1 Introduction

Spectral line shapes afford fundamental information for the physical processes in the scattering, fluorescence, absorption, and so on (Ott et al. 2013). The most common observed spectral line shapes are well-known symmetric Lorentzian line shapes, which can be explained by Lorentzian oscillators. However, in some atomic and optic systems, the asymmetric line shapes are usually observed, such as photoionization in atoms and Wood anomalies in gratings (Miroshnichenko et al. 2010). The theory breakthrough with a superposition principle from quantum mechanics was put forward by Ugo Fano in 1935, who suggested unique interference phenomena between a discrete state and a continuum state in atomic physics (Miroshnichenko et al. 2010). Unlike the Lorentzian resonance which can be fitted by a Lorentz formula, a Fano formula should be used for the asymmetric spectral line shape.

Recently, the blooming of asymmetric spectral line shapes in the transmission/absorption spectra in metamaterials as well as the scattering spectra in plasmonic systems regain the vitality of Fano resonance in optic systems. For instance, sharp Fano resonances have been observed in metamaterials with weak asymmetric split ring resonators (Luk’yanchuk et al. 2010; Singh et al. 2011). Fano resonances have also been observed in a plenty of novel plasmonic nanostructures such as plasmonic oligomers, heptamer, nanoclusters, nanocubes as well as the nanocavities (Luk’yanchuk et al. 2010; Fan et al. 2010a; Fang et al. 2011; Hao et al. 2008, 2009; Lassiter et al. 2010; Mirin et al. 2009; Sonnefraud et al. 2010; Verellen et al. 2009; Ye et al. 2012; Zhang et al. 2011). In physics, these structures can be designed and achieved quality factors of approximate 50 with good geometrical and chemical tunability. In applications, these sharp Fano resonances compared with Lorentz resonances could be exploited for notch filters, highly selective narrowband photo-emitters, biological and chemical sensors, and slow-light devices (Rahmani et al. 2013; Yanik et al. 2011; Fedotov et al. 2007). The most promising applications among them are the ultrasensitive sensing devices due to Fano resonances are highly sensitive to the dielectric environment. These also give us a chance to improve the surface enhanced infrared absorption (SEIRA) and surface enhanced Raman scattering (SERS) (Lee et al. 2007; Hao et al. 2008; Lal and Halas 2010; Banholzer et al. 2008; Liu et al. 2009) by the designed enhanced local electric field (ELEF) (Xu et al. 2011). As such, much effort has been taken for further excavate the Fano resonances as well as the applications based on both intensity change and phase shift in Fano resonances (Ott et al. 2013; Genet et al. 2003; Riffe 2011; Fano 1961).

On one hand, Fano resonances in metamaterials and plasmonic materials can be explained by the coupling of two classical Lorentzian oscillators with electromagnetic waves as driving force (Joe et al. 2006). On the other hand, Fano resonances suggest a good approximation of constructive and destructive interference of discrete states and continuum states [viewed as dark and bright states (Fan et al. 2010a)]. However, few works have been done on the transition from Lorentz resonances to Fano resonances. As Lorentz resonances can be tuned easily and understood well in metamaterials and plasmonic systems by geometry design, the Lorentz spectral parameters can be used to tune the Fano spectral parameters such as Fano line shape, Fano resonant frequency, as well as the Fano line-width artificially.

In this paper, the relationships of Lorentz parameters and Fano parameters have been studied. It is revealed that the influence of the damping constant of the Lorentz resonance is an approximate parabolic dependence with Fano spectral line shape parameter, Fano resonant frequency, and Fano line-width in the Fano resonance. Both Fano resonant frequency and line-width are parabolic dependent with the coupling coefficient of the Lorentz oscillators. However, Fano spectral line shape parameter demonstrates a saturation effect with the Lorentz coupling coefficient. A distinct transition can be observed in the Fano parameters when we tune the eigen-frequencies of the coupled Lorentz oscillators. Our results suggest that the controlling of Fano resonance can be done by the artificial designing of Lorentz resonance in metamaterials and plasmonic materials.

2 Coupling Lorentzian oscillators and Fano formula

2.1 Coupling Lorentzian oscillators

As a classic description, two coupled oscillators can be used to describe the Fano resonance. These coupled oscillators are equivalent to a discrete state and a continuum state in a quantum description. The coupled oscillators under the driving of external electromagnetic wave can be written as:

$$\begin{aligned} & {\text{x}}^{\prime \prime } +\upgamma_{1} {\text{x}}_{1}^{\prime } +\upomega_{1}^{2} {\text{x}}_{1} +\upupsilon_{12} {\text{x}}_{2} = {\text{Ae}}^{{{\text{i}}\upomega{\text{t}}}} \\ & {\text{x}}^{\prime \prime } +\upgamma_{2} {\text{x}}_{2}^{\prime } +\upomega_{2}^{2} {\text{x}}_{2} +\upupsilon_{21} {\text{x}}_{1} = 0 \\ \end{aligned}$$
(1)

where ω1 and ω2 are the eigen-frequency of independent oscillators. γ1 and γ2 represent the damping constants, and ν12 describes the coupling coefficient of the two oscillators. The amplitude of the first oscillator can be expressed as follows (Joe et al. 2006):

$${\text{c}}_{1} = {\text{A}}\frac{{\upomega_{2}^{2} + {\text{i}}\upomega \upgamma _{2} -\upomega^{2} }}{{\left( {\upomega_{1}^{2} + {\text{i}}\upomega \upgamma _{1} -\upomega^{2} } \right)\left( {\upomega_{2}^{2} + {\text{i}}\upomega \upgamma _{2} -\upomega^{2} } \right) -\upnu_{12}^{2} }}$$
(2)

In the region near ω1, the spectral response still shows a Lorentzian spectral line (Joe et al. 2006). However, in the region near ω2, the spectral response first goes into an anti-resonance near the ω2 due to the destructive interaction (Fig. 1). Then the spectral response goes into a peak position at an eigen-mode of the coupled system (Fig. 1).

Fig. 1
figure 1

Fano resonance line shape change with the damping constant γ2

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A Lorentz oscillator (Xu et al. 2006) is usually used for the localized response. In the classic analogy of Fano resonance (Joe et al. 2006), a continuum response is usually introduced by γ2 = 0. However, in the case of γ2 ≠ 0, Fano resonance can also happen as this case corresponds to the interaction of a discrete state with a quasi-continuum state. As shown in Fig. 1, when γ2 ≠ 0, the amplitude spectra is still asymmetric. However, the resonances become blur and broad with the increasing of γ2. As Fano resonances in plasmonic systems (Fan et al. 2010b) are usually broader than that in atomic systems (Ott et al. 2013), this suggests that Fano resonances in plasmonic systems can be viewed more likely the interaction of a discrete state with a quasi-continuum state.

2.2 Fano formula

In a quantum description, the couple of a continuum state with a discrete state gives rise to the destructive and constructive phenomena (Fano 1961; Miroshnichenko et al. 2010; Luk’yanchuk et al. 2010), which gives the asymmetric spectral line shape as follows (Miroshnichenko et al. 2010):

$$\upsigma = \frac{{(\upvarepsilon + {\text{q}})^{2} }}{{1 +\upvarepsilon^{2} }}$$
(3)

where q is a phenomenological spectral line shape parameter representing the excitation probability ratio between the continuum and discrete state. It depends on the phase angle ϕ as \(q(\varphi ) = - \cot (\varphi )\) (Giannini et al. 2011). The lower the value of q is, the higher the asymmetry of line shape is. In the extreme case, when q drops to zero, the line shape becomes a symmetric anti-resonance (Riffe 2011; Mirin et al. 2009). ε is the reduced energy, which can be expressed as:

$$\upvarepsilon = \frac{{{\text{E}} - {\text{E}}_{0}}}{{\left({\frac{\Gamma}{2}} \right) \times \hbar}}$$
(4)

ε depends on the energy of the incident electromagnetic wave \(({\text{E}} = \hbar\upomega),\) the Fano resonant energy \(({\text{E}}_{0} = \hbar\upomega_{0} )\) and its spectral line-width Γ. The Fano resonance mode is related to the discrete state (ω1) plus a shift ω0 = ω1 + Δ (where Δ is due to the interaction with the continuum state ω2) (Chang et al. 2005). The reciprocal of Γ is the lifetime of the Fano resonance.

3 Transition from Lorentz resonance to Fano resonance

Both classic (Lorentz) and quantum (Fano) models can be used to describe the asymmetric spectral line shapes (Gallinet and Martin 2011). The connection of q, ω0, Γ with the γ1, ν12, ω1, ω2, would be important for tuning and manipulating classic system to mimick the quantum phenomena in metamaterial and plasmonic materials. Numerical method is used to find the intrinsic relationship between them.

Figure 2a demonstrates the resonant behavior in the coupled oscillator system with different damping constant value γ1, while keeps the other parameters unchanged. It is evident that both anti-resonance and Fano resonance appear when γ1 ≠ 0. With the increasing of γ1, the intensity of anti-resonance increases, while the intensity of Fano resonance decreases. Although the frequency of anti-resonance (ω2) does not change anymore, the frequency of Fano resonance redshift with the increasing of γ1. Due to the coupling, the Fano resonance becomes broad gradually with the increasing of γ1. As the first oscillator (discrete state) sufferred a much stronger damping leading to a lower efficiency of coupling with the second oscillator (continuum state), the Fano resonance becomes broader at the same time. After numerical fitting with Fano formulas (3) and (4), we draw out the connections of q, ω0 and Γ of Fano resonances with γ1 (Fig. 2b). The q demonstrates nonlinear decrease with the increasing of damping constant γ1, which follows an approximately parabolic function. The resonant frequency ω0 decreases with the increasing of γ1, which can be described as a parabolic function. However, the line-width of ω0 increases with the increasing of γ1. These results are consistent with the spectral line-shape change in Fig. 2a. In one extreme case γ1 = 0, there just exist two coupled continuum states, which is symmetric in the spectral line shape. In another extreme case γ1 → \(\infty ,\) the spectral line shape becomes a symmetric anti-resonance (Mirin et al. 2009). These suggest that we can observe a gradual transition not only from a symmetric line shape to an asymmetric line shape but also from a resonance to an anti-resonance by tuning γ1. These afford a playground for mimicking quantum phenomenas by the classic system design.

Fig. 2
figure 2

a Resonant behavior of the coupled system under different γ1. b Mapping of Fano parameters q, ω0, Γ with the change of γ1

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Figure 3a demonstrates the resonant behavior of the coupled system under different coupling coefficients of the Lorentz oscillators ν12, while keeps the other parameters constant. It is shown that Fano resonant frequency moves to high frequency region, while the width becomes broader with the increasing of ν12. In the extreme case ν12 = 0, the discrete state does not couple to the continuum state with the spectral line shape as a Lorentzian symmetry (Riffe 2011). The relationship between ν12 and q, ω0, Γ are plotted in Fig. 3b. It is evident that q is increasing with the increasing of ν12 in the low value region, but it will saturate when we further increases the ν12. This means that the asymmetric spectral line-shape could reach the maximum, when the two states are most widespread overlap (Rau 2004). At the same time, Fano resonant frequency moves to high frequency (blue shift) with a parabolic dependence. The line-width of the Fano resonance also increases with the increasing of the coupling strength and can be described as a parabolic function. The results suggested that there exists a moderate coupling strength for better Fano resonance design.

Fig. 3
figure 3

a Resonant behavior of the coupled system under different ν12. b Mapping of Fano parameters q, ω0, Γ with the change of ν12

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Figure 4a shows the Fano resonant behavior of the coupled oscillators with the change of ω1. It is evident that there is a transition when ω1 moves close to ω2 as the Fano line-width increases firstly and then decreases when ω1 passes a transition position. The anti-resonance frequency also change as a dip on the left side of Fano resonance before ω1 = 1.21 to a right side of the Fano resonance after ω1 = 1.21. At the resonance position ω1 = ω2 = 1.21, the spectral line shapes show more Lorentz symmetry likely. Figure 4b shows the relationship between ω1 and q, ω0, Γ, which is also in accordance with Fig. 4a. When ω1 approaches ω2 from the low-frequency part, there is little change of the q, ω0, Γ with ω1. However, when ω1 is close to ω2, it can be regarded as a superposition of the two states and the spectral line shape is Lorentz symmetry likely. As ω1 moves away from ω2, the overlap of the coupled oscillators decrease and the values of q, ω0, Γ recover to Fano symmetry likely. It is also clearly that there exists an anti-crossing of Fano resonance with the ω1. At this state, the q and Γ demonstrate a peak with ω1. All these variations suggest that the transition from Lorentz resonance to Fano resonance (Sheikholeslami et al. 2011), which can be designed with the symmetry breaking. This is important as the plasmonic system can be used to mimic the transition effect as tuning ω1 by changing geometry size and geometric symmetry.

Fig. 4
figure 4

a Resonant behavior of the coupled system under different ω1. b Mapping of Fano parameters q, ω0, Γ with the change of ω1

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Similar to the tuning of ω1, we can also tune ω2 as shown in Fig. 5a. It is obvious that the anti-resonance locates at the right-side of the Fano resonance before ω2 = 1.0 and then shifts to the left-side as we further tune ω1. Fano resonant frequency is blue-shifted when the ω2 increases. In this process, the line-width of the Fano resonance has a maximum. Figure 5b shows the relationship between ω1 and q, ω0, Γ, which is in accordance with the change in Fig. 5a. The spectral line parameter q and the line-width Γ firstly increases with the increasing of ω2 and then decreases after they pass the transition position at ω1 = ω2. However, the Fano resonant frequency always increases with the increasing of ω 2 as a linear response. These also demonstrate that the Lorentz resonant frequency can be used to mimic the transition from Lorentz resonances to Fano resonances.

Fig. 5
figure 5

a Resonant behavior of the coupled system under different ω2. b Mapping of Fano parameters q, ω0, Γ with the change of ω2

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4 Conclusions

In summary, we carry out a numerical method to schematically investigate the transition from Lorentz resonances to Fano resonances. It is revealed that the influence of the damping constant of the Lorentz resonance is an approximate parabolic dependence with Fano spectral parameters. Both Fano resonant frequency and line-width are parabolic dependence with the coupling coefficient of the Lorentz oscillators. However, Fano spectral line shape parameter demonstrates a saturation effect with the Lorentz coupling coefficient. A distinct transition can be observed in the Fano parameters when we tune the eigen-frequencies of the coupled Lorentz oscillators. Our results suggest that the artificial controlling of Fano resonance can be done by designing of Lorentz resonance in metamaterials and plasmonic materials.