Tunable Fano resonance using weak-value amplification with asymmetric spectral response as a natural pointer

Weak measurement enables faithful amplification and high-precision measurement of small physical parameters and is under intensive investigation as an effective tool in metrology and for addressing foundational questions in quantum mechanics. Here we demonstrate weak-value amplification using the asymmetric spectral response of Fano resonance as the pointer arising naturally in precisely designed metamaterials, namely, waveguided plasmonic crystals. The weak coupling between the polarization degree of freedom and the spectral response of Fano resonance arises due to a tiny shift in the asymmetric spectral response between two orthogonal linear polarizations. By choosing the preselected and postselected polarization states to be nearly mutually orthogonal, we observe both real and imaginary weak-value amplifications manifested as a spectacular shift of the Fano-resonance peak and narrowing (or broadening) of the resonance linewidth, respectively. The remarkable control and tunability of Fano resonance in a single device enabled by weak-value amplification may enhance active Fano-resonance–based applications in the nano-optical domain. In general, weak measurements using Fano-type spectral response broadens the domain of applicability of weak measurements using natural spectral line shapes as a pointer in a wide range of physical systems.

Figure 1

Theoretical simulation of weak measurements using an asymmetric Fano spectral line shape for scheme 1 (a–c) and scheme 2 (d–f) for the case $q_x=q_y$. The spectral ($E=\hslash \omega$) variation of the intensity was simulated using preselection and postselection of the polarization states on the Jones matrix of Eq. (3). The input parameters were $E_{0x}=\hslash {\omega }_{0x}=1.756$ eV, $E_{0y}=\hslash {\omega }_{0y}=1.755$ eV, ${\gamma }_x={\gamma }_y=0.12$ eV, and $q_x=q_y=0.8$. (a) Spectra for postselections with two different angles $\pm \in$ using scheme 1. Also shown is the shift in the peaks of (b) the contributing narrow resonance $\delta E_0$ and (c) the Fano resonance $\delta E_m$ with varying $\in$ (open circles). The corresponding fits with Eq. (6) ($\sim \pm cot\in$) are for real weak-value amplification (solid lines). (d) Spectra for postselections with two different phase offsets $\pm \in$ using scheme 2. (e) Changes in the spectral width $\delta \gamma$ and (f) the shift of the Fano-resonance peak $\delta E_m$ with varying $\in$ (open circles). The corresponding fits with Eq. (8) ($\sim \pm cot\frac{\in }{2}$) are for imaginary weak-value amplification (solid lines).

Figure 2

Theoretical demonstration of the validity of weak measurements using an asymmetric Fano spectral line shape for the case $q_x\ne q_y$. Results of weak measurements using scheme 1 (a–c) and scheme 2 (d–f). The spectral ($E=\hslash \omega$) variation of the intensity was simulated using preselection and postselection of the polarization states on the Jones matrix of Eq. (3). The input parameters were $E_{0x}=1.756$ eV, $E_{0y}=1.755$ eV, width ${\gamma }_x={\gamma }_y=0.12$ eV, $q_x=0.77,$ and $q_y=0.8$. (a) Resulting spectra for postselections with two different small angles $\pm \in$ using scheme 1. Also shown is the shift in the peaks of (b) the contributing narrow resonance $\delta E_0$ and (c) the Fano resonance $\delta E_m$ with varying $\in$ shown by open circles. The corresponding fits with Eq. (6) $[\sim \pm cot(\in \left)\right]$ for real weak-value amplification are shown by solid lines. (d) Spectra $I_s\left(\omega \right)$ for postselections with two different small phase offsets $\pm \in$ using scheme 2. (e) Changes in the spectral width $\delta \gamma$ and (f) the shift of the Fano-resonance peak $E_m$ with varying $\in$ shown by open circles. The corresponding fits with Eq. (8) $[\sim \pm cot(\in \left)\right]$ for imaginary weak-value amplification are shown by solid lines.

Figure 3

(a) Schematic illustration of the concept. The preselected state is a $+{45}^{\circ }$ linear polarization. The plasmonic crystal generates a tiny shift in the asymmetric Fano spectral response for the $x$ and $y$ polarizations, providing the weak coupling. Postselections in $\pm \in$ angle away from the orthogonal linear polarization (${135}^{\circ }$) state lead to real weak-value amplification, manifesting as a shift of the Fano-resonance peak (scheme 1). Postselections in nearly orthogonal elliptical polarization states ($\pm \in$ phase offset from the orthogonal) lead to imaginary weak-value amplification, manifesting as narrowing (or broadening) of the resonance linewidth. (b) Typical SEM images of Au grating. (c) Scattering spectra ($E=1.650-1.940$ eV and $\lambda \approx 750-640$ nm) from the Au grating (red solid line). The theoretical fit with the Fano intensity expression is shown by a black dotted line. The fitted parameters of the resonance are $E_0=1.803$ eV, $E_m=1.837$ eV, width $\gamma =0.142$ eV, $q=0.274$, and $B=0.439$.

Figure 4

(a) Scattering spectral Mueller matrix $M\left(E\right)$ (for $E=1.650-2.250\mathrm{eV}$) of the Au grating. Also shown is the spectral variation of scattered intensities ($E=1.650-1.940\mathrm{eV}$) for (b) $x$ polarization (red solid line) and (c) $y$ polarization (magenta solid line). The spectra were obtained by preselection and postselection [using Eq. (9)] of the corresponding polarization states ($S:{\left[1100\right]}^T$ for $x$ and ${\left[1-100\right]}^T$ for $y$) on $M$. Theoretical fits with the Fano intensity expression are shown by black dotted lines. The fitted parameters of the resonance are $E_{0x}=1.805$ eV, $E_{0y}=1.800$ eV, width ${\gamma }_x=0.125$ eV, ${\gamma }_y=0.126$ eV, $q_x=0.329$, and $q_y=0.274$.

Figure 5

Experimental demonstration of weak measurements in plasmonic crystal using scheme 1 (a and b) and scheme 2 (c and d). (a) Spectra for postselections with four different angles $\in$ using scheme 1. The reference spectra for a strong measurement correspond to preselections and postselections at ${45}^{\circ }$ linear polarization (black solid line). (b) Shift in the peak of the narrow waveguide resonance ($\delta E_0$, left axis, blue open circles) and the Fano-resonance peak ($\delta E_m$ right axis, magenta closed circles) with varying $\in$. The corresponding fits with Eq. (6) ($\sim cot\in$) for real weak-value amplification of narrow waveguide resonance and the Fano-resonance peak are shown by solid and dashed lines, respectively. (c) Spectra for postselections with four different phase offset $\in$ using scheme 2. (d) Changes in the spectral width $\delta \gamma$ with varying $\in$(open circles). The corresponding fit with Eq. (8) ($\sim cot\frac{\in }{2}$) for imaginary weak-value amplification is shown by solid lines.

Figure 6

Theoretical simulation of the limit of weak-value amplification using an asymmetric Fano spectral line shape. The variations of the minimum value ${\in }_{min}$ (left axis, open circles) and the maximum achievable amplification factor $A^{max}$ (right axis, closed circles) (corresponding to ${\in }_{min}$) with varying $\mathrm{\Delta }q=q_x-q_y$ are shown for (a) real weak-value amplification using scheme 1, $A^{max}=\frac{\delta E_0^{max}}{2a}$, and (b) imaginary weak-value amplification using scheme 2, $A^{max}=\frac{\delta {\gamma }^{max}}{2a}$. The input parameters were $E_{0x}=1.756$ eV, $E_{0y}=1.755$ eV, and width ${\gamma }_x={\gamma }_y=0.12\mathrm{eV}$. The value for $q_x$ was varied in the range 0.5–0.8 and $q_y$ was kept constant ($q_y=0.8$). The ranges of ${\in }_{min}$ and $A^{max}$ typical of our experimental weak measurements (Fig. 5 for plasmonic crystal samples) are marked in the figures. Error bars in the marked regions represent typical variations in the $\mathrm{\Delta }q$ and $A^{max}$ parameters corresponding to typical fabrication tolerance of plasmonic crystals.