Analysis and modeling of Fano resonances using equivalent circuit elements

Fano resonance presents an asymmetric line shape formed by an interference of a continuum coupled with a discrete autoionized state. In this paper, we show several simple circuits for Fano resonances from the stable-input impedance mechanism, where the elements consisting of inductors and capacitors are formulated for various resonant modes, and the resistor represents the damping of the oscillators. By tuning the pole-zero of the input impedance, a simple circuit with only three passive components e.g. two inductors and one capacitor, can exhibit asymmetric resonance with arbitrary Q-factors flexiblely. Meanwhile, four passive components can exhibit various resonances including the Lorentz-like and reversely electromagnetically induced transparency (EIT) formations. Our work not only provides an intuitive understanding of Fano resonances, but also pave the way to realize Fano resonaces using simple circuit elements.

formations) can be imitated by four electric components. This approach has well-defined Fano-like effective properties and opens up the possibilities to construct extremely high-Q-factor devices while maintaining the simplification of the system. Besides, they can be a guidance for design in microwave or optical circuits and, in particular, for periodic artificial electromagnetic materials. Additionally, it is interesting to note that the passive circuit approach and the theoretical propositions presented in this work processes to achieve high-precision and compressed-composition components 19 .

Fano resonances without damping
First, we consider Fano resonances without damping, where the circuits are schematically shown in Fig. 1(a). We calculate the stable-input impedance of the circuit which is embedded in the single-input-single-output (SISO) system 20 , and tune its pole-zero 21 . The transmittance is defined as S 21 = P output /P input , where P input and P output are the incident and transmitted power, respectively. The stable-input impedance of series inductor-capacitor (LC) circuit and parallel LC circuit are given as in series s s s 0 0 and in parallel p p p 0 0 respectively, where the resonant frequencies ω s p s p s p 0/ 0 / / depend on the inductor L s/p = 1.0132 nH and the capactor C s/p = 1 pF. In Eq. (1), the stable-input impedance of series LC circuit Z in_series has the zeros ω = ± ω s0 and the poles ω = 0. Here, the negative frequency ω = −ω s0 are ignored due to its physical-meaningless. Then, the input impedance of series circuit is shorted to the ground at the zero ω = ω s0 , which leads the input energy total reflected and the transmittance is lowest S 21 = 0 as the solid line in Fig. 1(b). In Eq. (2), the input-impedance function of parallel LC circuit has the zero ω = 0, and the poles ω = ± ω p0 . Excluding the physical-meaningless pole ω = − ω p0 , the input-impedance of parallel LC circuit is infinite at the pole point ω = ω p0 and the transmittance is all-pass S 21 = 1 as the dashed line in Fig. 1(b). From Eq. (1,2), the steep in the vicinity of ω s0 and ω p0 is proportional to the inductor L s in the series circuit, and inversely proportional to the capacitor C p in the LC-parallel circuit. Therefore, the Q factor can be adjusted by the inductor L s and the capacitor C p as shown in Fig. 1(c,d), meanwhile, the corresponding capacitor C s and inductor L p is modified for the remaining of resonant frequency ω 0s/p = 5 GHz.
Here the Q-factor is expressed as where ω 0 is the central resonant frequency, and ω L , ω H are the half-amplitude frequencies lower and higher than ω 0 . In Fig. 1(c), the series-LC Q-factor are 10.8, 6.5 and 2 for the various inductor L s = 5 nH, 3 nH and 1 nH. which presents the series-LC resonance sharper with decreasing series inductor L s . In Fig. 1(d), the parallel-LC Q-factor are 2.27, 1.36 and 0.45 for the various capacitor C p = 5 pF, 3 pF and 1 pF, which presents the parallel-LC resonance sharper with increasing parallel capacitor C p .
Based on the above analysis, we can build the Fano-like asymmetric resonance by a series-LC circuit which represents the narrowband-dark mode coupling with a capacitor or an inductor as the broadband-bright mode, as shown in Fig. 2(a,d). Here we use the stable-input impedance method instead of oscillators-dynamic equations in spectra domain to reveal the mechanism of the asymmetric-coupling modes. In Fig. 2(a), the complementary capacitor C c is added parallel to the series-LC resonance, and the the stable-input impedance of this circuit system is: Abandoning the physical meaningless solutions, we get the pole of stable-input impedance ω which is greater than the zero ω s0 . In addition, the zeros and poles are corresponding to the reflect and transparent resonant frequencies respectively in main-energy thread. Therefore, the transmittance can steep down to zero ω s0 at the higher-frequent pole ω , and presents the formation of Fano-like asymmetric resonance and and high-Q factor. Further, we can get the infinite-Q-factor by turning the pole greatly close to the zero through increasing the complementary capacitor C c and decreasing the series capacitor C s . Here we maintain the series-resonant frequency ω s0 = 5 GHz, and increase the complementary capacitor C c = 20 pF, 50 pF, 100 pF, that leads to the transparent resonance 5.132 GHz, 5.050 GHz and 5.025 GHz closing to the reflect resonance ω s0 = 5 GHz gradually, and the resonance becomes sharper, as shown in Fig. 2(b). When the complementary capacitor C c = 20 pF and decreasing the series capacitor C s = 1 pF, 0.5 pF, 0.1 pF, under the conditions of the series inductor L s changing correspondingly for maintaining the series-resonant frequency ω s0 = 5 GHz, the transparent resonance is 5.132 GHz, 5.062 GHz, 5.013 GHz closing to the reflect resonance ω s0 = 5 GHz gradually, and the Q-factor becomes higher, as shown in Fig. 2(c).
The complementary inductor L c is parallel-added in the series-LC circuit, as shown in Fig. 2(d), and the the stable-input impedance is: 986 GHz closing to the reflect resonance ω s0 = 5 GHz gradually, and the Q-factor becomes higher, as shown in Fig. 2(f).
In Fig. 2(b), the Q-factor is 2513, 1263 and 197 for C c = 100 pF, 50 pF and 20 pF, which presents the resonance sharper with the increasing the complementary capactor C c . In Fig. 2(c), the Q-factor is 1671, 361.6 and 197.1 for C s = 0.1 pF, 0.5 pF and 1 pF, which presents the resonance sharper with decreasing the series capacitor C s . In Fig. 2(e), the Q-factor is 4976, 203.4 and 51.85 for L c = 0.01 nH, 0.05 nH and 0.1 nH, which presents the resonance sharper with decreasing the complementary inductor L c . In Fig. 2(f), the Q-factor is 831, 712 and 225 for L s = 20 nH, 15 nH and 5 nH, which presents the resonance sharper with increasing the complementary inductor L s .
We build the series and parallel resonant circuits parallel in the main-energy thread as shown in Fig. 3(a), and analyze the stable-input impedance: Here we maintain the series-circuit elements L s = 1.0132 nH, C s = 1 pF and thus the series-resonant frequency ω s0 = 5 GHz. Abandoning the physical meaningless solutions, when the series and parallel resonant frequencies satisfying ω p0 ≪ ω s0 , the pole ω B of Eq. (5) satisfies ω B ≈ 0, and the other pole ω A is little higher than the zero ω s0 . Therefore, the closing of pole and zero can construct transparent-asymmetric and high-Q-factor resonance. Based on the above analysis, we can set the parallel elements L p = 1.1032 nH, C p = 0.1 pFand thus the parallel-resonant frequency ω p0 = 0.503 GHz. which leads to the transparent resonant frequency ω B = 5.246 GHz closing to the zero ω s0 , shown as the dashed line in Fig. 3(b). When the series and parallel resonant frequencies satisfying ω p0 ≫ ω s0 , the pole ω ω ≈ 2 A p 0 is far from ω s0 , and the other pole ω B ≈ ω s0 is little lower than the zero ω s0 in Eq. (5) which constructs the transparent-asymmetric and high-Q-factor resonance. We set the parallel elements L p = 0.1 nH, C p = 0.1 pF and thus the parallel-resonant frequency ω p0 = 50.329 GHz, which leads to the transparent resonant frequency ω B = 4.768 GHz closing to the zero ω s0 , shown as the solid line in Fig. 3(b). When we set the parallel elements L p = 1.0132 nH, C p = 7 pF, and parallel-resonant frequency ω p0 = 1.9 GHz. Thus, from the solution of Eq. (5), the pole ω B = 4.753 GHz is located lower than the zero ω s0 which forms the Lorentz-like resonance, and the other pole ω A = 6.392 GHz is little higher than the zero ω s0 which forms the transparent-asymmetric and high-Q-factor resonance, shown as the dashed line in Fig. 3(c). When the poles ω s0 − ω B = ω A − ω s0 distribute even around the zero ω s0 , the two resonant frequencies locate asymmetric and a sharp reflect-resonance is formed at the zero ω s0 which is likely a converse reversely EIT formation. Here we set the parallel elements L p = 0.2993 nHand C p = 5 pF, thus, the the parallel resonant frequency ω p0 = 4.1144 GHz. The transparent resonant frequencies ω A = 5.895 GHz, ω B = 3.49 GHz, which forms two mirror symmetrical resonance, as shown the solid line in Fig. 3(c), and the reflect resonance ω s0 = 5 GHz is also like a reversely EIT phenomenon.

The asymmetric resonance with damping
Here we add the resistors R s , R c as damping in the series resonant circuit consisting of the series inductor L s = 1.0132 nH, the series capacitor C s = 1 pF and the complementary capacitor C s = 20 pF, as shown in Fig. 4(a).  Fig. 3(a) with different parallel-resonant frequencies ω p0 = 0.503 GHz, 50.329 GHz. (c) The transmittance of the circuit system in Fig. 3(a) with different parallelresonant frequencies ω p0 = 1.9 GHz, 11.18 GHz.
We set the resistor R s = 0.1 ohm and R c = 0.1 ohm respectively, which leads to the amplitude of the transmittance is lower than no-damping, but the shape of the asymmetric resonance remains unchanged, as shown in Fig. 4(b).

Conclusion
In conclusion, we show that the Fano resonance can be interpreted as an analogy with the stable-input impedance mechanism by taking passive circuit system as an example. Based on the circuit theory, only three passive components (such as two inductors and one capacitor) can mimic arbitrary Q-factor asymmetric resonance flexiblely by adjusting the pole-zero of the impedance. Furthermore, four passive components can imitate the various resonance (such as Lorentz-like and reversely EIT formations). Besides, our work provides an intuitive understanding of the Fano resonance using briefly electric formation and processes to achieve high-precision by compressed-composition components.