Abstract
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.
Highlights
Fano resonance has received much attention due to the interesting physics such as distinctly asymmetric shape and high quality-factor (Q-factor)[1]
The circuit system which is an effective-mapping image of the classical mechanics can be devoted to the mechanism of Fano resonance[18]
We formulate the series and parallel circuits consisting of inductors and capacitors for various-resonant modes, and the resistor represents the damping of the oscillators
Summary
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 S21 =Poutput/Pinput, where Pinput and Poutput are the incident and transmitted power, respectively. The stable-input impedance of series inductor-capacitor (LC) circuit and parallel LC circuit are given as respectively, where the resonant frequencies ωs0/p0 = 1/ Ls/pCs/p = 5GHz depend on the inductor Ls/p = 1.0132 nH and the capactor Cs/p = 1 pF. In Eq (1), the stable-input impedance of series LC circuit Zin_series has the zeros ω =±ωs[0] and the poles ω = 0. Here, the negative frequency ω =−ωs[0] are ignored due to its physical-meaningless. Then, the input impedance of series circuit is shorted to the ground at the zero ω =ωs[0], which leads the input energy total reflected and the transmittance is lowest S21 = 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 = ±ωp[0]. Excluding the physical-meaningless pole ω =−ωp[0], the input-impedance of parallel LC circuit is infinite at the pole point ω =ωp[0] and the transmittance is all-pass S21 = 1 as the dashed line in Fig. 1(b). From Eq (1, 2), the steep in the vicinity of ωs[0] and ωp[0] is proportional to the inductor Ls in the series circuit, and inversely proportional to the capacitor Cp in the LC-parallel circuit. Therefore, the Q factor can be adjusted by the inductor Ls and the capacitor Cp as shown in Fig. 1(c,d), meanwhile, the corresponding capacitor Cs and inductor Lp is modified for the remaining of resonant frequency ω0s/p = 5 GHz. Here the Q-factor is expressed as Q =ω0/(ωH −ωL), where ω0 is the central resonant frequency, and ωL, ωH are the half-amplitude frequencies lower and higher than ω0. In Fig. 1(c), the series-LC Q-factor are 10.8, 6.5 and 2 for the various inductor Ls = 5 nH, 3 nH and 1 nH. which presents the series-LC resonance sharper with decreasing series inductor Ls. In Fig. 1(d), the parallel-LC Q-factor are 2.27, 1.36 and 0.45 for the various capacitor Cp = 5 pF, 3 pF and 1 pF, which presents the parallel-LC resonance sharper with increasing parallel capacitor Cp. 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 Cc is added parallel to the series-LC resonance, and the the stable-input impedance of this circuit system is: Zin_series_C Abandoning the physical meaningless solutions, we get the pole of stable-input impedance ωs[0] (Cs/Cc) + 1 in Eq (3) which is greater than the zero ωs[0]. 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 ωs[0] at the higher-frequent pole ωs[0] (Cs/Cc) + 1 with the coefficient (Cs/Cc) + 1 → 1, 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 Cc and decreasing the series capacitor Cs. Here we maintain the series-resonant frequency ωs0 = 5 GHz, and increase the complementary capacitor Cc = 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 ωs0 = 5 GHz gradually, and the resonance becomes sharper, as shown in Fig. 2(b). When the complementary capacitor Cc = 20 pF and decreasing the series capacitor Cs = 1 pF, 0.5 pF, 0.1 pF, under the conditions of the series inductor Ls changing correspondingly for maintaining the series-resonant frequency ωs0 = 5 GHz, the transparent resonance is 5.132 GHz, 5.062 GHz, 5.013 GHz closing to the reflect resonance ωs0 = 5 GHz gradually, and the Q-factor becomes higher, as shown in Fig. 2(c). The complementary inductor Lc is parallel-added in the series-LC circuit, as shown in Fig. 2(d), and the the stable-input impedance is: Zin_series_L = Abandoning the physical meaningless solutions, we get the pole ωs0/ (Lc/Ls) + 1 in Eq (4) lower than the zero ωs[0]. Therefore, the transmittance can steep down to zero at the pole ωs0/ (Lc/Ls) + 1 located lower than the zero ωs[0] when the coefficient (Lc/Ls) + 1 → 1, and presents the formation of Fano-like asymmetric resonance. Further, we can get the infinite-Q-factor by turning the pole point close to the zero point through decreasing the complementary inductor Lc and decreasing the series inductor Ls. Here we maintain the series-resonant frequency ωs0 = 5 GHz, and decrease the complementary inductor Lc = 0.1 nH, 0.05 nH, 0.01 nH, that leads the transparent resonance 4.770 GHz, 4.881 GHz, 4.976 GHz closes to the reflect resonance ωs0 = 5 GHz gradually, and the Q-factor becomes higher, as shown in Fig. 2(e). When the complementary capacitor Lc = 0.1 nH is constant and increasing the series inductor Ls = 5 nH, 15 nH, 20 nH, under the conditions of the series inductor changing correspondingly for maintaining the series-resonant frequency ωs0 = 5 GHz, the transparent resonance is 4.951 GHz, 4.976 GHz, 4.986 GHz closing to the reflect resonance ωs0 = 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 Cc = 100 pF, 50 pF and 20 pF, which presents the resonance sharper with the increasing the complementary capactor Cc. In Fig. 2(c), the Q-factor is 1671, 361.6 and 197.1 for Cs = 0.1 pF, 0.5 pF and 1 pF, which presents the resonance sharper with decreasing the series capacitor Cs. In Fig. 2(e), the Q-factor is 4976, 203.4 and 51.85 for Lc = 0.01 nH, 0.05 nH and 0.1 nH, which presents the resonance sharper with decreasing the complementary inductor Lc. In Fig. 2(f), the Q-factor is 831, 712 and 225 for Ls = 20 nH, 15 nH and 5 nH, which presents the resonance sharper with increasing the complementary inductor Ls. 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: Zseries_parallel jω Cp (ω − ωs0)(ω1 + ωs0) ωA)(ω − ωB)(ω + ωA)(ω ωB ) ( ) ( ) where the poles expressed asωA = ωs20 + 2ωp20 + ωs40 + 4ωp40 /2 and ωB = ωs20 + 2ωp20 − ωs40 + 4ωp40 /2. Here we maintain the series-circuit elements Ls = 1.0132 nH, Cs =1 pF and thus the series-resonant frequency ωs0 = 5 GHz. Abandoning the physical meaningless solutions, when the series and parallel resonant frequencies satisfying ωp0 ≪ ωs[0], the pole ωB of Eq (5) satisfies ωB ≈ 0, and the other pole ωA is little higher than the zero ωs[0]. 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 Lp = 1.1032 nH, Cp =0.1 pFand thus the parallel-resonant frequency ωp0 = 0.503 GHz. which leads to the transparent resonant frequency ωB = 5.246 GHz closing to the zero ωs[0], shown as the dashed line in Fig. 3(b). When the series and parallel resonant frequencies satisfying ωp0 ≫ ωs[0], the pole ωA ≈ 2 ω p0 is far from ωs[0], and the other pole ωB ≈ωs[0] is little lower than the zero ωs[0] in Eq (5) which constructs the transparent-asymmetric and high-Q-factor resonance. We set the parallel elements Lp = 0.1 nH, Cp = 0.1 pF and thus the parallel-resonant frequency ωp0 = 50.329 GHz, which leads to the transparent resonant frequency ωB = 4.768 GHz closing to the zero ωs[0], shown as the solid line in Fig. 3(b). When we set the parallel elements Lp = 1.0132 nH, Cp = 7 pF, and parallel-resonant frequency ωp0 = 1.9 GHz. Thus, from the solution of Eq (5), the pole ωB = 4.753 GHz is located lower than the zero ωs[0] which forms the Lorentz-like resonance, and the other pole ωA = 6.392 GHz is little higher than the zero ωs[0] which forms the transparent-asymmetric and high-Q-factor resonance, shown as the dashed line in Fig. 3(c). When the poles ωs0 −ωB =ωA −ωs[0] distribute even around the zero ωs[0], the two resonant frequencies locate asymmetric and a sharp reflect-resonance is formed at the zero ωs[0] which is likely a converse reversely EIT formation. Here we set the parallel elements Lp = 0.2993 nHand Cp = 5 pF, thus, the the parallel resonant frequency ωp0 = 4.1144 GHz. The transparent resonant frequencies ωA = 5.895 GHz, ωB = 3.49 GHz, which forms two mirror symmetrical resonance, as shown the solid line in Fig. 3(c), and the reflect resonance ωs0 = 5 GHz is also like a reversely EIT phenomenon.
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