Physical Review 124, 1866 (1961). [CrossRef]

]. It can arise in the context of an interaction between a localized or discrete resonance with a background or continuous channel. If the background is completely reflecting or transmitting, the effect of the resonance is a Lorentzian peak or dip. If the background is partially reflecting, the more general transition with two extrema is obtained (see e.g. Fig. 2(b)).

Fig. 1. (a) Schematic of the two-cavity device (bottom) and electric field plot of a device with three blocking rods per cavity (top). (b) Transmission T (black line) and reflection amplitude r=1T (red line) versus number of blocking rods in the PhC waveguide at wavelength a/0.367.

In the optical domain, these resonances are commonplace, as beams in free-space or wave-guides couple with cavities. The single peak Lorentzian case is most studied, both in the linear and nonlinear domain for filtering and switching purposes, respectively. It was realized that the more general resonances lead to different switching possibilities in one-cavity devices [2

2. V. Lousse and J.P. Vigneron, “Use of Fano resonances for bistable optical transfer through photonic crystal films,” Phys. Rev. E 69, 155106 (2004).

]. Coupling of multiple Lorentz-type cavities also proved to be a promising field for nonlinear functionality [3

3. J.E. Heebner, R.W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19, 722–731 (2002). [CrossRef]

, 4

4. B. Maes, P. Bienstman, and R. Baets, “Switching in coupled nonlinear photonic-crystal resonators,” J. Opt. Soc. Am. B 22, 1778–1784 (2005). [CrossRef]

].

Recently we proposed a switching device that employs a symmetry-breaking bifurcation [5

5. B. Maes, M. Soljačić, J.D. Joannopoulos, P. Bienstman, R. Baets, S-P. Gorza, and M. Haelterman, “Switching through symmetry breaking in coupled nonlinear micro-cavities,” Opt. Express 14, 10678–10683 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-22-10678. [CrossRef] [PubMed]

]. This breaking occurs because of the feedback between two nonlinear cavities. In the linear regime the structure is left-right symmetric, see Fig. 1(a), and we excite the system with equal powers from both sides. Linearly this results in equal output powers to the left and to the right. At higher powers, however, an instability can appear, and the structure becomes asymmetric: one cavity will be more excited than the other, resulting in index differences through the Kerr effect, leading to feedback away from the symmetric solution. In such a structure the two equal input powers result in two different output powers, as the fields of both inputs interfere constructively on one side and destructively on the other.

Because of the symmetry of the linear system, there are always two equivalent asymmetric states: one where the left output power is larger than the right output power, and the mirrored state, where the right output is larger than the left output. By increasing or decreasing one of the inputs, it is possible to switch between these two states. Thus, this scheme provides the option of switching with positive pulses, which is not obvious in single-cavity devices. In addition, the dynamical interaction between two cavities can be tuned judiciously, so interesting regimes are available, as we show later.

In this paper we broaden the study of the symmetry-breaking device by considering the entire class of Fano resonances, instead of the single Lorentzian case. A general analytical criterion for the existence of the effect is derived. The generalization leads to additional optimization possibilities, and we show e.g. that low-power effects are feasible.

We mainly consider the context of photonic crystal (PhC) waveguides and cavities. These structures offer the possibility for highly compact, efficient devices. Single cavity switching has already been demonstrated experimentally [6

6. P.E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express 13, 801–820 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-3-801. [CrossRef] [PubMed]

, 7

7. 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, 2678–2687 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-7-2678. [CrossRef] [PubMed]

, 8

8. T. Uesugi, B-S. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express 14, 377–386 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-1-377. [CrossRef] [PubMed]

]. The results are more general however, as the generic coupled-mode theory (CMT) is employed. Indeed, Fano resonances have also been demonstrated in other contexts such as ring resonators [9

9. Y. Lu, J. Yao, X. Li, and P. Wang, “Tunable asymmetrical Fano resonance and bistability in a microcavity-resonator-coupled Mach-Zehnder interferometer,” Opt. Lett. 30, 3069–3071 (2005). [CrossRef] [PubMed]

] and PhC films [2

2. V. Lousse and J.P. Vigneron, “Use of Fano resonances for bistable optical transfer through photonic crystal films,” Phys. Rev. E 69, 155106 (2004).

].

Fig. 2. Transmission T in three different cavity structures: (a) no blocking rods (Tbackground≈1); (b) one blocking rod with side 0.2a (Tbackground≈0.5); (c) three blocking rods (Tbackground≈0).

2. Fano resonances

A PhC system with a single-cavity coupled to a waveguide (one side of the field plot in Fig. 1(a)) can show the whole range of Fano resonances [10

10. S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80, 908–910 (2002). [CrossRef]

]. The shape of the transmission curve in this configuration is determined by the waveguide properties. This direct channel is adjustable by adding or removing rods in the waveguide. Fine-tuning is possible by changing rod radii. The result of such a modeling exercise is shown in Fig 1(b). This presents the transmission T through the waveguide, without the cavity, in function of the ‘number’ of blocking rods. If this number is between 0 and 1 we increase the radius of a single blocking rod. In contrast, between 1 and 3, we increase the radii of two rods adjacent to a rod which is already the same size as the lattice rods. The discontinuous connection follows from this approach. Clearly, the system is tunable between complete and zero transmission.

The modeled two-dimensional PhC system is embedded in a square lattice with period a of square rods with side 0.4a. The rods have index 3.5 in an air background. The waveguide is formed by removing a row of rods. The cavity is created by a rectangular defect rod with dimensions a×0.48a. Calculations are performed with the mode expansion method [11

11. P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001). [CrossRef]

].

3. General criterion

The coupled-mode equations for this structure are [12

12. S. Fan, W. Suh, and J.D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003). [CrossRef]

, 4

4. B. Maes, P. Bienstman, and R. Baets, “Switching in coupled nonlinear photonic-crystal resonators,” J. Opt. Soc. Am. B 22, 1778–1784 (2005). [CrossRef]

]:

dajdt=[i(ω0+δωj)1τ]aj+dfj+dbj+1,
(1)
[bjfj+1]=exp(iϕ)[rititr]·[fjbj+1]+daj[11],
(2)

for j=1, 2. The complex aj are cavity mode amplitudes, so |aj|2 is the energy in the mode. The fj and bj are waveguide mode amplitudes, thus |fj|2 (resp. |bj|2) represents the power flowing in the (single-mode)waveguide in the forward (resp. backward) direction, see Fig. 1(a). Both cavities have the same resonance frequency ω 0 and lifetime τ. The nonlinear frequency shift is δωj=-|aj|2/(P 0τ2), with P 0 the ‘characteristic nonlinear power’ of the cavities [13

13. M. Soljačić, M. Ibanescu, S.G. Johnson, Y. Fink, and J.D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E66, 055601(R) (2002).

], which we discuss below. The coupling parameter in this system is d=iexp(i(ϕ+β)/2)/√τ, where β=atan(t/r). The real constants ϕ, t and r determine the direct channel properties. The reflection and transmission are coupled by r 2+t 2=1.

The input power on the left side is PLin≡|f 1|2, on the right side we have as input PRin≡|b 3|2. In this paper the input powers are equal, so we note PinPLin=PRin. The output powers are PLout ≡|b 1|2 and PRout≡|f 3|2 on the left and right side, respectively. For symmetry-breaking states one has PLoutPRout.

The parameter P 0 allows to evaluate the power levels for the operation of nonlinear devices. In [13

13. M. Soljačić, M. Ibanescu, S.G. Johnson, Y. Fink, and J.D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E66, 055601(R) (2002).

] it is defined for a single cavity as P 0=c/(κQ 2 ω 0 n 2), with c the speed of light, Q=ω 0τ/2 the quality factor, and n 2 the nonlinear Kerr material coefficient. κ is a dimensionless nonlinear strength parameter, which is proportional to the overlap of the resonator mode with the nonlinear material region [13

13. M. Soljačić, M. Ibanescu, S.G. Johnson, Y. Fink, and J.D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E66, 055601(R) (2002).

]. We normalize all our powers with respect to P 0. Indeed, this P 0 indicates the power level at which nonlinear effects appear in single cavity nonlinear devices. E.g. in a structure with a single cavity that is blocked (r=1), one can show that the minimum input power for the standard bistability effect is √3P 0 [13

13. M. Soljačić, M. Ibanescu, S.G. Johnson, Y. Fink, and J.D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E66, 055601(R) (2002).

]. Further on we will obtain that significantly lower switching powers are achievable in the two-cavity devices.

In the following we obtain a criterion for this general system. It determines for which frequencies one expects asymmetric states. The analysis is an extension of the method in [5

5. B. Maes, M. Soljačić, J.D. Joannopoulos, P. Bienstman, R. Baets, S-P. Gorza, and M. Haelterman, “Switching through symmetry breaking in coupled nonlinear micro-cavities,” Opt. Express 14, 10678–10683 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-22-10678. [CrossRef] [PubMed]

]. We work in the frequency domain so d/dt becomes . Eliminating f 2 and b 2 we obtain:

[i(ω0ω+δω1)1τ]a1+κd(c1a1+a2)=(d+κc1c2)f1κc2b3,
(3)
[i(ω0ω+δω2)1τ]a2+κd(c1a2+a1)=(d+κc1c2)b3κc2f1,
(4)

with c 1=rexp(), c 2=itexp() and κ=d/(1-c 2 1).We equate the left sides of Eqs. 3 and 4, as we assume equal input power and phase from both sides, so f 1=b 3. We take the modulus squared, and after factoring we get:

(AB)[(A2+AB+B2)+2(Δ+Γ1)(A+B)+Δ2+2ΔΓ1+Γ2]=0,
(5)
Fig. 3. (a) Critical detunings Δ± for r=1 (black lines), r=0.8 (blue lines) and r=0 (red lines), respectively. The largest value is Δ+, the smallest corresponds to Δ-. (b) Minimum input power Pminin and corresponding Pmaxin versus ϕ for r=1.0 (black lines) and r=0.8 (red lines), respectively. For these results Pminin was minimized. All powers in units of P 0.

with dimensionless cavity energies A=α|a 1|2 and B=α|a 2|2, where α=-1/(P 0τ). In addition, the detuning is Δ=τ(ω 0-ω) and

Γ1=(rt+rsinϕ+tcosϕ)(1+2rcosϕ+r2),
(6)
Γ2=(1+2tsinϕ+t2)(1+2rcosϕ+r2).
(7)

Starting with the discriminant from Eq. 5 one obtains the critical detunings

Δ±=Γ1±3(Γ2Γ12).
(8)

4. Optimization

In this section we provide insight into optimal parameter choices. In addition we show how some features of the nonlinear switching behavior are correlated with the linear properties.We note that this system has many parameters with a profound influence on the characteristics. Furthermore, the figure of merit that one may define depends on the envisaged application.

In this paper our first focus is on the minimum input power Pminin that is needed to observe symmetry breaking states. Above this power, the symmetric solution becomes unstable. Subsequently, we examine the maximum input power Pmaxin at which point the nonlinear asymmetric ‘eye’ closes (see e.g. Fig.4(a), also discussed below). Above this input power, the symmetric solution is stable again. Thus, asymmetric states are observable in the range of input powers between Pminin and Pmaxin. If this Pmaxin is too large, it will be difficult to switch by using positive pulses. On the other hand, if Pmaxin is too close to Pminin the range of symmetry breaking input powers is too narrow for comfortable switching.

Fig. 4. (a) Asymmetric states for r=1, ϕ=0.5π with Δ=1.02 (black dots) and Δ=0.62 (red dots), respectively. The symmetric states are shown with a line. (b) Linear transmission T versus detuning Δ with r=1 for ϕ=0.5π (ϕ=0.1π) in black (red), respectively.

To examine Pminin we perform a parameter sweep: For a fixed ϕ and r value we iterate over a range of detunings Δ (where, according to the criterion, symmetry breaking appears). Every ϕ and r combination corresponds to one Δ with the smallest input power for symmetry breaking Pminin.

4.1. Perfect reflection

First, we examine the perfect reflection case, or r=1. Figure 3(b)(black lines) shows the sweep results in this case; it plots the minimum Pminin versus ϕ. It is evident that ϕ=0.5π is the most interesting point, as it has the lowest Pminin. We also plot the corresponding Pmaxin in Fig. 3(b). Remark that Pminin is quite insensitive to the exact phase, whereas Pmaxin shows a stronger variation.

For ϕ in [-π, 0] we see that PmininPmaxin, so that the range of asymmetry is too small. On the contrary, at ϕ=0.5π there is a comfortable order of magnitude difference between Pminin and Pmaxin. This operational point is therefore very advantageous. Two nonlinear switching curves for r=1 are shown in Fig. 4(a). These curves show PLout and PRout versus Pin. The straight line through the origin presents the symmetric solutions, where PLout=PRout. The curves formed by the dots are the asymmetric solutions, meaning PLoutPRout. Note the mirrored nonlinear solutions: if there is a state with PLout>PRout, there is an equivalent one with PLout<PRout. The linear solutions are unstable when the symmetric straight line lies inside the asymmetric eye, as linear stability analysis shows, indicating the experimental legitimacy of the asymmetric solutions. The nonlinear curves can have interesting shapes, see the black curve in Fig. 4(a). It was determined that the parts with different curvature around Pin/P 0=0.125 are unstable, leading to four stable asymmetric states [5

5. B. Maes, M. Soljačić, J.D. Joannopoulos, P. Bienstman, R. Baets, S-P. Gorza, and M. Haelterman, “Switching through symmetry breaking in coupled nonlinear micro-cavities,” Opt. Express 14, 10678–10683 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-22-10678. [CrossRef] [PubMed]

].

The system with the smallest Pminin has Δ=1.02 and is shown in black. If we decrease the detuning, we also find we can obtain flat, high-contrast output powers (Fig. 4(a)(red dots)). The latter devices provide useful contrast for digital logic applications. In addition, the necessary Pminin is still adequate.

Fig. 5. Asymmetric states for (a) r=0.8, ϕ=0.6π, Δ=1.25 and (b) r=0.8, ϕ=0.23π, Δ=2.1. The symmetric states are shown with a line. (c) Linear transmission T versus detuning Δ with r=0.8 for ϕ=0.6π (ϕ=0.23π) in black (red), respectively.

4.2. Imperfect reflection

An important difference between the perfect and imperfect reflection cases is the appearance of ϕ values with extremely low Pminin. For r=0.8 this is the case around ϕ=0.8π and ϕ=0.2π. However, in [0, 0.5π] the value of Pmaxin becomes high, so positive pulse switching is unlikely (one needs to go beyond Pmaxin). With ϕ around 0.23π e.g. it is possible to have low power asymmetric states, see Fig. 5(b). The Pminin drops by an order of magnitude, below 10-3×P 0. A disadvantage in these cases is that the contrast between the output powers is often limited.

5. Conclusions

We extended the description of symmetry breaking in two coupled nonlinear cavities. We describe the appearance of the effect for the larger class of Fano resonances. As was the case with coupled Lorentzian resonators an elegant analytical description is employed and a criterion for the emergence of asymmetric states is derived. In addition, we optimize the switching scheme: we determine the parameter regimes where low input powers or high contrast output powers appear. This indicates which structures would be most promising for experimental verification. The perfectly reflecting structure with an inter-cavity phase around ϕ=π/2 is appropriate for switching operations with positive pulses and has a good contrast. Extremely low bifurcation powers, much lower than for single cavity devices, are obtained in a structure with partial reflection. This corresponds to the appearance of very narrow transmission peaks in the linear spectrum.

Acknowledgments

BM acknowledges a postdoctoral fellowship from the Funds for Scientific Research - Flanders (FWO-Vlaanderen).We acknowledge the Belgian IUAP project Photonics@be.

References and links

1.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Physical Review 124, 1866 (1961). [CrossRef]

2.

V. Lousse and J.P. Vigneron, “Use of Fano resonances for bistable optical transfer through photonic crystal films,” Phys. Rev. E 69, 155106 (2004).

3.

J.E. Heebner, R.W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19, 722–731 (2002). [CrossRef]

4.

B. Maes, P. Bienstman, and R. Baets, “Switching in coupled nonlinear photonic-crystal resonators,” J. Opt. Soc. Am. B 22, 1778–1784 (2005). [CrossRef]

5.

B. Maes, M. Soljačić, J.D. Joannopoulos, P. Bienstman, R. Baets, S-P. Gorza, and M. Haelterman, “Switching through symmetry breaking in coupled nonlinear micro-cavities,” Opt. Express 14, 10678–10683 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-22-10678. [CrossRef] [PubMed]

6.

P.E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express 13, 801–820 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-3-801. [CrossRef] [PubMed]

7.

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, 2678–2687 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-7-2678. [CrossRef] [PubMed]

8.

T. Uesugi, B-S. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express 14, 377–386 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-1-377. [CrossRef] [PubMed]

9.

Y. Lu, J. Yao, X. Li, and P. Wang, “Tunable asymmetrical Fano resonance and bistability in a microcavity-resonator-coupled Mach-Zehnder interferometer,” Opt. Lett. 30, 3069–3071 (2005). [CrossRef] [PubMed]

10.

S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80, 908–910 (2002). [CrossRef]

11.

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001). [CrossRef]

12.

S. Fan, W. Suh, and J.D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003). [CrossRef]

13.

M. Soljačić, M. Ibanescu, S.G. Johnson, Y. Fink, and J.D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E66, 055601(R) (2002).

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

ToC Category:
Nonlinear Optics

History
Original Manuscript: October 26, 2007
Revised Manuscript: December 12, 2007
Manuscript Accepted: December 16, 2007
Published: February 20, 2008

Citation
Björn Maes, Peter Bienstman, and Roel Baets, "Symmetry breaking with coupled Fano resonances," Opt. Express 16, 3069-3076 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-3069


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References

  1. U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866 (1961). [CrossRef]
  2. V. Lousse, and J. P. Vigneron, "Use of Fano resonances for bistable optical transfer through photonic crystal films," Phys. Rev. E 69, 155106 (2004).
  3. J. E. Heebner, R. W. Boyd, and Q-H. Park, "SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides," J. Opt. Soc. Am. B 19, 722-731 (2002). [CrossRef]
  4. B. Maes, P. Bienstman, and R. Baets, "Switching in coupled nonlinear photonic-crystal resonators," J. Opt. Soc. Am. B 22, 1778-1784 (2005). [CrossRef]
  5. B. Maes, M. Solja¡ci’c, J.D. Joannopoulos, P. Bienstman, R. Baets, S-P. Gorza, and M. Haelterman, "Switching through symmetry breaking in coupled nonlinear micro-cavities," Opt. Express 14, 10678-10683 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-22-10678. [CrossRef] [PubMed]
  6. P. E. Barclay, K. Srinivasan, and O. Painter, "Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper," Opt. Express 13, 801-820 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-3-801. [CrossRef] [PubMed]
  7. 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, 2678-2687 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-7-2678. [CrossRef] [PubMed]
  8. T. Uesugi, B-S. Song, T. Asano, and S. Noda, "Investigation of optical nonlinearities in an ultrahigh-Q Si nanocavity in a two-dimensional photonic crystal slab," Opt. Express 14, 377-386 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-1-377. [CrossRef] [PubMed]
  9. Y. Lu, J. Yao, X. Li, and P. Wang, "Tunable asymmetrical Fano resonance and bistability in a microcavityresonator-coupled Mach-Zehnder interferometer," Opt. Lett. 30, 3069-3071 (2005). [CrossRef] [PubMed]
  10. S. Fan, "Sharp asymmetric line shapes in side-coupled waveguide-cavity systems," Appl. Phys. Lett. 80, 908-910 (2002). [CrossRef]
  11. P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001). [CrossRef]
  12. S. Fan, W. Suh, and J. D. Joannopoulos, "Temporal coupled-mode theory for the Fano resonance in optical resonators," J. Opt. Soc. Am. A 20, 569-572 (2003). [CrossRef]
  13. M. Solja¡cic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601(R) (2002).

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