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]. To date, Fano resonance can be observed not only in atomic resonance systems but also in artificial photonic structures involving plasmonic nanostructures [2], photonic crystals [3–5], electromagnetic metamaterials [6, 7], and coupled resonators [8–10], 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]; on the other hand, tunable Fano resonances that hold significant implications for device applications such as optical switches, modulators and routers [3, 4, 12–16] can readily be realized by virtue of flexible tunability of these photonic structures [17–21]. Among a variety of tunable Fano resonances, one unique category of which the slope or the asymmetric parameter [1, 11] $q$ 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, 23]. For brevity, we term this category of tunable Fano resonances a “reversible Fano resonance”.

Generally speaking, to realize tunable photonic transport properties such as tunable group delay or tunable bandwidth, optical coupled resonators are a more popular and feasible approach in comparison to other photonic structures, since abundant simple and ultrafast tuning schemes such as refractive index induced phase shift tuning [19–21, 24], loss tuning [25, 26], and dispersion tuning [27] in optical coupled resonator systems have been provided. Nevertheless, the tunability of the Fano resonance in coupled resonator systems is always accomplished by only the phase shift (between resonant and non-resonant pathways) tuning scheme [16, 28]. For example, in indirectly coupled resonators, the reversible (tunable) Fano resonance is observed when the phase shift of light field in the waveguide connecting two indirectly resonators is tuned [28]. Similarly, the reversible (tunable) Fano resonance is experimentally performed in a resonator coupled Mach-Zehnder interferometer (RCMZI) when the phase difference between the resonant and non-resonant pathways of the RCMZI is adjusted [16]. In this paper, we introduce and perform a novel dispersion tuning scheme in a coupled-resonator-induced transparency (CRIT) structure coupled Mach-Zehnder interferometer (MZI) in order to tune the Fano resonance of the RCMZI, demonstrating that the slope and the asymmetric parameter $q$ describing the Fano resonance spectral line shape of the RCMZI experience a sign reversal when the dispersion transition of the CRIT structure from abnormal dispersion (fast light) to normal dispersion (slow light) occurs. Therefore, the theoretical and experimental results in this paper indicate that the profile of the Fano resonance in the RCMZI is determined by the dispersion of the CRIT structure in the RCMZI besides the phase shift difference between the two arms of the RCMZI, and the reversible Fano resonance that has profound implications for optical sensing and optical information process such as optical switching and routing can be realized by the dispersion tuning scheme in the RCMZI.

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 , if the resonator system in the RCMZI is a coupled-resonator-induced transparency [29] (CRIT) structure and the phase shift difference between the reference arm and the resonance arm is $\Delta \varphi$, the normalized interference transmission spectrum (Fano resonance spectral line shape) $T_{out}(\omega )$ at the output port of the Output coupler of the RCMZI can be described by

 

Fig. 1 Schematic of a CRIT structure coupled Mach-Zehnder interferometer.

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Utilizing the transfer matrix theory [30, 31],the response (complex transmission coefficient) of the CRIT structure depending on the angular frequency $\omega$ of light field can be expressed by $t_2(\omega )=\sqrt{T_2}\exp (i{\theta }_2)=[{\rho }_2-a_2{t}_1\exp (i\omega {\tau }_2)]/[1-{\rho }_2{a}_2{t}_1\exp (i\omega {\tau }_2)]$, where ${\rho }_2$, $T_2(\omega )=\left|t_2^2(\omega )\right|$, ${\theta }_2(\omega )=\arg [t_2(\omega )]$, $a_2$, ${\tau }_2$, and $t_1$ 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 $t_1(\omega )=\sqrt{T_1}\exp (i{\theta }_1)=[{\rho }_1-a_1\exp (i\omega {\tau }_1)]/[1-{\rho }_1{a}_1\exp (i\omega {\tau }_1)]$, where ${\rho }_1$, $T_1(\omega )=\left|t_1^2(\omega )\right|$, ${\theta }_1(\omega )=\arg [t_1(\omega )]$, $a_1$, and ${\tau }_1$ 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 $t_i({\omega }_0)$ and ${\theta }_i({\omega }_0)$ by $t_{i0}$ and ${\theta }_{i0}$ $(i=1,2)$, where ${\omega }_0$ (${\omega }_0$ satisfies ${\omega }_0{\tau }_1\mathrm{mod}2\text{π}=0$ and ${\omega }_0{\tau }_2\mathrm{mod}2\text{π}=0$) 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 $\omega$ of light field, one can obtain the slope $S$ of the interference transmission spectrum $T_{out}(\omega )$ at ${\omega }_0$ as follows:

Furthermore, the conventional asymmetric parameter [1, 11] $q$ can be exploited to describe the output asymmetric interference transmission spectral line shape of the RCMZI. Considering only the first order intracavity dispersion [27] of the CRIT structure such as $t_1(\omega )\exp (i\omega {\tau }_2)=t_{10}+i{t}_{10}({\tau }_{g10}+{\tau }_2)(\omega -{\omega }_0)$, we can approximately rewrite the interference transmission profile $T_{out}(\omega )$ in terms of the following Fano resonance formula in the vicinity of ${\omega }_0$:

 

Fig. 2 (a) Dependence of the asymmetry parameter $q$ (solid curve) and the slope $S({\omega }_0)$ (dashed curve) of the Fano resonance of the RCMZI on the loss parameter $a_1$ for different $\delta \varphi$. (b) Dependence of the asymmetry parameter $q$ (solid curve) and the slope $S({\omega }_0)$ (dashed curve) of the Fano resonance of the RCMZI on the loss parameter $a_1$ for $\delta \varphi =0$ and different ${\rho }_2$. (c) Dependence of the asymmetry parameter $q$ (solid curve) and the slope $S({\omega }_0)$ (dashed curve) of the Fano resonance of the RCMZI on the loss parameter $a_1$ for $\delta \varphi =0$ and different ${\rho }_1$. 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 ${\rho }_1=0.938$, ${\rho }_2=0.755$, $a_2=0.79$, ${\tau }_1=15.2\text{ns}$, and ${\tau }_2=18.5\text{ns}$.

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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 $a_1$ of Ring 1 needs to cover either the critical value ${a^{\prime }}_1$ of the loss parameter corresponding to the weak dispersion ${\tau }_{g20}=0$ or the critical value ${a^{″}}_1$ of the loss parameter corresponding to the strong dispersion ${\tau }_{g20}=\infty$. Thus it is instructive to derive the expressions of ${a^{\prime }}_1$ and ${a^{″}}_1$. Using the relation between the group delay ${\tau }_{g2}(\omega )$ and the transfer function $t_2$ of the CRIT structure such as ${\tau }_{g2}(\omega )=(t_2^{*}\partial t_2/\partial \omega -t_2\partial t_2^{*}/\partial \omega )/(2i{t}_2{t}_2^{*})$, the group delay ${\tau }_{g20}$ can be obtained:

For the presented experiment parameters in this paper as follows: ${\rho }_1=\sqrt{0.88}=0.938$, ${\rho }_2=\sqrt{0.57}=0.755$, the measured loss parameter value $a_2=0.79$ of Ring 2, ${\tau }_1=15.2\text{ns}$, and ${\tau }_2=18.5\text{ns}$, one can obtain only ${a^{\prime }}_1=0.720$ located in the variation range $0.58\le a_1\le 0.84$ of $a_1$ in the presented experiment while calculating Eqs. (6), (7) and (8). Thus, one may probe the group delay ${\tau }_{g20}$ depending on $a_1$ in the vicinity of ${a^{\prime }}_1$ 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 $T_2(\omega )$ of the CRIT structure in the vicinity of ${\omega }_0$ exhibits an approximately flat-top (or flat-bottom) profile while the tuned loss parameter $a_1$ approaches ${a^{\prime }}_1$ or the dispersion of the CRIT structure approaches the weak dispersion ${\tau }_{g20}=0$, since Eq. (6) determining ${a^{\prime }}_1$ is also the condition that a white light cavity (WLC) should satisfy [34]. In addition, the interference transmission spectrum $T_{out}(\omega )$ 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 ${\omega }_0$ due to the vanishing slope $S({\omega }_0)$ caused by the weak dispersion ${\tau }_{g20}=0$, when $a_1={a^{\prime }}_1$. As shown in Fig. 2, the discontinuous step change of $q$ corresponding to $a_1={a^{\prime }}_1$ can be negligible in that the asymmetry parameter $q$ is not applicable to describe the symmetry interference transmission spectrum of $S({\omega }_0)=0$ corresponding to the weak dispersion ${\tau }_{g20}=0$.

As shown in Fig. 3 , for the CRIT structure in the respective undercoupled regimes $({\rho }_1>a_1,{\rho }_2>a_2\left|t_1{}_0\right|)$ of Ring 1 and Ring 2, as the loss parameter $a_1$ of Ring 1 that satisfies $a_1<{a^{\prime }}_1=0.720$ (the critical value ${a^{\prime }}_1$ corresponding to the weak dispersion ${\tau }_{g20}=0$) is increased, the group delay exhibits fast light ${\tau }_{g20}<0$ (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 $a_1$ approximates ${a^{\prime }}_1$. Finally, when $a_1$ continues to be increased beyond the critical value ${a^{\prime }}_1$, the group delay exhibits slow light ${\tau }_{g20}>0$ (normal dispersion), and $q$ turns out to be negative and is slowly boosted. Therefore, when $a_1$ increases, on one hand, owing to the dispersion transition, the asymmetry parameter $q$ (and the slope $S({\omega }_0)$) 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 $t_{10}$.

 

Fig. 3 The asymmetry parameter $q$ (solid curve) of the Fano resonance of the RCMZI, the slope $S({\omega }_0)$ (dashed curve) of the Fano resonance of the RCMZI, and the group delay ${\tau }_{g20}$ (dotted curve) of the CRIT structure depending on the loss parameter $a_1$. The corresponding system parameters are ${\rho }_1=0.938$, ${\rho }_2=0.755$, $a_2=0.79$, ${a^{\prime }}_1=0.720$, ${\tau }_1=15.2\text{ns}$, and ${\tau }_2=18.5\text{ns}$.

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3. Experiment setup and results

The whole experimental setup that we use to observe the dispersion transition and the reversible Fano resonance is schematically shown in Fig. 4 . In the setup, we implement a RCMZI by coupling a CRIT structure to an arm of a nearly balanced fiber Mach-Zehnder interferometer (MZI). The resonance (upper) arm and the reference (lower) arm which constitute the MZI are fabricated by telecom single mode optical fiber (SMF-28). The CRIT structure that consists of two fiber ring resonators (Ring 1 of circumference $L_1=3.10\text{m}$ and Ring 2 of circumference $L_2=3.79\text{m}$) is side-coupled to the resonance arm. Er³⁺ doped optical fiber (the doping concentration of Er³⁺ is $5\times {10}^{24}{\text{m}}^{-3}$, and the length of the Er³⁺ doped optical fiber is $1.02\text{m}$) and a wavelength division multiplexer (WDM) are embedded in Ring 1. Thus the loss parameter of Ring 1 can be modulated by varying the input power of a 980 nm laser which pumps the Er³⁺ doped optical fiber. The two single mode fiber couplers Coupler 1 of ${\rho }_1=0.938$ and Coupler 2 of ${\rho }_2=0.755$ constituting Ring 1 and Ring 2 are carefully chosen in order to ensure the respective undercoupled regimes of Ring 1 and Ring 2. The incident light is launched from a 1550 nm tunable laser of narrow linewidth (10KHz) into the RCMZI. To scan the probe wavelength and hence obtain the interference transmission spectra (Fano resonance spectral lines) of the RCMZI, the wavelength of the incident light is linearly tuned by a driven triangular voltage (frequency: 10 Hz, amplitude: 5 V) generated by a signal generator connecting to the piezoelectric ceramic of the 1550 nm laser. The polarization of the incident light is carefully adjusted by a polarization controller next to an attenuator so as to excite one of the eigen polarization modes of each resonator (Ring 1 and Ring 2) and guarantee the interference interaction between the output light waves exiting from the resonance arm and the reference arm. For the two resonators (Ring 1 and Ring 2) of similar size in our experiment, if only two resonance modes exist in the free spectral range (FSR) of Ring 1 (or Ring 2) in the transmission spectrum (indicated by light intensity $I_1$) of the CRIT structure and the on/off ratio of the interference transmission spectrum (indicated by light intensity $I_2$) is highest, the adjusted polarization of the incident light is appropriate for the observations of the transmission spectrum of the CRIT structure and the interference transmission spectrum (Fano resonance) of the RCMZI. Propagating through the optical isolator, the attenuator, the polarization controller, and the 98%-2% coupler in Fig. 4, the incident light field turns out to be the input light field $E_{in}$ of the input light intensity $I_0=\left|E_{in}^2\right|$ corresponding to the input optical power 3.15 mW for the RCMZI. To acquire the transmission spectrum $T_2(\omega )$, the Fano resonance profile $T_{out}(\omega )$, and the group delay ${\tau }_{g2}(\omega )$ of the CRIT structure, the light intensities $I_1$, $I_2$, and $I_3$ shown in Fig. 4 need to be simultaneously measured by the three InGaAs photodetectors DET 1, DET 2, and DET 3 of good linearity depending on optical power, respectively, while the loss parameter of Ring 1 is being modulated. Additionally, the InGaAs photodetector DET 4 is employed to monitor the power fluctuation of the incident light. More detailed experiment process and methods are described in Sec. 4.

 

Fig. 4 Experiment setup. WDM: wavelength division multiplexer. Red and blue lines represent 1550 nm and 980 nm channels of WDMs; DET: InGaAs photodetector; OSC: digital oscilloscope; Input coupler and Output coupler: 3 dB coupler; Coupler A and Coupler B: 3 dB coupler. The configuration in the dashed box is a nearly balanced RCMZI that is implemented by coupling a CRIT structure to an arm of a fiber Mach–Zehnder interferometer.

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Figure 5(a) shows the measured transmission spectra $T_2(\omega )$ 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 ${\omega }_0$, as shown by the red solid curve in Fig. 5(a). Subsequently, when we add the pump power gradually and thus the loss parameter $a_1$ 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]. 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 ${a^{\prime }}_1$ of the loss parameter $a_1$ 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 ($0.047\le T_2(0)\le 0.329$) of the CRIT structure at ${\omega }_0$ is not high in Fig. 5(a). Generally, to obtain the nearly transparent transmission of the CRIT structure at ${\omega }_0$, 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 $({\rho }_2=a_2{t}_1)$ [35]. For the presented experiment in Fig. 5, since the complex transmission coefficient of Ring 1 at ${\omega }_0$ satisfies $0.463\le t_{10}\le 0.785$, the approximately destructive interference through Ring 1 does not occur, and hence $T_2(0)$ is not too high. Nevertheless, $T_2(0)$ can be readily increased, for example, one may choose Coupler 1 of lower ${\rho }_1$ or Coupler 2 of higher ${\rho }_2$.

 

Fig. 5 The experimental (a) and theoretical (b) transmission spectra $T_2(\Delta )$ of the CRIT structure depending on frequency detuning $\Delta =(\omega -{\omega }_0)/2\text{π}$ for the five different loss parameter $a_1$ 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 $a_1$ 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).

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As expected, the experiment results of the group delay of the CRIT structure shown in Fig. 6(a) demonstrate the dispersion transition from abnormal dispersion (fast light ${\tau }_{20}<0$) to normal dispersion (slow light ${\tau }_{20}>0$) in the vicinity of ${\omega }_0$. Figure 6(a) shows that, as the pump power increases, the experimental group delay at ${\omega }_0$ is tuned from a negative value ${\tau }_{g20}=-35.0\text{ns}$(the red curve in Fig. 6(a)) to an approximately vanishing value (weak dispersion) ${\tau }_{g20}=0.8\text{ns}$(the orange curve in Fig. 6(a)) associated with the approximately flat-bottom transmission, and to positive values such as ${\tau }_{g20}=8.0\text{ns}$(the yellow curve in Fig. 6(a)), ${\tau }_{g20}=18.1\text{ns}$(the green curve in Fig. 6(a)), and ${\tau }_{g20}=23.2\text{ns}$(the blue curve in Fig. 6(a)), along with the preceding variation of the transmission spectra shown in Fig. 5(a). In addition, as shown in Fig. 6(b), the experiment group delay curves of the CRIT structure are generally in agreement with those theoretical corresponding results obtained by the transfer matrix theory despite a bit of difference between them. According to Eq. (2) and the experiment method described in the next section, the experimental result of the group delay in Fig. 6(a) is primarily determined by the experiment data (the light intensities $I_1$ and $I_2$) associated with the transmission spectrum $T_2(\omega )$ of the CRIT structure and the interference transmission spectrum $T_{out}(\omega )$ of the RCMZI system. In contrast to the group delay, the experimental result of $T_2(\omega )$ in Fig. 5(a) coincides better with the corresponding theoretical result in Fig. 5(b). Thus, the difference between the experimental and theoretical group delay may be mainly attributed to the interference process between the light fields from the resonance and reference arms (or the experiment data associated with $T_{out}(\omega )$). As shown in Table 1 , since the experimental result of the modulus of the slope $S({\omega }_0)$ is always slightly lower than the corresponding theoretical result (except for $a_1=0.75$), we speculate that there are two sources which degenerate the perfect interference interaction between the light waves from the resonance and reference arms: one is the slightly different propagation losses of the two light waves through the resonance and reference pathways; the other is the slightly different polarization states of the two light waves. Apparently, the two sources can reduce the modulus of the slope $S({\omega }_0)$, for example, when the interference between the two light waves of extremely different amplitudes or two approximately orthogonal polarization states occurs, the slope of the interference transmission spectrum becomes declining and even vanishing. Furthermore, since the group delay is proportional to the slope of the interference transmission spectrum as described by Eq. (2), the two sources can also reduce the experiment result of the modulus of the group delay, which is actually demonstrated as shown in Figs. 6(a) and 6(b) and Table 1. Therefore, the preceding two sources that degenerate the perfect interference interaction may lead to the slight difference between the experimental and theoretical group delay (and the slope $S({\omega }_0)$ of the interference transmission spectrum) shown in Figs. 6(a) and 6(b).

 

Fig. 6 The experimental (a) and theoretical (b) group delay ${\tau }_{g2}(\Delta )$ curves of the CRIT structure depending on frequency detuning $\Delta =(\omega -{\omega }_0)/2\text{π}$ for the five different loss parameter $a_1$ values shown in Fig. 5. The solid curves in Figs. 5(a) and 6(a) that are of same color are simultaneously measured results with the identical loss parameter $a_1$. In Fig. 6(b), each dashed curve is the corresponding theoretical simulation result of the solid curve of same color in Fig. 6(a).

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Table 1. The experimental results of $q$and$S({\omega }_0)$.

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Figure 7(a) illustrates the simultaneously measured interference transmission spectra $T_{out}(\omega )$ of the RCMZI system. For the experiment results in Fig. 7(a), the phase difference $\Delta \varphi$ that satisfies $\text{1}\text{.60}\le \Delta \varphi \le \text{1}\text{.67}$ is approximately $\text{π}/2$, and Ring 1 and Ring 2 are in the respective undercoupled regimes since the measured loss and coupling parameters satisfy $a_1<{\rho }_1$ and $\left|t_{10}\right|a_2<{\rho }_2$. As shown in Fig. 7(a), the experiment results of $T_{out}(\omega )$ 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).

 

Fig. 7 The experimental (a) and theoretical (b) interference transmission spectra $T_{out}(\Delta )$ of the RCMZI versus frequency detuning $\Delta =(\omega -{\omega }_0)/2\text{π}$. (c) The experimental (solid curves) and fitting (dashed curves) interference transmission spectra $T_{out}(\epsilon )$ of the RCMZI versus the reduced frequency detuning $\epsilon$. 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 $a_1$. In Fig. 7(a), for the five different loss parameter $a_1$ values, the phase shift difference $\Delta \varphi$ 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.

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To characterize the tunable Fano resonance quantitatively, we infer the slope $S({\omega }_0)$ of the experimental interference transmission spectra in Fig. 7(a), and fit these spectral line shapes in the vicinity of ${\omega }_0$ 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 $a_1$ is initially 0.58 without pump and the dispersion of the CRIT structure is abnormal dispersion due to ${\tau }_{g20}=-35.0\text{ns}$, the resulting interference transmission spectrum exhibits an asymmetry Fano resonance spectral line shape of a positive slope value $S({\omega }_0)=0.149{\text{rad}}^{-1}$ and a positive asymmetry parameter value $q=0.338$. Also, the dip (peak) of the Fano resonance profile at the long (short) wavelength wing $\Delta <0$ $(\Delta >0)$, which results from the positive $S({\omega }_0)$ and $q$, can be observed. Once the loss parameter $a_1$ exceeds the critical value ${a^{\prime }}_1=0.720$ due to the increase of the pump power of the 980 nm laser, the tuned group delay becomes positive (${\tau }_{g20}=8.0\text{ns}$ for $a_1=0.75$, ${\tau }_{g20}=18.1\text{ns}$ for $a_1=0.82$, and ${\tau }_{g20}=23.2\text{ns}$ for $a_1=0.84$ 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 ($S({\omega }_0)=-0.0641{\text{rad}}^{-1}$ of the yellow curve for $a_1=0.75$, $S({\omega }_0)=-0.145{\text{rad}}^{-1}$ of the green curve for $a_1=0.82$, and $S({\omega }_0)=-0.197{\text{rad}}^{-1}$ of the blue curve for $a_1=0.84$ in Fig. 7(a)) and negative asymmetry parameters $q$ ($q=-0.353$ of the yellow curve for $a_1=0.75$, $q=-0.345$ of the green curve for $a_1=0.82$, and $q=-0.343$ of the blue curve for $a_1=0.84$ in Fig. 7(a)). For these three Fano resonance spectra with the negative $S({\omega }_0)$ and $q$, the peaks at the long wavelength wing $\Delta <0$ and the dips at the short wavelength wing $\Delta >0$ as a result of the negative $S({\omega }_0)$ 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 (${\tau }_{g20}=0.8\text{ns}$) and its associated flat interference transmission spectrum of $S({\omega }_0)=-0.00282{\text{rad}}^{-1}$ in the vicinity of ${\omega }_0$, when the loss parameter $a_1$ approaches the critical value ${a^{\prime }}_1=0.720$. In short, the experiment results demonstrate that both the slope $S({\omega }_0)$ 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 ${a^{\prime }}_1$ and its resulting flat interference transmission spectrum with the vanishing slope $S({\omega }_0)$ 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 $S({\omega }_0)$ 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 $a_1=0.73$ approaches the critical value ${a^{\prime }}_1=0.720$, since the observed flat interference transmission spectrum of $S({\omega }_0)=-0.00282{\text{rad}}^{-1}$ corresponding to the approximately weak dispersion (${\tau }_{g20}=0.8\text{ns}$) is not a typical asymmetry Fano resonance spectral line shape as discussed above. The theoretical model upon the sign reversals of $S({\omega }_0)$ and $q$ introduced in Sec. 2 is verified, since these theoretical $S({\omega }_0)$ and $q$ values agree well with those corresponding experimental values despite the small deviation between them. For the small deviation between the theoretical ($S({\omega }_0)$,$q$) values and the experimental ($S({\omega }_0)$,$q$) 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, 23] can be realized by the dispersion tuning scheme in the RCMZI. To obtain the reversible Fano resonance of a higher value of $\left|q\right|$, 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]. 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 $S({\omega }_0)$ 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 $S({\omega }_0)$ and $q$ is unrelated to the phase shift difference deviation $\delta \varphi$ (or the phase shift difference $\Delta \varphi$). Therefore, one can first tune the phase shift to make $\left|q\right|$ approach unity, and then tune the group delay to obtain the sign reversals of $S({\omega }_0)$ 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, 24]. For silicon resonators, loss tuning can be accomplished by Raman-induced loss tuning [26] or free-carrier injection effect [25] in silicon. Since the response time of the loss tuning using these two effects in silicon can be less than 1 ns [25, 26], 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 $T_2(\omega )$, $T_{out}(\omega )$, and ${\tau }_{g2}(\omega )$, it is crucial to measure some additional assistant parameters besides the corresponding voltage values $V_1$, $V_2$, and $V_3$ of the preceding simultaneously measured light intensities ($I_1$, $I_2$, and $I_3$) 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 $T_2(\omega )$, we measure the additional light intensity $I_1$ 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 $I_1=I_0{\rho }_{in}^2{T}_2(\omega ){\rho }_A^2{a}_{tr}$ and ${I^{\prime }}_1=I_0{\rho }_{in}^2{\rho }_2^2{\rho }_A^2{a}_{tr}$ of $I_1$ for the RCMZI with the loaded and unloaded CRIT structure, the more precise normalized transmission spectrum $T_2(\omega )$ can be deduced by the formula $T_2(\omega )={\rho }_2^2{I}_1/{I^{\prime }}_1$ in that the effective propagation loss $a_{tr}$ which exists in $I_1$ and ${I^{\prime }}_1$ cancels each other out, where ${\rho }_{in}$, ${\rho }_A$ and $a_{tr}$ 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 $T_{out}(\omega )$ from the impact of the similar loss, the additional light intensity $I_2$ 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 $I_2$ can be considered as $I_2=I_0{T}_{out}(\omega )a_{ref}$ and ${I^{\prime }}_2=I_0(1-{\rho }_{in}^2){\rho }_B^2(1-{\rho }_{out}^2)a_{ref}$ for the RCMZI with and without the resonance arm, respectively. The accurate normalized interference transmission spectrum $T_{out}(\omega )$ can be obtained by $T_{out}(\omega )=(1-{\rho }_{in}^2){\rho }_B^2(1-{\rho }_{out}^2)I_2/{I^{\prime }}_2$ without the deviation caused by the effective propagation loss represented by $a_{ref}$, where ${\rho }_{out}$, ${\rho }_B$ and $a_{ref}$ 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 ${\tau }_{g2}(\omega )$, the required voltage values ($V_1$, $V_2$, and $V_3$) of $I_1$, $I_2$, and $I_3$ 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 ($V_1$, $V_2$, and $V_3$) are not only determined by the light intensity values of $I_1$, $I_2$, and $I_3$ 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 $I_1$ and $I_2$ by DET 1 and DET 2 for the ratio ${\chi }_A={V^{\prime }}_2/{V^{\prime }}_1$ between the voltage value ${V^{\prime }}_1$ of $I_1$ and the voltage value ${V^{\prime }}_2$ of $I_2$, respectively, when the reference arm is isolated from the RCMZI; secondly, the additional light intensities $I_2$ and $I_3$ are detected by DET 2 and DET 3 for the ratio ${\chi }_B={V^{″}}_2/{V^{″}}_3$ between the voltage value ${V^{″}}_2$ of $I_2$ and the voltage value ${V^{″}}_3$ of $I_3$, 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

Moreover, to obtain the accurate loss parameter $a_2$ of Ring 2, we also measure the transmission spectrum of Ring 2 by the foregoing measurement method applied for $T_2(\omega )$ 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 $\left|{T^{\prime }}_2(\omega )\right|={\left|[{\rho }_2-a_2{\rho }_1\exp (i\omega {\tau }_2)]/[1-{\rho }_2{a}_2{\rho }_1\exp (i\omega {\tau }_2)]\right|}^2$ of the transmission spectrum from the transfer matrix theory, the accurate loss parameter $(a_2=0.79)$ 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.