## Symmetry breaking with coupled Fano resonances

Optics Express, Vol. 16, Issue 5, pp. 3069-3076 (2008)

http://dx.doi.org/10.1364/OE.16.003069

Acrobat PDF (295 KB)

### 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

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

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]

*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.

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]

## 2. Fano resonances

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

*a*of square rods with side 0.4

*a*. 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*×0.48

*a*. 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

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

*j*=1, 2. The complex

*a*are cavity mode amplitudes, so |

_{j}*a*|

_{j}^{2}is the energy in the mode. The

*f*and

_{j}*b*are waveguide mode amplitudes, thus |

_{j}*f*|

_{j}^{2}(resp. |

*b*|

_{j}^{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

*δω*=-|

_{j}*a*|

_{j}^{2}/(

*P*

_{0}τ

^{2}), with

*P*

_{0}the ‘characteristic nonlinear power’ of the cavities [13], which we discuss below. The coupling parameter in this system is

*d*=

*i*exp(

*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.

*P*≡|

^{L}_{in}*f*

_{1}|

^{2}, on the right side we have as input

*P*≡|

^{R}_{in}*b*

_{3}|

^{2}. In this paper the input powers are equal, so we note

*P*≡

_{in}*P*=

^{L}_{in}*P*. The output powers are

^{R}_{in}*P*≡|

^{L}_{out}*b*

_{1}|

^{2}and

*P*≡|

^{R}_{out}*f*

_{3}|

^{2}on the left and right side, respectively. For symmetry-breaking states one has

*P*≠

^{L}_{out}*P*.

^{R}_{out}*ϕ*plays a central role in the dynamical behavior of the system, and thus in our analysis. It depends on the precise reflection and transmission effects because of the blocking rods (if any) next to the resonator. But, more importantly, it is fully controllable by adjusting the length of the waveguide in between the resonators. In PhC systems this length is adjusted by changing the number of periods. However, this is not limiting: The phase

*ϕ*is invariant with period 2

*π*, and the propagation factor

*k*of PhC waveguides is not difficult to tune. If there are no blocking rods, we have

_{z}*ϕ*=-

*mk*-

_{z}a*π*/2, with

*m*the number of periods in one half of the device.

*P*

_{0}allows to evaluate the power levels for the operation of nonlinear devices. In [13] 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*

_{0}τ/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]. 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 √3

*P*

_{0}[13]. Further on we will obtain that significantly lower switching powers are achievable in the two-cavity devices.

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]

*t*becomes

*iω*. Eliminating

*f*

_{2}and

*b*

_{2}we obtain:

*c*

_{1}=

*r*exp(

*iϕ*),

*c*

_{2}=

*it*exp(

*iϕ*) 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:

*A*=

*α*|

*a*

_{1}|

^{2}and

*B*=

*α*|

*a*

_{2}|

^{2}, where

*α*=-1/(

*P*

_{0}τ). In addition, the detuning is Δ=τ(

*ω*

_{0}-

*ω*) and

*α*<0) one needs

*A*<0, which means that the asymmetric solution (

*A*≠

*B*) exists if Δ>Δ

_{+}. For negative nonlinearity one obtains that Δ<Δ

_{-}. For

*r*=1 Eq. 8 reduces to the previous criterion [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]

*r*-values, except for some phases where the critical detuning becomes infinite. However, the usefulness of the nonlinear switching curves depends on the application, the power levels etc. Therefore, in the next section an analysis is performed to identify practical parameter regions.

## 4. Optimization

*P*that is needed to observe symmetry breaking states. Above this power, the symmetric solution becomes unstable. Subsequently, we examine the maximum input power

^{min}_{in}*P*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

^{max}_{in}*P*and

^{min}_{in}*P*. If this

^{max}_{in}*P*is too large, it will be difficult to switch by using positive pulses. On the other hand, if

^{max}_{in}*P*is too close to

^{max}_{in}*P*the range of symmetry breaking input powers is too narrow for comfortable switching.

^{min}_{in}*P*we perform a parameter sweep: For a fixed

^{min}_{in}*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

*P*.

^{min}_{in}### 4.1. Perfect reflection

*r*=1. Figure 3(b)(black lines) shows the sweep results in this case; it plots the minimum

*P*versus

^{min}_{in}*ϕ*. It is evident that

*ϕ*=0.5

*π*is the most interesting point, as it has the lowest

*P*. We also plot the corresponding

^{min}_{in}*P*in Fig. 3(b). Remark that

^{max}_{in}*P*is quite insensitive to the exact phase, whereas

^{min}_{in}*P*shows a stronger variation.

^{max}_{in}*ϕ*in [-

*π*, 0] we see that

*P*≈

^{min}_{in}*P*, so that the range of asymmetry is too small. On the contrary, at

^{max}_{in}*ϕ*=0.5

*π*there is a comfortable order of magnitude difference between

*P*and

^{min}_{in}*P*. This operational point is therefore very advantageous. Two nonlinear switching curves for

^{max}_{in}*r*=1 are shown in Fig. 4(a). These curves show

*P*and

^{L}_{out}*P*versus

^{R}_{out}*P*. The straight line through the origin presents the symmetric solutions, where

_{in}*P*=

^{L}_{out}*P*. The curves formed by the dots are the asymmetric solutions, meaning

^{R}_{out}*P*≠

^{L}_{out}*P*. Note the mirrored nonlinear solutions: if there is a state with

^{R}_{out}*P*>

^{L}_{out}*P*, there is an equivalent one with

^{R}_{out}*P*<

^{L}_{out}*P*. 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

^{R}_{out}*P*/

_{in}*P*

_{0}=0.125 are unstable, leading to four stable asymmetric states [5

**14**, 10678–10683 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-22-10678. [CrossRef] [PubMed]

*P*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

^{min}_{in}*P*is still adequate.

^{min}_{in}*r*=1, so the background (without cavities) is perfectly reflecting. However, in the presence of cavities the light can tunnel through the resonators and pass the barriers, which leads to frequencies with perfect transmission. Note that the transmission curves for

*ϕ*and

*ϕ*+

*π*look the same. However, the phases of the underlying amplitudes are different. Therefore, the nonlinear dynamics of

*ϕ*and

*ϕ*+

*π*are dissimilar, which is already clear from Fig. 3(b) by comparing the

*ϕ*-ranges [-

*π*, 0] and [0,

*π*]. However, there are similar trends, see e.g. Fig. 3(b): both have extrema at

*ϕ*=-0.5

*π*and

*ϕ*=0.5

*π*.

*ϕ*[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]

*ϕ*=±0.5

*π*are special cases, because then the two peaks coalesce into one broader peak, see Fig. 4(b)(black curve). This broader peak has steeper sides, which is often advantageous for nonlinear effects, and thus here too for low power symmetry breaking. Indeed, from the definition of the characteristic power we note that P

_{0}~ 1/Q

^{2}. So, in the single cavity case, there is a direct link between steepness of the linear transmission curve (

*Q*) and nonlinear switching power levels (

*P*

_{0}). Although, we do not define

*Q*and

*P*

_{0}for two-cavity devices, it is clear that a steep transmission curve correlates with a low switching power. An interpretation of this is that for nonlinear effects the linear curve shifts towards the operating frequency, and large slopes result in low powers to instigate this shift.

### 4.2. Imperfect reflection

*r*≠1. Figure 3(b)(red lines) shows

*P*and

^{min}_{in}*P*for

^{max}_{in}*r*=0.8, which means that 64%(=

*r*

^{2}) of the power is reflected. This has to be compared with the curves for

*r*=1 (black lines in Fig. 3(b)). In the range [-

*π*, 0] we see again that

*P*and

^{min}_{in}*P*follow each other closely, which makes observation of asymmetry difficult.

^{max}_{in}*π*,

*π*] the minimum input power

*P*is larger for

^{min}_{in}*r*=0.8 than for

*r*=1. In addition,

*P*becomes smaller in this range. In the range [0.5

^{max}_{in}*π*, 0.75

*π*] it is common to see acceptable nonlinear switching curves, see Fig. 5(a).

*P*is not as small as for r=1, but

^{min}_{in}*P*can be tuned favorably. The linear transmission curve for this situation, see Fig. 5(c)(black curve), shows that the peaks coalesce into one broader peak, similar to the previous total reflection case.

^{max}_{in}*ϕ*values with extremely low

*P*. For

^{min}_{in}*r*=0.8 this is the case around

*ϕ*=0.8

*π*and

*ϕ*=0.2

*π*. However, in [0, 0.5

*π*] the value of

*P*becomes high, so positive pulse switching is unlikely (one needs to go beyond

^{max}_{in}*P*). With

^{max}_{in}*ϕ*around 0.23

*π*e.g. it is possible to have low power asymmetric states, see Fig. 5(b). The

*P*drops by an order of magnitude, below 10

^{min}_{in}^{-3}×

*P*

_{0}. A disadvantage in these cases is that the contrast between the output powers is often limited.

*P*is apparent from the linear transmission curves, see Fig. 5(c)(red curve). By coupling two cavities with

^{min}_{in}*r*≠1 one can obtain vanishingly narrow transmission peaks, which evidently have their impact on nonlinear switching properties [4

**22**, 1778–1784 (2005). [CrossRef]

*P*

_{0}. Thus, the extremely narrow peaks are easily shifted by nonlinearity. This interference effect is not available in the perfectly reflecting case.

## 5. Conclusions

*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

## References and links

1. | U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Physical Review |

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

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 |

4. | B. Maes, P. Bienstman, and R. Baets, “Switching in coupled nonlinear photonic-crystal resonators,” J. Opt. Soc. Am. B |

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 |

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 |

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 |

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 |

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. |

10. | S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. |

11. | P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. |

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 |

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

- U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866 (1961). [CrossRef]
- V. Lousse, and J. P. Vigneron, "Use of Fano resonances for bistable optical transfer through photonic crystal films," Phys. Rev. E 69, 155106 (2004).
- 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]
- B. Maes, P. Bienstman, and R. Baets, "Switching in coupled nonlinear photonic-crystal resonators," J. Opt. Soc. Am. B 22, 1778-1784 (2005). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- S. Fan, "Sharp asymmetric line shapes in side-coupled waveguide-cavity systems," Appl. Phys. Lett. 80, 908-910 (2002). [CrossRef]
- 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]
- 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]
- 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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