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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 7 — Apr. 8, 2013
  • pp: 8570–8586
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Reversible Fano resonance by transition from fast light to slow light in a coupled-resonator-induced transparency structure

Yundong Zhang, Xuenan Zhang, Ying Wang, Ruidong Zhu, Yulong Gai, Xiaoqi Liu, and Ping Yuan  »View Author Affiliations


Optics Express, Vol. 21, Issue 7, pp. 8570-8586 (2013)
http://dx.doi.org/10.1364/OE.21.008570


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Abstract

We theoretically propose and experimentally perform a novel dispersion tuning scheme to realize a tunable Fano resonance in a coupled-resonator-induced transparency (CRIT) structure coupled Mach-Zehnder interferometer. We reveal that the profile of the Fano resonance in the resonator coupled Mach-Zehnder interferometers (RCMZI) is determined not only by the phase shift difference between the two arms of the RCMZI but also by the dispersion (group delay) of the CRIT structure. Furthermore, it is theoretically predicted and experimentally demonstrated that the slope and the asymmetry parameter ( q ) describing the Fano resonance spectral line shape of the RCMZI experience a sign reversal when the dispersion of the CRIT structure is tuned from abnormal dispersion (fast light) to normal dispersion (slow light). These theoretical and experimental results indicate that the reversible Fano resonance which holds significant implications for some attractive device applications such as highly sensitive biochemical sensors, ultrafast optical switches and routers can be realized by the dispersion tuning scheme in the RCMZI.

© 2013 OSA

1. Introduction

Fano resonance, which is first introduced in order to explain the asymmetric spectral line shapes in the photoionization of an atom, stems from the interference interaction between a discrete level and a continuum level [1

1. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]

]. To date, Fano resonance can be observed not only in atomic resonance systems but also in artificial photonic structures involving plasmonic nanostructures [2

2. F. Hao, Y. Sonnefraud, P. Van Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry breaking in plasmonic nanocavities: subradiant LSPR sensing and a tunable Fano resonance,” Nano Lett. 8(11), 3983–3988 (2008). [CrossRef] [PubMed]

], photonic crystals [3

3. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002). [CrossRef]

5

5. X. Yang, M. Yu, D.-L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102(17), 173902 (2009). [CrossRef] [PubMed]

], electromagnetic metamaterials [6

6. G. Shvets and Y. A. Urzhumov, “Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances,” Phys. Rev. Lett. 93(24), 243902 (2004). [CrossRef] [PubMed]

, 7

7. V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99(14), 147401 (2007). [CrossRef] [PubMed]

], and coupled resonators [8

8. C. Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003). [CrossRef]

10

10. B. B. Li, Y. F. Xiao, C. L. Zou, Y. C. Liu, X. F. Jiang, Y. L. Chen, Y. Li, and Q. H. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. 98(2), 021116 (2011). [CrossRef]

], since the similar interference interaction between resonant (discrete) modes and continua also exists in these photonic structures. Recently, the Fano resonance based on these photonic structures has attracted more and more research interest due to its two advantages: on one hand, the Fano resonance enables many promising applications such as slow light, biochemical sensing, and filters [11

11. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010). [CrossRef]

]; on the other hand, tunable Fano resonances that hold significant implications for device applications such as optical switches, modulators and routers [3

3. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002). [CrossRef]

, 4

4. S. H. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80(6), 908–910 (2002). [CrossRef]

, 12

12. M. F. Yanik, S. H. Fan, and M. Soljačić, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83(14), 2739–2741 (2003). [CrossRef]

16

16. L. J. Zhou and A. W. Poon, “Fano resonance-based electrically reconfigurable add-drop filters in silicon microring resonator-coupled Mach-Zehnder interferometers,” Opt. Lett. 32(7), 781–783 (2007). [CrossRef] [PubMed]

] can readily be realized by virtue of flexible tunability of these photonic structures [17

17. Y. Tanaka, J. Upham, T. Nagashima, T. Sugiya, T. Asano, and S. Noda, “Dynamic control of the Q factor in a photonic crystal nanocavity,” Nat. Mater. 6(11), 862–865 (2007). [CrossRef] [PubMed]

21

21. Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys. 3(6), 406–410 (2007). [CrossRef]

]. Among a variety of tunable Fano resonances, one unique category of which the slope or the asymmetric parameter [1

1. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]

, 11

11. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010). [CrossRef]

] q can change their signs during tuning process is of substantial interest, since the features of the category of tunable Fano resonances can be exploited for device applications in highly sensitive biochemical sensors, optical switches and optical routers [22

22. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010). [CrossRef] [PubMed]

, 23

23. F. Cheng, H. F. Liu, B. H. Li, J. Han, H. Xiao, X. F. Han, C. Z. Gu, and X. G. Qiu, “Tuning asymmetry parameter of Fano resonance of spoof surface plasmons by modes coupling,” Appl. Phys. Lett. 100(13), 131110 (2012). [CrossRef]

]. For brevity, we term this category of tunable Fano resonances a “reversible Fano resonance”.

This paper is organized as follows. In Sec. 2, we derive the expressions describing the relationship between the dispersion of the CRIT structure and the profile of the Fano resonance spectral line shape in the CRIT structure coupled MZI. According to the expressions, we reveal that dispersion transition of the CRIT structure from abnormal dispersion to normal dispersion or from normal dispersion to abnormal dispersion may lead to the reversible Fano resonance in the RCMZI. Also, in this section, we derive the expression of the group delay of the CRIT structure and hence obtain the condition under which the dispersion transition of the CRIT structure can occur. In Sec. 3, we implement the RCMZI by coupling a CRIT structure to an arm of a fiber Mach-Zehnder interferometer (MZI), and observe the reversible Fano resonance due to the dispersion transition from abnormal dispersion to normal dispersion in this RCMZI. Moreover, we infer the slope and extract the asymmetric parameter q from the experiment results of the Fano resonance spectral line shapes, and compare them with the corresponding theoretical values in order to verify the theory introduced in Sec. 2. In Sec. 4, we describe the experiment method and process in detail. Finally, we summarize in Sec. 5.

2. Theory

A typical balanced RCMZI consists of two arms of nearly equal length. One of the two arms coupled to a resonator system is the resonance arm, and the other is the reference arm. Two 3 dB directional couplers (Input and Output couplers) at the ends of the two arms are employed to split and recombine light field. Therefore, the interference interaction between the resonant mode and the non-resonant mode at the output port of the Output coupler may give rise to Fano resonances. For concreteness, as shown by the dashed box in Fig. 1
Fig. 1 Schematic of a CRIT structure coupled Mach-Zehnder interferometer.
, if the resonator system in the RCMZI is a coupled-resonator-induced transparency [29

29. D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004). [CrossRef]

] (CRIT) structure and the phase shift difference between the reference arm and the resonance arm is Δϕ, the normalized interference transmission spectrum (Fano resonance spectral line shape) Tout(ω) at the output port of the Output coupler of the RCMZI can be described by
Tout(ω)=|Eout(ω)/Ein(ω)|2=|t2(ω)exp(iΔϕ)|2/8,
(1)
where Eout(ω) and Ein(ω) represent the output light field of the RCMZI and the input light field, and t2(ω) denotes the complex transmission coefficient of the CRIT structure, and the factor (1/8) characterizes the impact of the four 3 dB couplers (Input coupler, Output coupler, Coupler A, and Coupler B) on the interference transmission.

Utilizing the transfer matrix theory [30

30. J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. 24(12), 847–849 (1999). [CrossRef] [PubMed]

, 31

31. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12(1), 90–103 (2004). [CrossRef] [PubMed]

],the response (complex transmission coefficient) of the CRIT structure depending on the angular frequency ω of light field can be expressed by t2(ω)=T2exp(iθ2)=[ρ2a2t1exp(iωτ2)]/[1ρ2a2t1exp(iωτ2)], where ρ2, T2(ω)=|t22(ω)|, θ2(ω)=arg[t2(ω)], a2, τ2, and t1 represent the reflection coefficient of Coupler 2, the normalized transmission (normalized transmission spectrum) of the CRIT structure, the phase shift induced by the CRIT structure, the loss parameter (round-trip attenuation factor) of Ring 2, the round-trip time of Ring 2, and the response (complex transmission coefficient) of Ring 1, respectively. Likewise, using the transfer matrix theory, the response of Ring 1 can be given by t1(ω)=T1exp(iθ1)=[ρ1a1exp(iωτ1)]/[1ρ1a1exp(iωτ1)], where ρ1, T1(ω)=|t12(ω)|, θ1(ω)=arg[t1(ω)], a1, and τ1 represent the reflection coefficient of Coupler 1, the normalized transmission (normalized transmission spectrum) of Ring 1, the phase shift induced by Ring 1, the loss parameter (round-trip attenuation factor) of Ring 1, and the round-trip time of Ring 1, respectively. For brevity, we respectively denote ti(ω0) and θi(ω0) by ti0 and θi0 (i=1,2), where ω0 (ω0 satisfies ω0τ1mod2π=0 and ω0τ2mod2π=0) is the coincident resonant angular frequency of the CRIT structure.

To characterize the output (asymmetric) interference transmission spectral line shape quantitatively, we first derive the expression of the slope of the interference transmission. Taking the derivative of Eq. (1) with respect to the angular frequency ω of light field, one can obtain the slope S of the interference transmission spectrum Tout(ω) at ω0 as follows:
S(ω0)=1τ2(Toutω|ω0)=τg20|t20|sin(θ20Δϕ)4τ2,
(2)
where τg20 represents the group delay induced by the CRIT structure at ω0. The slope S(ω0) in Eq. (2) is a crucial performance parameter of a RCMZI in that it is relevant and proportional to the sensitivity of the RCMZI-based sensor [32

32. M. Terrel, M. J. F. Digonnet, and S. H. Fan, “Ring-coupled Mach-Zehnder interferometer optimized for sensing,” Appl. Opt. 48(26), 4874–4879 (2009). [CrossRef] [PubMed]

]. The modulus of S(ω0) relies on the relative delay τg20/τ2, the transmission amplitude |t20|, and Δϕ according to Eq. (2). Generally, when |t20|0 and Δϕ±2πn, |S(ω0)| will not be vanishing, which implies that the interference transmission exhibits an asymmetric spectral line shape. Also, one may implement the RCMZI with the phase shift difference Δϕ=±π/2 to obtain the asymmetric spectral line shape of the steepest slope |S(ω0)| or the highest sensitivity for sensing [32

32. M. Terrel, M. J. F. Digonnet, and S. H. Fan, “Ring-coupled Mach-Zehnder interferometer optimized for sensing,” Appl. Opt. 48(26), 4874–4879 (2009). [CrossRef] [PubMed]

], since θ20 always equals to either 0 or π. Additionally, Eq. (2) elucidates that the sign of S(ω0) is determined by Δϕ and τg20. Thus, the phase shift (Δϕ) tuning scheme as used in [16

16. L. J. Zhou and A. W. Poon, “Fano resonance-based electrically reconfigurable add-drop filters in silicon microring resonator-coupled Mach-Zehnder interferometers,” Opt. Lett. 32(7), 781–783 (2007). [CrossRef] [PubMed]

] or the dispersion (τg20) tuning scheme can result in the sign reversal of S(ω0), provided that the phase shift Δϕ(Δϕ) satisfies ΔϕΔϕ=±π or the dispersion τg20(τg20) satisfies τg20×τg20<0 for a constant θ20, where Δϕ and τg20 are the tuned phase shift difference and the tuned group delay, respectively. For the dispersion tuning scheme, the condition τg20×τg20<0 implies the dispersion transition of the CRIT structure tuned from abnormal dispersion τg20<0 to normal dispersion τg20>0 (or from normal dispersion τg20>0 to abnormal dispersion τg20<0). For the CRIT structure shown in Fig. 1, the undercoupled regimes (ρ1>a1,ρ2>a2|t10|) of Ring 1 and Ring 2 under which the CRIT structure can yield either slow or fast light with a constant phase shift θ20=0 may be a feasible situation to realize the dispersion transition and the resulting sign reversal of S(ω0).

Furthermore, the conventional asymmetric parameter [1

1. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]

, 11

11. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010). [CrossRef]

] q can be exploited to describe the output asymmetric interference transmission spectral line shape of the RCMZI. Considering only the first order intracavity dispersion [27

27. Y. Dumeige, S. Trebaol, and P. Feron, “Intracavity coupled-active-resonator-induced dispersion,” Phys. Rev. A 79(1), 013832 (2009). [CrossRef]

] of the CRIT structure such as t1(ω)exp(iωτ2)=t10+it10(τg10+τ2)(ωω0), we can approximately rewrite the interference transmission profile Tout(ω) in terms of the following Fano resonance formula in the vicinity of ω0:
Tout(ε)=(1+ρ222ρ2δϕ)(ε+q)28ρ22(ε2+1)+(1+t2022t20δϕ)8(ε2+1)+(1ρ22)28(ε2+1)(1ρ2a2t10)2(1+ρ222ρ2δϕ),
(3)
provided that Δϕ approaches π/2 and Ring 1 is in the undercoupled regime (ρ1>a1), where ε=(ωω0)|τg10+τ2|ρ2a2t10/(1ρ2a2t10), δϕ=π/2Δϕ, and τg10 denote the reduced frequency detuning, the phase shift difference deviation from π/2, and the group delay induced by Ring 1 at ω0, respectively. Equation (3) elucidates that the interference transmission profile Tout(ω) will exhibit an asymmetry Fano resonance spectral line shape if Δϕπ/2 and ρ1>a1. The parameter q in Eq. (3) describing the degree of the asymmetry of the Fano resonance can be given by
q=(1ρ22)ρ2|τg20(a2|t10|ρ2)|[(1+ρ222ρ2δϕ)(1ρ2a2t10)τg20(a2|t10|ρ2)].
(4)
On one hand, Eq. (4) indicates that the modulus of the asymmetry parameter q depends on the term of (1ρ22)ρ2/[(1+ρ222ρ2δϕ)(1ρ2a2t10)], which describes the ratio between the resonant and non-resonant amplitudes [1

1. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]

, 33

33. M. Galli, S. L. Portalupi, M. Belotti, L. C. Andreani, L. O'Faolain, and T. F. Krauss, “Light scattering and Fano resonances in high-Q photonic crystal nanocavities,” Appl. Phys. Lett. 94(7), 071101 (2009). [CrossRef]

]. Thus, the modulus of q independent of the group delay τg20 is dependent on δϕ (or Δϕ), t10, ρ2, and a2. As a result, to obtain a strongly asymmetric resonance (high |q|), one can appropriately tune the phase shift difference Δϕ (or δϕ) as shown in Fig. 2(a)
Fig. 2 (a) Dependence of the asymmetry parameter q (solid curve) and the slope S(ω0) (dashed curve) of the Fano resonance of the RCMZI on the loss parameter a1 for different δϕ. (b) Dependence of the asymmetry parameter q (solid curve) and the slope S(ω0) (dashed curve) of the Fano resonance of the RCMZI on the loss parameter a1 for δϕ=0 and different ρ2. (c) Dependence of the asymmetry parameter q (solid curve) and the slope S(ω0) (dashed curve) of the Fano resonance of the RCMZI on the loss parameter a1 for δϕ=0 and different ρ1. In (a), or (b), or (c), the solid and dashed curves that are of same color are the results with the identical parameters. The other system parameters are ρ1=0.938, ρ2=0.755, a2=0.79, τ1=15.2ns, and τ2=18.5ns.
, or can choose Coupler 1 of high ρ1 or Coupler 2 of low ρ2 as shown in Figs. 2(b) and 2(c). For concreteness, in Fig. 2(a), when δϕ(0) is slightly increased (the phase shift tuning as in [17

17. Y. Tanaka, J. Upham, T. Nagashima, T. Sugiya, T. Asano, and S. Noda, “Dynamic control of the Q factor in a photonic crystal nanocavity,” Nat. Mater. 6(11), 862–865 (2007). [CrossRef] [PubMed]

] is carried out), |q| is boosted, and hence a strongly asymmetric resonance (|q|1) in [33

33. M. Galli, S. L. Portalupi, M. Belotti, L. C. Andreani, L. O'Faolain, and T. F. Krauss, “Light scattering and Fano resonances in high-Q photonic crystal nanocavities,” Appl. Phys. Lett. 94(7), 071101 (2009). [CrossRef]

] can be obtained. On the other hand, the sign of q is determined by the signs of τg20 and the term of (a2|t10|ρ2) according to Eq. (4). For the CRIT structure in the respective undercoupled regimes (ρ1>a1,ρ2>a2|t10|) of Ring 1 and Ring 2, the normal dispersion τg20>0 is associated with a negative q value and the abnormal dispersion τg20<0 is associated with a positive value q, since the sign of (a2|t10|ρ2) is negative for the undercoupled Ring 2 (ρ2>a2|t10|). As a consequence, the dispersion transition τg20×τg20<0 of the CRIT structure can lead to a sign reversal of the asymmetry parameter q. In summary, our theoretical investigation reveals that the signs of the slope and the asymmetry parameter (q) characterizing the profile of the Fano resonance in the RCMZI are dependent on the dispersion of the CRIT structure besides the relative amplitude and the phase shift difference between the resonant mode and the non-resonant mode, and the reversible Fano resonance that requires the sign reversals of S(ω0) and q can be realized by the dispersion transition of the CRIT structure in the RCMZI.

In our experiment, we fabricate a CRIT structure, of which the dispersion can be tuned by modulating the loss of active optical fiber embedded in Ring 1. To acquire the dispersion transition from abnormal (normal) dispersion to normal (abnormal) dispersion, the variation range of the loss parameter a1 of Ring 1 needs to cover either the critical value a1 of the loss parameter corresponding to the weak dispersion τg20=0 or the critical value a1 of the loss parameter corresponding to the strong dispersion τg20=. Thus it is instructive to derive the expressions of a1 and a1. Using the relation between the group delay τg2(ω) and the transfer function t2 of the CRIT structure such as τg2(ω)=(t2*t2/ωt2t2*/ω)/(2it2t2*), the group delay τg20 can be obtained:
τg20=(1ρ22)a2|t10|(τg10+τ2)(a2|t10|ρ2)(1ρ2a2|t10|),
(5)
where τg10=a1(1ρ12)τ1/[(a1ρ1)(1ρ1a1)]. According to Eq. (5), we can derive the equations that a1 and a1 satisfy as follows:
0=τ2τ1+a1(1ρ12)(a1ρ1)(1ρ1a1),
(6)
a1=(ρ1a2±ρ2)(a2±ρ1ρ2),
(7)
or

a1=ρ1.
(8)

For the presented experiment parameters in this paper as follows: ρ1=0.88=0.938, ρ2=0.57=0.755, the measured loss parameter value a2=0.79 of Ring 2, τ1=15.2ns, and τ2=18.5ns, one can obtain only a1=0.720 located in the variation range 0.58a10.84 of a1 in the presented experiment while calculating Eqs. (6), (7) and (8). Thus, one may probe the group delay τg20 depending on a1 in the vicinity of a1 by Eq. (5), and confirms that the transition from abnormal dispersion to normal dispersion can occur in the presented experiment. It is noteworthy that the transmission spectrum T2(ω) of the CRIT structure in the vicinity of ω0 exhibits an approximately flat-top (or flat-bottom) profile while the tuned loss parameter a1 approaches a1 or the dispersion of the CRIT structure approaches the weak dispersion τg20=0, since Eq. (6) determining a1 is also the condition that a white light cavity (WLC) should satisfy [34

34. G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett. 99(13), 133601 (2007). [CrossRef] [PubMed]

]. In addition, the interference transmission spectrum Tout(ω) will also appear to be a flat and symmetry spectral line shape rather than an asymmetry Fano resonance spectral line shape in the vicinity of ω0 due to the vanishing slope S(ω0) caused by the weak dispersion τg20=0, when a1=a1. As shown in Fig. 2, the discontinuous step change of q corresponding to a1=a1 can be negligible in that the asymmetry parameter q is not applicable to describe the symmetry interference transmission spectrum of S(ω0)=0 corresponding to the weak dispersion τg20=0.

As shown in Fig. 3
Fig. 3 The asymmetry parameter q (solid curve) of the Fano resonance of the RCMZI, the slope S(ω0) (dashed curve) of the Fano resonance of the RCMZI, and the group delay τg20 (dotted curve) of the CRIT structure depending on the loss parameter a1. The corresponding system parameters are ρ1=0.938, ρ2=0.755, a2=0.79, a1=0.720, τ1=15.2ns, and τ2=18.5ns.
, for the CRIT structure in the respective undercoupled regimes (ρ1>a1,ρ2>a2|t10|) of Ring 1 and Ring 2, as the loss parameter a1 of Ring 1 that satisfies a1<a1=0.720 (the critical value a1 corresponding to the weak dispersion τg20=0) is increased, the group delay exhibits fast light τg20<0 (abnormal dispersion) as depicted by the dotted curve, and the asymmetry parameter q (as depicted by the solid curve) maintains a positive value and is slightly reduced. Then, a step change of q from a positive value to a negative value emerges once a1 approximates a1. Finally, when a1 continues to be increased beyond the critical value a1, the group delay exhibits slow light τg20>0 (normal dispersion), and q turns out to be negative and is slowly boosted. Therefore, when a1 increases, on one hand, owing to the dispersion transition, the asymmetry parameter q (and the slope S(ω0)) of the Fano resonance experiences a sign reversal from a positive value to a negative value; on the other hand, the modulus of q gradually declines due to the decrease of t10.

3. Experiment setup and results

Figure 5(a)
Fig. 5 The experimental (a) and theoretical (b) transmission spectra T2(Δ) of the CRIT structure depending on frequency detuning Δ=(ωω0)/2π for the five different loss parameter a1 values. In Fig. 5(a), the red, orange, yellow, green, and blue curves are measured under the 980 nm pump powers of 0 mW, 2.01 mW, 2.73 mW, 7.49 mW, and 12.4 mW, respectively. The corresponding inferred loss parameter a1 values are 0.58 (red), 0.73 (orange), 0.75 (yellow), 0.82 (green), and 0.84 (blue). In Fig. 5(b), each dashed curve is produced by the experiment parameters of the solid curve of same color in Fig. 5(a).
shows the measured transmission spectra T2(ω) of the CRIT structure for five different pump powers of the 980 nm laser. Prior to pump, the transmission spectral line shape exhibits a dip rather than a transparent peak in the vicinity of the coincident resonant angular frequency ω0, as shown by the red solid curve in Fig. 5(a). Subsequently, when we add the pump power gradually and thus the loss parameter a1 of Ring 1 increases, the transmission spectra experience a variation from a resonance dip (represented by the red curve in Fig. 5(a)) to an approximately flat-bottom profile (represented by the orange curve in Fig. 5(a)), and to typical CRIT (EIT-like) spectra (represented by the yellow, green and blue curves in Fig. 5(a)) due to the mode splitting [29

29. D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004). [CrossRef]

]. The emergence of the approximately flat-bottom profile implies that the transition from abnormal dispersion to normal dispersion may occur, since the flat-bottom transmission is always associated with the weak dispersion corresponding to the critical value a1 of the loss parameter a1 determined by Eq. (6) according to the preceding discussion. Furthermore, as shown in Fig. 5(b), the theoretical transmission spectra of the CRIT structure obtained by the transfer matrix theory coincide well with those corresponding experiment results in Fig. 5(a). Note that the transmission (0.047T2(0)0.329) of the CRIT structure at ω0 is not high in Fig. 5(a). Generally, to obtain the nearly transparent transmission of the CRIT structure at ω0, the approximately vanishing complex transmission coefficient of Ring 1 resulting from the destructive interference through Ring 1 can move Ring 2 away from the low transmission in the critically coupled regime (ρ2=a2t1) [35

35. Y. Dumeige, T. K. N. Nguyen, L. Ghisa, S. Trebaol, and P. Feron, “Measurement of the dispersion induced by a slow-light system based on coupled active-resonator-induced transparency,” Phys. Rev. A 78(1), 013818 (2008). [CrossRef]

]. For the presented experiment in Fig. 5, since the complex transmission coefficient of Ring 1 at ω0 satisfies 0.463t100.785, the approximately destructive interference through Ring 1 does not occur, and hence T2(0) is not too high. Nevertheless, T2(0) can be readily increased, for example, one may choose Coupler 1 of lower ρ1 or Coupler 2 of higher ρ2.

Figure 7(a)
Fig. 7 The experimental (a) and theoretical (b) interference transmission spectra Tout(Δ) of the RCMZI versus frequency detuning Δ=(ωω0)/2π. (c) The experimental (solid curves) and fitting (dashed curves) interference transmission spectra Tout(ε) of the RCMZI versus the reduced frequency detuning ε. The solid curves in Figs. 5(a), 6(a), and 7(a) that are of same color are simultaneously measured results with the identical loss parameter a1. In Fig. 7(a), for the five different loss parameter a1 values, the phase shift difference Δϕ is 1.67(red), 1.62(orange), 1.61(yellow), 1.61(green), and 1.60(blue), respectively. In Fig. 7(b), each dashed curve is the corresponding theoretical simulation result of the solid curve of same color in Fig. 7(a).The theoretical (dashed curves in Fig. 7(b)) and fitting (dashed curves in Fig. 7(c)) interference transmission spectra are achieved by using Eqs. (1) and (3), respectively. In Fig. 7(c), each solid curve is obtained from the one of identical color in Fig. 7(a) by transforming the horizontal axis.
illustrates the simultaneously measured interference transmission spectra Tout(ω) of the RCMZI system. For the experiment results in Fig. 7(a), the phase difference Δϕ that satisfies 1.60Δϕ1.67 is approximately π/2, and Ring 1 and Ring 2 are in the respective undercoupled regimes since the measured loss and coupling parameters satisfy a1<ρ1 and |t10|a2<ρ2. As shown in Fig. 7(a), the experiment results of Tout(ω) confirm the prediction of Eq. (3), for the observed interference transmission spectra of the RCMZI indeed exhibit asymmetry Fano resonance spectral line shapes. As a consequence of the dispersion tuning of the CRIT structure shown in Fig. 6(a), the asymmetry Fano resonance spectral line shape is drastically varied, as shown in Fig. 7(a). Moreover, the Fano resonance still originates from the interference interaction between the resonant state in the resonance arm and the non-resonant state in the reference arm, since all the experimental curves are reproduced by the theoretical simulation of Eq. (1) describing this interference interaction, as shown in Fig. 7(b).

To characterize the tunable Fano resonance quantitatively, we infer the slope S(ω0) of the experimental interference transmission spectra in Fig. 7(a), and fit these spectral line shapes in the vicinity of ω0 by Eq. (3) in Fig. 7(c) in order to extract the asymmetry parameter q of the Fano resonance. As shown by the red curves in Fig. 7(a), when the loss parameter a1 is initially 0.58 without pump and the dispersion of the CRIT structure is abnormal dispersion due to τg20=35.0ns, the resulting interference transmission spectrum exhibits an asymmetry Fano resonance spectral line shape of a positive slope value S(ω0)=0.149rad1 and a positive asymmetry parameter value q=0.338. Also, the dip (peak) of the Fano resonance profile at the long (short) wavelength wing Δ<0 (Δ>0), which results from the positive S(ω0) and q, can be observed. Once the loss parameter a1 exceeds the critical value a1=0.720 due to the increase of the pump power of the 980 nm laser, the tuned group delay becomes positive (τg20=8.0ns for a1=0.75, τg20=18.1ns for a1=0.82, and τg20=23.2ns for a1=0.84 shown by the yellow, green, and blue curves in Fig. 6(a)), and the resulting interference transmission spectra exhibit asymmetry Fano resonance spectral line shapes with negative slopes (S(ω0)=0.0641rad1 of the yellow curve for a1=0.75, S(ω0)=0.145rad1 of the green curve for a1=0.82, and S(ω0)=0.197rad1 of the blue curve for a1=0.84 in Fig. 7(a)) and negative asymmetry parameters q (q=0.353 of the yellow curve for a1=0.75, q=0.345 of the green curve for a1=0.82, and q=0.343 of the blue curve for a1=0.84 in Fig. 7(a)). For these three Fano resonance spectra with the negative S(ω0) and q, the peaks at the long wavelength wing Δ<0 and the dips at the short wavelength wing Δ>0 as a result of the negative S(ω0) and q are experimentally demonstrated, as shown by the yellow, green, and blue curves in Fig. 7(a). Furthermore, as shown by the orange curves in Figs. 6(a) and 7(a), we also observe the approximately weak dispersion (τg20=0.8ns) and its associated flat interference transmission spectrum of S(ω0)=0.00282rad1 in the vicinity of ω0, when the loss parameter a1 approaches the critical value a1=0.720. In short, the experiment results demonstrate that both the slope S(ω0) and the asymmetry parameter q of the Fano resonance in the RCMZI experience a sign reversal from positive to negative when the dispersion of the CRIT structure is tuned from abnormal dispersion to normal dispersion. Between the abnormal dispersion and the normal dispersion, the weak dispersion corresponding to the critical value a1 and its resulting flat interference transmission spectrum with the vanishing slope S(ω0) value are also observed.

In addition to the preceding experiment results, using the corresponding experiment parameters, we calculate Eqs. (2) and (4), and obtain the values of the theoretical slope S(ω0) and the theoretical asymmetry parameter q, as shown in Table 1. In Table 1, the value of q is not given when the loss parameter a1=0.73 approaches the critical value a1=0.720, since the observed flat interference transmission spectrum of S(ω0)=0.00282rad1 corresponding to the approximately weak dispersion (τg20=0.8ns) is not a typical asymmetry Fano resonance spectral line shape as discussed above. The theoretical model upon the sign reversals of S(ω0) and q introduced in Sec. 2 is verified, since these theoretical S(ω0) and q values agree well with those corresponding experimental values despite the small deviation between them. For the small deviation between the theoretical (S(ω0),q) values and the experimental (S(ω0),q) values, it can also be attributed to the two preceding reasons: one is the slightly different propagation losses of the light fields through the resonance and reference pathways; the other is the slightly different polarization states of the light fields through the resonance and reference pathways.

Therefore, the experimental results indicate that the profile of the Fano resonance in the RCMZI is determined by the dispersion of the CRIT structure in the RCMZI, demonstrating that the reversible Fano resonance that has profound implications for device applications such as biochemical sensors or optical switches [22

22. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010). [CrossRef] [PubMed]

, 23

23. F. Cheng, H. F. Liu, B. H. Li, J. Han, H. Xiao, X. F. Han, C. Z. Gu, and X. G. Qiu, “Tuning asymmetry parameter of Fano resonance of spoof surface plasmons by modes coupling,” Appl. Phys. Lett. 100(13), 131110 (2012). [CrossRef]

] can be realized by the dispersion tuning scheme in the RCMZI. To obtain the reversible Fano resonance of a higher value of |q|, both the phase shift tuning scheme and the dispersion tuning scheme can be integrated into the RCMZI, provided that one replaces Input coupler by a Y-branch waveguide integrated optical (phase) modulator [36

36. H. C. Lefevre, The Fiber-Optic Gyroscope (Artech House, 1993).

]. Using the electrically driven phase modulator and the CRIT structure of tunable group delay, the phase shift and dispersion tuning schemes can independently be realized for the RCMZI. The dispersion transition and the resulting sign reversals of S(ω0) and q are not influenced by the phase shift tuning as shown in Fig. 2(a), since the requirement of the dispersion transition and the resulting sign reversals of S(ω0) and q is unrelated to the phase shift difference deviation δϕ (or the phase shift difference Δϕ). Therefore, one can first tune the phase shift to make |q| approach unity, and then tune the group delay to obtain the sign reversals of S(ω0) and q. Moreover, to attain larger bandwidth and more compact size, one may fabricate a silicon-based RCMZI according to the configuration introduced in this paper on silicon-on-insulator (SOI) substrate by such fabrication processes in [20

20. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]

, 24

24. L. J. Zhou and A. W. Poon, “Silicon electro-optic modulators using p-i-n diodes embedded 10-micron-diameter microdisk resonators,” Opt. Express 14(15), 6851–6857 (2006). [CrossRef] [PubMed]

]. For silicon resonators, loss tuning can be accomplished by Raman-induced loss tuning [26

26. Y. H. Wen, O. Kuzucu, T. G. Hou, M. Lipson, and A. L. Gaeta, “All-optical switching of a single resonance in silicon ring resonators,” Opt. Lett. 36(8), 1413–1415 (2011). [CrossRef] [PubMed]

] or free-carrier injection effect [25

25. S. Sandhu, M. L. Povinelli, and S. H. Fan, “Stopping and time reversing a light pulse using dynamic loss tuning of coupled-resonator delay lines,” Opt. Lett. 32(22), 3333–3335 (2007). [CrossRef] [PubMed]

] in silicon. Since the response time of the loss tuning using these two effects in silicon can be less than 1 ns [25

25. S. Sandhu, M. L. Povinelli, and S. H. Fan, “Stopping and time reversing a light pulse using dynamic loss tuning of coupled-resonator delay lines,” Opt. Lett. 32(22), 3333–3335 (2007). [CrossRef] [PubMed]

, 26

26. Y. H. Wen, O. Kuzucu, T. G. Hou, M. Lipson, and A. L. Gaeta, “All-optical switching of a single resonance in silicon ring resonators,” Opt. Lett. 36(8), 1413–1415 (2011). [CrossRef] [PubMed]

], it implies that the ultrafast reversible Fano resonance and the reversible Fano resonance-based ultrafast optical switches or routers in the silicon RCMZI may be feasible and promising. Furthermore, due to the large modulation depth of the loss tuning based on the two effects, the tunable Fano resonance with the slope and the asymmetry parameter q of wider tuned range in the silicon RCMZI can be expected.

4. Experiment method

For the presented experiment, to obtain the accurate experiment results of T2(ω), Tout(ω), and τg2(ω), it is crucial to measure some additional assistant parameters besides the corresponding voltage values V1, V2, and V3 of the preceding simultaneously measured light intensities (I1, I2, and I3) recorded by DET 1, DET 2 and DET 3 in Fig. 4. For example, in order to avoid the influence of propagation loss of light field on the experiment result of T2(ω), we measure the additional light intensity I1 by the photodetector DET 1, when the CRIT structure is unloaded into the RCMZI (in this case, only Coupler 2 that is not connected to Coupler 1 is embedded into the RCMZI). According to the expressions I1=I0ρin2T2(ω)ρA2atr and I1=I0ρin2ρ22ρA2atr of I1 for the RCMZI with the loaded and unloaded CRIT structure, the more precise normalized transmission spectrum T2(ω) can be deduced by the formula T2(ω)=ρ22I1/I1 in that the effective propagation loss atr which exists in I1 and I1 cancels each other out, where ρin, ρA and atr represent the reflection coefficient of Input coupler, the reflection coefficient of Coupler A and the effective propagation loss of light field in the pathway from Input coupler to WDM 2 (this loss includes the propagation loss of light field from Input coupler to WDM 2 and the total insert loss induced by Input coupler, Coupler 2, Coupler A, and WDM 2), respectively. Likewise, to prevent the measurement of Tout(ω) from the impact of the similar loss, the additional light intensity I2 needs to be detected by the photodetector DET 2, when the resonance arm is isolated from the RCMZI (in this case, the RCMZI only consists of Input coupler, the reference arm, and Output coupler). Assume that the effective propagation losses of light field in the resonance and reference pathways are identical. Thus, the measured I2 can be considered as I2=I0Tout(ω)aref and I2=I0(1ρin2)ρB2(1ρout2)aref for the RCMZI with and without the resonance arm, respectively. The accurate normalized interference transmission spectrum Tout(ω) can be obtained by Tout(ω)=(1ρin2)ρB2(1ρout2)I2/I2 without the deviation caused by the effective propagation loss represented by aref, where ρout, ρB and aref denote the reflection coefficient of Output coupler, the reflection coefficient of Coupler B and the effective propagation loss of light field in the reference pathway (including the propagation loss through the reference arm from Input coupler to WDM 3 and the total insert loss induced by Input coupler, Coupler B, Output coupler, and WDM 3). To infer the accurate experiment result of τg2(ω), the required voltage values (V1, V2, and V3) of I1, I2, and I3 that are simultaneously recorded by the three different photodetectors (DET 1, DET 2 and DET 3) in the dispersion tuning process should be utilized with caution, since the recorded values (V1, V2, and V3) are not only determined by the light intensity values of I1, I2, and I3 but also influenced by the similar effective propagation loss of light field and the sensitivities of the three photodetectors. To circumvent the complex respective measurements of the effective propagation loss and the sensitivity, we first detect the additional light intensites I1 and I2 by DET 1 and DET 2 for the ratio χA=V2/V1 between the voltage value V1 of I1 and the voltage value V2 of I2, respectively, when the reference arm is isolated from the RCMZI; secondly, the additional light intensities I2 and I3 are detected by DET 2 and DET 3 for the ratio χB=V2/V3 between the voltage value V2 of I2 and the voltage value V3 of I3, respectively, when the resonance arm is isolated from the RCMZI. Taking the sensitivities of the photodetectors (DET 1, DET 2 and DET 3) and the effective propagation loss into account, these voltage values can be expressed by
V1=η1I1,
(9)
V2=η2I2=η2I1(1ρA2)ρout2aA/ρA2,
(10)
V2=η2I2=η2I3ρB2(1ρout2)aB/(1ρB2),
(11)
V3=η3I3,
(12)
where aA, aB, and ηi describe the effective propagation loss in the pathway from Coupler A to WDM 3, the effective propagation loss in the pathway from Coupler B to WDM 3, and the sensitivity of DET i (i = 1,2,3), respectively. In Eqs. (9)-(12), I1 and I2 are the light intensity values of I1 and I2 for the RCMZI without the reference arm, respectively, whereas I2 and I3 are the light intensity values of I2 and I3 for the RCMZI without the resonance arm, respectively. Substituting the voltage values V1, V2, V2, and V3 into the ratios χA=V2/V1 and χB=V2/V3, we derive
χA=η2(1ρA2)ρout2aA/(η1ρA2),
(13)
χB=η2ρB2(1ρout2)aB/[η3(1ρB2)].
(14)
Additionally, while the dispersion of the CRIT structure of the whole RCMZI being tuned, the recorded voltage values V1, V2, and V3 corresponding to I1, I2, and I3 are
V1=η1I1=η1I0ρin2T2(ω)ρA2atr,
(15)
V2=η2I2=η2I1(1ρA2)ρout2aAρA2+η2I3ρB2(1ρout2)aB(1ρB2)+2η2I1I3(1ρA2)ρout2aAρB2(1ρout2)aBρA2(1ρB2)×cos(θ2Δϕ),
(16)
V3=η3I3=η3I0(1ρin2)(1ρB2)aB/aref.
(17)
Using Eqs. (13)-(17) and the measured ratios χA=V2/V1 and χB=V2/V3, we can avoid the complex measurements and readily acquire the accurate experiment result of the phase shift of the CRIT structure by the formula as follows:
θ2(ω)=Δϕ+arccos[(V2χAV1χBV3)2χAV1χBV3].
(18)
Therefore, the accurate experiment result of the group delay τg2(ω) can be inferred by taking the derivative of the phase shift θ2(ω) acquired by Eq. (18) with respect to angular frequency ω of light field.

Moreover, to obtain the accurate loss parameter a2 of Ring 2, we also measure the transmission spectrum of Ring 2 by the foregoing measurement method applied for T2(ω) and the experiment setup shown in Fig. 4, when the CRIT structure is not completely fabricated and consists of only Ring 2. By fitting the measured transmission spectrum using the formula |T2(ω)|=|[ρ2a2ρ1exp(iωτ2)]/[1ρ2a2ρ1exp(iωτ2)]|2 of the transmission spectrum from the transfer matrix theory, the accurate loss parameter (a2=0.79) of Ring 2 can be achieved.

5. Conclusion

In conclusion, our theoretical investigation reveals that the slope and the asymmetry parameter (q) characterizing the Fano resonance in a CRIT structure coupled MZI are dependent on the dispersion of the CRIT structure, and the dispersion transition of the CRIT structure may give rise to the reversible Fano resonance. Moreover, we fabricate the CRIT structure coupled fiber MZI, and experimentally observe the dispersion transition of the CRIT structure tuned from abnormal dispersion to normal dispersion and the resulting reversible Fano resonance. These experiment results verify our theoretical model, demonstrating that the reversible Fano resonance which may be exploited in some attractive device applications such as biochemical sensors, optical switches and routers can be realized by the dispersion tuning scheme in the RCMZI. It is noteworthy that the reversible Fano resonance in the RCMZI has more significant implications for optical information process such as ultrafast optical switching and routing, if one implements the silicon-based RCMZI on SOI.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (NSFC) under Grants No. 61078006 and No. 61275066.

References and links

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X. Yang, M. Yu, D.-L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102(17), 173902 (2009). [CrossRef] [PubMed]

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G. Shvets and Y. A. Urzhumov, “Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances,” Phys. Rev. Lett. 93(24), 243902 (2004). [CrossRef] [PubMed]

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V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99(14), 147401 (2007). [CrossRef] [PubMed]

8.

C. Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003). [CrossRef]

9.

W. Liang, L. Yang, J. K. S. Poon, Y. Y. Huang, K. J. Vahala, and A. Yariv, “Transmission characteristics of a Fabry-Perot etalon-microtoroid resonator coupled system,” Opt. Lett. 31(4), 510–512 (2006). [CrossRef] [PubMed]

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B. B. Li, Y. F. Xiao, C. L. Zou, Y. C. Liu, X. F. Jiang, Y. L. Chen, Y. Li, and Q. H. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. 98(2), 021116 (2011). [CrossRef]

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M. F. Yanik, S. H. Fan, and M. Soljačić, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83(14), 2739–2741 (2003). [CrossRef]

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Y. Lu, J. Q. Yao, X. F. Li, and P. Wang, “Tunable asymmetrical Fano resonance and bistability in a microcavity-resonator-coupled Mach-Zehnder interferometer,” Opt. Lett. 30(22), 3069–3071 (2005). [CrossRef] [PubMed]

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X. Yang, C. Husko, C. W. Wong, M. B. Yu, and D.-L. Kwong, “Observation of femtojoule optical bistability involving Fano resonances in high-Q/Vm silicon photonic crystal nanocavities,” Appl. Phys. Lett. 91(5), 051113 (2007). [CrossRef]

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L. J. Zhou and A. W. Poon, “Fano resonance-based electrically reconfigurable add-drop filters in silicon microring resonator-coupled Mach-Zehnder interferometers,” Opt. Lett. 32(7), 781–783 (2007). [CrossRef] [PubMed]

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23.

F. Cheng, H. F. Liu, B. H. Li, J. Han, H. Xiao, X. F. Han, C. Z. Gu, and X. G. Qiu, “Tuning asymmetry parameter of Fano resonance of spoof surface plasmons by modes coupling,” Appl. Phys. Lett. 100(13), 131110 (2012). [CrossRef]

24.

L. J. Zhou and A. W. Poon, “Silicon electro-optic modulators using p-i-n diodes embedded 10-micron-diameter microdisk resonators,” Opt. Express 14(15), 6851–6857 (2006). [CrossRef] [PubMed]

25.

S. Sandhu, M. L. Povinelli, and S. H. Fan, “Stopping and time reversing a light pulse using dynamic loss tuning of coupled-resonator delay lines,” Opt. Lett. 32(22), 3333–3335 (2007). [CrossRef] [PubMed]

26.

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OCIS Codes
(130.6010) Integrated optics : Sensors
(260.2030) Physical optics : Dispersion
(260.5740) Physical optics : Resonance
(230.4555) Optical devices : Coupled resonators
(130.4815) Integrated optics : Optical switching devices

ToC Category:
Integrated Optics

History
Original Manuscript: January 7, 2013
Revised Manuscript: March 25, 2013
Manuscript Accepted: March 25, 2013
Published: April 1, 2013

Citation
Yundong Zhang, Xuenan Zhang, Ying Wang, Ruidong Zhu, Yulong Gai, Xiaoqi Liu, and Ping Yuan, "Reversible Fano resonance by transition from fast light to slow light in a coupled-resonator-induced transparency structure," Opt. Express 21, 8570-8586 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8570


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References

  1. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev.124(6), 1866–1878 (1961). [CrossRef]
  2. F. Hao, Y. Sonnefraud, P. Van Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry breaking in plasmonic nanocavities: subradiant LSPR sensing and a tunable Fano resonance,” Nano Lett.8(11), 3983–3988 (2008). [CrossRef] [PubMed]
  3. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B65(23), 235112 (2002). [CrossRef]
  4. S. H. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett.80(6), 908–910 (2002). [CrossRef]
  5. X. Yang, M. Yu, D.-L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett.102(17), 173902 (2009). [CrossRef] [PubMed]
  6. G. Shvets and Y. A. Urzhumov, “Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances,” Phys. Rev. Lett.93(24), 243902 (2004). [CrossRef] [PubMed]
  7. V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett.99(14), 147401 (2007). [CrossRef] [PubMed]
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