# Turning a hot spot into a cold spot: polarization-controlled Fano-shaped local-field responses probed by a quantum dot

Sep 21, 2020

### Mechanism for local-field interferences and Fano-shaped local-field responses

To make the local-field interferences nearly fully controllable, the nanoantenna should have some local region where the local field can be excited by both orthogonal polarizations, and at the same time, the local-field polarization should have a negligible dependence on the excitation polarization. Let us consider a nanoantenna on a substrate in the xy plane, and consider normally incident plane-wave excitation light Eexc with an arbitrary elliptical polarization and a unity amplitude, which can be expressed as a coherent superposition of its x-polarization component and y-polarization component

$${mathbf{E}}_{mathrm {exc}} = costheta ,{mathbf{e}}_x + {mathrm {e}}^{{mathrm{i}}varphi }sintheta ,{mathbf{e}}_{it{y}}$$

(1)

where ex and ey are the unit vectors in the x and y directions, and tanθ and φ define the amplitude ratio and phase difference between the y– and x-polarization components of the excitation light. Under the excitation by Eexc, the local field can be expressed as

$${mathbf{E}} = costheta ,{mathbf{E}}^ leftrightarrow + {mathrm{e}}^{{mathrm{i}}varphi }sintheta ,{mathbf{E}}^ updownarrow$$

(2)

which is the coherent superposition of the local fields excited by the x– and y-polarization components of Eexc. Here E (E) denotes the local field under the excitation by x-polarized (y-polarized) light with a unity amplitude. It should be noted that E, E and E include not only the scattered field but also the incident field. In this work, when we discuss the local field or local-field response, we refer to the total local field and not only to the scattered local field. If the nanoantenna has some local region where E and E have similar local-field polarizations (denoted ({hat{mathbf u}})), then the local field excited by the x-polarization component of Eexc and that excited by the y-polarization component of Eexc can effectively interfere in this region, and the interference can be controlled by the excitation polarization. Nearly complete destructive local-field interference (i.e., nearly complete local-field suppression) can be achieved in this region when the excitation polarization satisfies the conditions

$$theta = arctanleft| {{mathbf{E}}^ leftrightarrow } right|/left| {{mathbf{E}}^ updownarrow } right|,,varphi = phi ^ leftrightarrow – phi ^ updownarrow + 180^ circ$$

(3)

where |E| (|E|) is the magnitude of E (E), and ϕ (ϕ) is the phase of the component of E (E) along the major axis of its polarization ellipse. Equation (3) gives the excitation polarization for optimal local-field suppression. The orthogonal polarization (θ′, φ′) = (90° − θ, φ − 180°) leads to optimal local-field enhancement.

Now we have elucidated the conditions for achieving nearly fully controllable local-field interferences, but without consideration of the spectral dispersion. To make the spectral dispersion exhibit a Fano lineshape, the local-field response should be resonant for one excitation polarization (here, we assume y-polarization) but nonresonant for the orthogonal excitation polarization (here we assume x-polarization). Then, the local-field response under y-polarized excitation can be approximately described by a Lorentz resonance ({mathbf{E}}^ updownarrow = Ae^{{mathrm{i}}delta _{mathrm{A}}}omega _0/left( {omega – omega _0 + {mathrm{i}}gamma {mathrm{/}}2} right){hat{mathbf u}}), while the local-field response under x-polarized excitation can be approximately described by a flat response ({mathbf{E}}^ leftrightarrow = B{mathrm e}^{{mathrm{i}}delta _{mathrm{B}}}{hat{mathbf u}}). Here, A, B, δA and δB are real-valued constants in the spectral range of interest, and ω0 and γ are the resonant frequency and linewidth of the Lorentz resonance, respectively. The resonant response E and flat response E interfere with each other in the near field according to Eq. (2), with their amplitude ratio and phase difference controlled by the excitation polarization. This is the near-field analogy of the Fano resonance, where resonant scattering interferes with flat scattering to produce an asymmetric spectral lineshape35,36,37,38. Indeed, if complete destructive interference (or local-field suppression) is achieved at an arbitrarily specified frequency ωs using the excitation polarization given by Eq. (3), then the spectral dispersion of the local-field response is simply the well-known Fano lineshape (see Supplementary Section S1 for the derivation)

$$left| {mathbf{E}} right|^2 propto frac{{left( {{it{Omega }} + q} right)^2}}{{{it{Omega }}^2 + 1}}$$

(4)

where Ω = 2(ω − ω0)/γ is the dimensionless frequency and q = 2(ω0 − ωs)/γ is the Fano asymmetry parameter. A Fano resonance features high-dynamic-range dispersion, quickly changing from an enhanced response (due to constructive interference) to a suppressed response (due to destructive interference) in the spectral domain. The spectral position of the Fano dip is determined by q via Ω = −q. The relation between q and ωs indicates that q can be dynamically tuned by simply controlling the excitation polarization.

### Antenna design for Fano-shaped local-field responses

According to the design rules elucidated above, we propose a QD-loaded nanoantenna as a proof-of-principle design, as shown in Fig. 1a. The nanoantenna consists of two chemically synthesized colloidal GNRs, G1 and G2. The two GNRs form an asymmetric dimer with a nanogap. A laser beam with controlled polarization excites the antenna modes. A single colloidal QD is placed in the nanogap to spectrally probe the local-field response. The QD-loaded nanoantenna can be assembled using the nanomanipulation technique10. Relevant optical characterizations can be performed with the experimental setup shown in Fig. 1b, including polarization-controlled and wavelength-controlled excitation and single QD fluorescence detection (see Supplementary Section S2 for details of the experimental setup).

The antenna is designed to allow polarization-controlled effective interference between a resonant local-field response and a flat local-field response in the nanogap, which then enables tuneable Fano-shaped local-field responses. On the one hand, the local-field response is resonant for y-polarized excitation but roughly flat for x-polarized excitation, as indicated by the numerically simulated local-field spectral responses in Fig. 1c. Under y-polarized excitation, the intensity response (red curve) shows a strong enhancement peak, and the phase response (red curve in the inset) undergoes an abrupt change by ~π across the resonance peak, while under x-polarized excitation, both the intensity response (blue curve) and phase response (blue curve in the inset) are roughly flat. As shown by the local-field intensity distributions at the resonant wavelength of ~680 nm, under y-polarized excitation, the nanogap hosts a hot spot with strong local-field enhancement (upper-right inset), while under x-polarized excitation, the local-field enhancement is moderate (lower inset). On the other hand, the local-field polarization in the nanogap is nearly independent of the excitation polarization, as indicated by the numerically simulated local-field polarizations in Fig. 1d. In the x–y plane at the same height as the centre of the QD, the z-components of the local-field responses are negligible (Supplementary Fig. S1), i.e., E and E are polarized in the x–y plane. Thus the local-field polarizations can be described by a polarization ellipse in the x–y plane with ellipse parameters ψ and χ, where ψ is the orientation angle of the polarization ellipse defined as the angle between the major axis of the polarization ellipse and the x-axis, and χ is the ellipticity angle of the polarization ellipse defined as the arc tangent of the ratio of the ellipse’s minor axis to major axis, whose sign defines the rotation direction. The curves of χ are close to 0°, which indicates that E and E are both nearly linearly polarized. The curves of ψ show that the local-field polarization angles of E and E are similar in the spectral range of interest. Around the resonant wavelength, the local-field polarizations of E and E are nearly identical, as is also indicated by the instantaneous electric field vectors plotted in the insets of Fig. 1d.

The resonant response E and flat response E can then effectively interfere in the near field according to Eq. (2), with their amplitude ratio and phase difference controlled by the excitation polarization. With the numerically simulated dispersions of the amplitude and phase of E and E, we can readily calculate according to Eq. (3) the required excitation polarization for achieving optimal destructive local-field interference (or local-field suppression) at an arbitrarily specified excitation wavelength. The required excitation polarization parameters and the corresponding local-field suppression factors, as functions of the specified excitation wavelength, can be found in Supplementary Section S4. Figure 1e, f shows the local-field spectral responses for a series of excitation polarizations. They can indeed be fitted by Fano lineshapes (dashed curves with the q values given; see Supplementary Section S1 for the fitting method), with Fano dips at the specified wavelengths (see Supplementary Fig. S3 with the zoomed-in vertical axis) and Fano asymmetry parameters tuneable from negative to positive values. Note that at the Fano dips the local field is nearly vanishing, which means that the backgrounds of the Fano lineshapes are very low. The low background and high tunability of the Fano lineshapes indicate that local-field interference can be designed to be nearly fully controllable. The residual local-field responses at the Fano dips are due to the slight misalignments between E and E (Supplementary Section S5). Local-field distributions (cf. the inset of Fig. 1f) show that at the Fano dips the nanogap hosts a deep-subwavelength cold spot with strongly suppressed local field. This indicates that by changing the excitation polarization, the hot spot with strong local-field enhancement (cf. Fig. 1c) can be turned into a cold spot with strong local-field suppression (cf. Fig. 1f).

If the local field is excited by a broadband ultrafast pulse, then the spectrum of the local field can be tuned by controlling the local-field spectral response, as demonstrated above. Since coherence is sustained in the local-field response25,26,27, and the spectral and temporal properties are related by Fourier’s principle, the temporal dynamics of the local field can then be accordingly tuned by simply controlling the excitation polarization. The polarization-controlled spectral distributions and temporal evolutions of the ultrafast local field are given in Supplementary Section S6.

### Fabrication of the QD-loaded nanoantenna

The designed QD-loaded nanoantenna is deterministically fabricated based on the nanomanipulation technique (see ‘Methods’ for the fabrication process). Figure 2a shows an atomic force microscope (AFM) image of the fabricated sample. The two GNRs (Nanopartz Inc.) have similar diameters (~32 nm for G1; ~34 nm for G2) and similar lengths (~78 nm for G1; ~80 nm for G2). The GNRs have similar plasmonic resonance peaks around the wavelength of 638 nm, as indicated by the measured dark-field scattering spectra shown in Fig. 2b. The QD is a colloidal CdSeTe/ZnS core–shell QD (Invitrogen, Qdot 800 ITK carboxyl) encapsulated in a silica shell with a thickness of ~10 nm46. The total diameter of the silica-encapsulated QD is ~31 nm. The width of the nanogap between G1 and G2 is ~36 nm. Due to fabrication imperfections, the position of the QD deviates slightly from the centre of the nanogap by ~2 nm towards G1. The method for estimating the structural parameters of the fabricated sample can be found in Supplementary Section S7. Clear binary fluorescence intermittency can be identified in the fluorescence time trajectory of the QD (inset of Fig. 2b), with its ‘off’-state fluorescence signal down to the background signal level, which is characteristic of a single QD47. The silica shell protects the QD from being quenched by surface defects, which makes the QD quantitatively reliable as a local-field probe. The intrinsic ‘on’-state quantum yield of the QD on the substrate can be assumed to be unity (Supplementary Section S9). The QD has a broadband and smooth absorption spectrum (Fig. 2b), which makes it suitable for spectral probing. The absorption spectrum is measured through photoluminescence excitation (PLE) spectroscopy48,49. The emission spectrum of the QD exhibits a peak at ~808 nm (Fig. 2b), which has no noticeable dependence on the excitation wavelength in the spectral range of interest (560–750 nm).

### Fluorescence tailoring by the nanoantenna

Since local-field probing relies on the detection of the fluorescence of the QD, it is important to know the fluorescence properties, which may be significantly influenced by the nanoantenna due to the Purcell effect10,50,51,52. Antenna modes can provide a large local density of photonic states (LDOS), and then, according to Fermi’s golden rule, the radiative decay rate of the QD will be enhanced via energy transfer to the antenna modes53,54. Owing to the broad bandwidth of the antenna resonance, although the emission wavelength of the QD (~808 nm) is far detuned from the resonant peak of the nanoantenna (~645 nm), the measured fluorescence lifetime of the QD is still shortened by ~7-fold (Fig. 2c), which can be explained by the simulated Purcell factor of ~8.5 (Supplementary Section S10). With such a significant Purcell effect, the fluorescence of the QD should predominantly transfer to the antenna modes and the far-field radiation is expected to be tailored by the antenna. Far-field polarization analysis (Fig. 2d) shows that the collected photons have a high degree of linear polarization of ~0.97, in stark contrast to the unpolarized photon emission before the QD is coupled to the nanoantenna. The polarization angle agrees with the simulation. This indicates that the fluorescence indeed predominantly transfers to the antenna modes and the far-field radiation properties are governed by the antenna. Note that here, the shortened lifetime and tailored far-field radiation polarization are both well explained by the Purcell effect; nonradiative energy transfer to surface recombination centres or carrier tunnelling to metal is unlikely to play a significant role in our experiment owing to the thick silica shell of the QD (see Supplementary Section S11 for a detailed analysis).

### Local-field spectral probing using the QD

Through PLE spectroscopy, i.e., measuring the photoluminescence intensity as a function of excitation wavelength, the absorption spectrum can be obtained as σ(λ) = Iem(λ)/[ρex(λ)ηξ], where ρex(λ) is the intensity of the excitation light (photons s−1 cm−2) as a function of excitation wavelength, Iem(λ) is the detected photon count rate (photons s−1) as a function of excitation wavelength, η is the ‘on’-state quantum yield and ξ is the fluorescence detection efficiency48. Before the QD is coupled to the antenna, η is taken as unity (Supplementary Section S9) and ξ is determined to be ~0.97% (Supplementary Section S8). After the QD is coupled to the antenna, its fluorescence is tailored by the antenna, and therefore η and ξ may change. According to numerical simulations, η is reduced to ~56% due to the finite radiation efficiencies of the antenna modes (Supplementary Section S10). Although the effective radiation dipole is governed by the antenna modes, ξ will not change because the effective radiation dipoles of the antenna modes are still in the horizontal plane. The experimental details for measurement of absorption spectra can be found in the ‘Methods’.

By measuring the absorption spectra of the QD before and after it couples to the antenna, the substrate-normalized local-field spectral response of the nanoantenna can be obtained as the ratio σant(λ)/σsub(λ), where σant(λ) denotes the absorption spectrum after the QD has been coupled to the antenna and σsub(λ) denotes the absorption spectrum when the QD is on the substrate before being coupled to the antenna. The experimentally probed substrate-normalized local-field response corresponds to the quantity |Eant|2/|Esub|2, where Eant and Esub are the incidence-normalized local field at the antenna and at the substrate surface, respectively. The local-field response at the substrate surface is featureless, so it can be reliably theoretically or numerically estimated. Then, the incidence-normalized local-field response can be calculated from the substrate-normalized local-field response. For experimental studies, substrate normalization is usually used, so in our experimental demonstrations, we still use the substrate-normalized local-field response, which is also denoted as excitation enhancement factor. An excitation enhancement factor less than one means that the local field is suppressed, and its reciprocal is denoted as the excitation suppression factor.

For local-field probing, the QD is excited far from its band edge, and here, its bright plane is made nearly horizontal, so the excitation of the QD is nearly polarization independent in the x–y plane55. Thus, the QD can probe the intensity of the local field in the xy plane in an isotropic manner. In our study, regardless of whether the QD is excited directly by the incident light or has been coupled to the antenna, the local field at the QD is predominantly in the xy plane (Supplementary Section S3), so the QD can reliably probe the local-field response. If otherwise, the z-component of the local field were significant, then the QD probe would somewhat overestimate the local-field intensity, because the absorption coefficient along its dark axis (which is in the z direction) is expected to be slightly larger than that in its bright plane due to its slightly elongated shape.

### Experimental demonstration of Fano-shaped local-field responses

With the fabricated QD-loaded nanoantenna and the local-field probing technique, we are now ready to experimentally demonstrate polarization-controlled Fano-shaped local-field responses as predicted by our theoretical and numerical study. As one of the requirements for producing a Fano-shaped local-field response, a resonant local-field spectral response and a flat local-field spectral response must be experimentally shown first. To this end, we measure the excitation enhancement spectra under y– and x-polarized excitation, as shown in Fig. 3. As expected, under y-polarized excitation, the measured enhancement spectrum (black data points) shows a strong resonance peak at the wavelength of ~645 nm, with an excitation enhancement factor of ~80; under x-polarized excitation, the measured spectrum (blue data points) is rather flat, with an enhancement factor of ~5. The measured spectra agree well with the numerically simulated local-field enhancement spectra (solid black and blue curves). Note that all the numerical simulations for comparison with the experimental results use the structural parameters estimated for the fabricated sample (Supplementary Section S7).

Then, the interference between the resonant and flat local-field responses needs to be controlled by the excitation polarization. Similar to the theoretical study, we need to find the required excitation polarizations for achieving optimal destructive local-field interference (or local-field suppression) at specified wavelengths. For any specified wavelength, we experimentally find the required excitation polarization by successively searching the elliptical polarization parameters φ and θ for a minimized photoluminescence. The experimentally obtained values of θ and φ are shown in Fig. 4a. These values agree well with the numerically calculated values. Since the numerically obtained values are calculated according to Eq. (3), this indicates that our system indeed works according to the proposed local-field interference mechanism.

The correspondingly achieved minimal enhancement factors (the reciprocals of the enhancement factors are the excitation suppression factors) are plotted as square data points in Fig. 4b. The solid black curve shows the minimal enhancement factors obtained from numerical simulations. Significant excitation suppression is experimentally achieved for all excitation wavelengths. Notably, for the excitation wavelength of 725 nm, the excitation is suppressed to 2%, i.e., an excitation suppression factor as high as 50 is experimentally achieved. The significantly higher suppression factor at 725 nm than at 645 nm is attributed to the fact that for the fabricated QD-loaded nanoantenna the local-field polarizations of E and E in the nanogap are more similar at 725 nm than at 645 nm (Supplementary Section S12). Although the local-field response is probed at one spatial point, numerical simulations show that a three-dimensional local-field cold spot is hosted in the nanogap (Supplementary Fig. S12).

Having experimentally obtained the required excitation polarizations, the excitation enhancement spectrum can then be measured for these excitation polarizations. The measured excitation enhancement spectra are shown in Fig. 4c. The measured enhancement spectra (square data points) agree well with the numerically calculated results (solid curves). Importantly, these experimentally measured spectra can indeed be well fitted by Fano lineshapes (dashed curves) of different q values depending on the excitation polarization. The Fano dips are at the specified local-field suppression wavelengths (see the inset with the zoomed-in vertical axis). Thus, we have already experimentally demonstrated that the local-field response in the nanogap can exhibit dynamically tuneable Fano lineshapes. By simply controlling the excitation polarization, the Fano asymmetry parameter q can be tuned from negative to positive values, and correspondingly, the Fano dip can be tuned across a broad wavelength range. Moreover, the nearly vanishing Fano dips indicate that the Fano lineshapes have low backgrounds. Fano lineshapes with a sharp change between the Fano peak (with a local-field enhancement) and the nearly vanishing Fano dip (with a local-field suppression) imply a high dynamic range in the spectral domain.

It is worth noting that if the Fano dip is far from the resonant wavelength of 645 nm (e.g., at 700, 725 nm), the response spectrum around the dip can be rather flat, which allows broadband local-field suppression. For instance, for the excitation spectrum with a dip at 725 nm, the measured excitation suppression factors all exceed 20 for excitation wavelengths spanning from 710 to 740 nm, as shown by the blue-coloured spectrum in Fig. 4c. Such broadband suppression is attributed to the relatively weak spectral dispersions of the amplitudes and phases of E and E at wavelengths significantly longer than the resonant wavelength of 645 nm. This behaviour is expected for a Fano-shaped response with a large value of q.

### Switching between enhancement and suppression

Once the excitation polarization for optimal excitation suppression has been found, optimal excitation enhancement can then be obtained by simply using the orthogonal polarization, as shown by the polarization ellipses in Fig. 5a. The achieved maximal excitation enhancement factors are plotted as square data points in Fig. 5a. Figures 4b and 5a together indicate that the local-field response can be switched between enhancement and suppression in a broad wavelength range, implying that the hot spot in the nanogap can be turned into a cold spot. The achievable lowest excitation enhancement factor (the reciprocal of which is the achievable highest excitation suppression factor) and highest excitation enhancement factor depend on the excitation wavelength. Here, we define the wavelength-specified dynamic range of local-field response as the ratio of the highest and lowest excitation enhancement factors that can be achieved at the specified excitation wavelength. Figure 5b shows the wavelength-specified dynamic ranges for excitation wavelengths from 625 to 725 nm. At wavelengths longer than the resonant wavelength of 645 nm, although the achievable highest excitation enhancement factor is less than that achievable at the resonant wavelength (Fig. 5a), a high dynamic range can still be achieved because the achievable highest suppression factor is greater than that achievable at the resonant wavelength (Fig. 4b). For instance, at 725 nm, while the experimentally achieved highest excitation enhancement factor is ~18, the excitation suppression factor can be as high as ~50, so the dynamic range can reach ~900, which is even higher than the dynamic range at the resonant wavelength of 645 nm. The experimentally achieved wavelength-specified dynamic ranges for excitation wavelengths between 645 and 725 nm all exceed 400. Theoretically, a wavelength-specified dynamic range over 9000 can be achieved at ~740 nm, where the theoretically obtained suppression factor is as high as ~700. Experimentally, the highest suppression factor is limited to ~50 due to the limited fabrication and measurement precision, and consequently, the highest wavelength-specified dynamic range is limited to ~900. Note that the dynamic range defined above is defined at a single spatial point, different from the spatial contrast ratio. To obtain a high spatial contrast ratio, more antennas are needed. For instance, if two identical antennas, each with a high dynamic range of the local-field response, are placed together with orthogonal orientations, then we can achieve a high spatial contrast ratio, which is roughly the value of the dynamic range of each antenna (Supplementary Section S13).