1. Introduction

Light-matter interaction at molecular level is of fundamental importance concerning molecules are the basic building blocks for cells and networks. Plasmonic nanostructures have received a thriving growth due to their unique capability of confining light far below the diffraction limit. Initially, synthesized nanostructures such as nanoparticles (NPs) [1–3], nanorods [4,5], nanodisks [6], nanoprisms [7,8] or even metallic flakes [9] and roughness [10,11] were all proved as effective media, which based on the lightning rod effect, provide local electromagnetic (EM) field enhancement of ~10². With the advent of sophisticated fabrication techniques such as electron beam lithography (EBL) and focused ion beam (FIB) milling, the EM field enhancement, basing on the formation of the plasmonic gap mode, can be further raised up to 10⁴ utilizing specially tailored dimer or nanoantenna structures [12]. Theoretically in classical regime, the EM enhancement scales super-exponentially with the decreasing of the gap size [13]. Whereas in quantum regime, the plasmonic tunneling and nonlocal effect [14] both hindered EM enhancement further, setting a fundamental limit for the optimal gap size, typically on the order of several tenths of a nanometer [15]. Obviously, the state of the art fabrication techniques based on EBL or FIB can neither reach such a critical dimension with high repeatability nor provide reasonably large scalability. This sets a hurdle preventing the study of quantum and nonlinear plasmonics at low light levels. Considering this, conventional thin film growth techniques such as sputtering and evaporation may cope with this problem that made nanostructures with quantum size achievable at the cost of producing randomness. Namely during the deposition process, the originally discrete nanograins become bigger and bigger and upon coalescence, randomly distributed nano-voids and nano-interstices between interconnected clusters can be stochastically formed. Electrically, the sheet resistivity shows an abrupt decrease as soon as the discrete grains evolves into semi-continuous film [16], and such film thickness was widely termed as percolation threshold [17, 18]. Optically, clear-cut determination of the percolation threshold is not straightforward [19], not only depends on the types of clusters [20] formed but the exact optical process involved in characterization. Nonetheless, giant optical responses based on the significantly enhanced local EM field of percolated films were reported: single molecule surface enhanced Raman scattering (SMSERS) [21,22], low threshold random lasers [23], harmonic generations [24] and ultra-broadband light emission [25], to name a few.

To use the percolated film as an enhancement substrate, develop mehods to sieve, localize and manipulate plasmonic hot spots for the desired wavelengths is vital. Techniques based on linear optical effects provide good ensemble average over all existing hot spots, yielding an effective enhancement. However, the actual EM enhancement is highly inhomogeneous, localized, and dispersive which has not yet fully studied. Therefore, there is an immediate need to establish the abovementioned capabilities particularly for applications of single molecule spectroscopy and super-localization nanoscopy. Contrary to linear optics, hot spots generated via nonlinear optical process cannot simply be summed up with equal power contribution to yield an overall enhancement, in particular for extremely small gap sites where the signal may be largely amplified [26]. With higher order nonlinear process involved, few hot spots may overwhelmingly dominate the overall signal, bias the centroid of the object point spread function (PSF) at far field, and exhibit the so-called “less is more” signature. Up to date, high precision direct observation of the hot spots were only successfully made by the scanning near field optical microscope (SNOM) [27] and photo-excited emission microscope (PEEM) [28]. In parallel to this, indirect evidences were commonly collected using single molecule localization microscopy (SMLM) techniques. Essentially, fluorescent molecules were used as a local probe. Through the Brownian motion of the fluorescent molecules on the substrate, the position dependent fluorescence was statistically recorded, acting as an indicator of the strength of the local field [29]. The abovementioned methods either require special experimental conditions (e.g. ultrahigh vacuum for PEEM) or relatively longer times to acquire data so as to produce reasonably large signals over noise, hindering the characterization in real time.

In this study, we demonstrate an efficient way to sieve, localize and steer plasmonic hot spot without applying fluorescent tags on fractal metallic films. Through scanning the relative time delay between the transform limited femtosecond pump and the linearly chirped supercontinuum probe, spectrally dispersed FWM mediated by nano-interstices of the percolated Au films were recorded spatial-temporally. Of particular importance, the intrinsic optical filtering provides excellent isolation between the signal and the excitation wavelengths. Also, the 3-photon nonlinear process not only provides a minimal excitation volume but produces a much enhanced signal which is proportional to the sixth power of the local EM field [30]. Consequently, a moderately enhanced local field may result in orders of magnitude amplification of the FWM signal. These advantages altogether provide excellent background suppression and much enhanced signal to noise (S/N) ratio, enabling one to sieve the most intense hot spot associated with the finest morphological features on the random film. With proper threshold of the incident intensity, the number of hot spot per unit area can be reduced and isolated emitting subsets results. Moreover, the hot spot steering achieved by scanning various time delays provides a brand new switching mechanism that enables one to combine with superresolution algorithms, realizing multicolor superlocalization with spatial resolution far beyond one photon diffraction limit. Due to the broadband nature of the chirped probe beam, the dispersive property of the hot spot correlated with particular hot site on the nanostructured random film can be mapped out. This gives additional advantage in probing nanostructures without using specially designed fluorescent molecules and suffers no bleaching or blinking as those based on fluorophore or Raman tags. The temporal behavior of the spectrum also provides useful information for dynamic nanoscopy.

The paper is organized as follows: in section 2, we present the fabrication and characterization of the deposited Au films. Percolated film was obtained by the signpost of a broad and flat transmission spectrum. In section 3, femtosecond chirp-manipulated four wave mixing (FWM) on the percolated film was performed, and the enhanced 3rd-order nonlinear process was verified. Section 4 presents the polarization anisotropy measurement which ensures the plasmonic gap mode is the main cause providing the enhancement. In section 5, plasmon superlocalization based on superresolved FWM images was demonstrated. The region of localization and the position accuracy are justified.

2. Fabrication, morphology, and transmission spectrum

To reach percolation, nanostructured random films with various thicknesses were deposited by sputtering technique. Microscope glass slides were ultrasonically cleaned by acetone, isopropanol, and deionized water consecutively. After baking, Au films with various thicknesses were sputtered (ULVAC, GLD-051) onto glass substrate at fixed current I = 20 mA and different sputtering time. The morphology of the sample was characterized by SEM (HITACHI, S-4300), and the filling fraction was calculated by binarizing and thresholding the SEM images. Fig. 1(a) depicts some selected SEM images which show that with the increase of the deposition time, the morphology evolves from isolated nanoparticles (NPs) to semi-continuous clusters, and further into continuous film. The normalized transmission spectra were depicted in Fig. 1(b) where typical signposts such as (1) a single drop at λ≅550nm, (2) doublet drops with one red-shifted with the increase of the sputtering time, (3) broad and flat transmittance for λ>600nm, and (4) a single peak located at the interband transition wavelength λ = 505nm were observed. These phenomena indicate film morphologies consist of isolated grains, coupled grains, interconnected clusters at percolation, and bulk-like continuous film, respectively. Initially, the transmission spectrum (i) shows a single drop at λ≅550nm, corresponding to particle plasmon resonance. When the deposition time was gradually increased, the spectra (ii) and (iii) both exhibit an additional transmission drop at longer wavelength. Unlike typical particle plasmon resonance which exhibits slightly red shift as the film becomes thicker, the resonance located at such longer wavelength exhibits relatively shallow drop but larger red shift. This reveals the onset of the coupling between isolated particles or clusters. Further increase the thickness of the film results in interconnected clusters that weakens the coupling and the shallow drop at longer wavelength cease to appear. Spectra (iv)-(vi) show gradually decreased transmission at longer wavelengths and the anomalous absorption band located around the particle plasmon resonance remains there. The typical signature for films reaching percolation threshold is a broadband and flat window where the transmission is wavelength independent [31], as shown in (vi). When the thickness of the film is further increased beyond the percolation threshold, bulk property dominates the transmission spectra which are clearly in evidence by the interband transition peaked at λ = 505 nm, as shown in (vii) and (viii). The transition from isolated NPs to percolated film can be easily distinguished from spectrum (v) and (vi).

 

Fig. 1 (a) Selected SEM images of the nanostructured Au films with various deposition time. (b) The transmission spectra for films with various deposition time. Note that the particle plasmon resonance was located at λ = 550nm and the interband transition occurred at λ = 505 nm. The scale bar in the SEM image is 1μm.

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3. Spectral-temporal trace of FWM

To sieve and localize the plasmonic hot spot at various wavelengths on the film, home-built femtosecond chirp-manipulated four wave mixing (FWM) method was applied. Essentially, the pump beam was delivered from a mode-locked Ti-Sapphire laser (Tsunami, Spectra Physics) operated at λ = 760nm with pulse duration of 80 fs. The probe beam was produced by coupling part of the pump energy into a 12-cm-long photonic crystal fiber (SCG 800, Newport), generating a supercontinuum (SC) spectrum from λ = 500 nm to 1200 nm. Depending on the in-coupled power of the pump, the level of the chirp of the SC probe can be varied. In our experiment, a chirp coefficient C = 43nm/ps was used which gives a moderate spectral resolution and reasonably high signal upon rapid scan. The pump and probe with various time delays were then recombined, appropriately filtered, and focused onto the sample via a 100 × microscope objective lens (N.A. = 0.9, Mitsutoyo). The back-reflected FWM signal was collected with the same objective lens, passing through a short pass filter, and directed to the spectrometer (Zolix Omni-λ300) equipped with a cooled CCD (DV401A-BV, Andor) for spectrum analysis, and the EMCCD (iXon Ultra 897, Andor) for superlocalization imaging. The experimental setup was depicted in Appendix A. Fig. 2(a) shows the measured FWM trace. The slanted distribution arises from positively chirped probe beam, and the maximum FWM signal appears at λ_FWM = 610 nm, corresponding to wave mixing of the pump beam centered at λ_pump = 760 nm and the spatial-temporally overlapped probe at λ_probe = 1007 nm on the percolated Au film. The temporal duration of the FWM signal over the measured wavelength range equals approximately to that of the pump beam indicating an instantaneously nonlinear response from the percolated film. To clarify the nonlinear process, we fixed the pump power and found that the FWM signal scales linearly with the probe power. When the probe power fixed instead, the FWM signal scales quadratically with the pump power, as shown in Fig. 2(b). It is found that all measured FWM signal follows well with the energy conservation relation: ω_FWM = 2ω_pump-ω_probe, ensuring the wave mixing was indeed originated from the 3rd-order nonlinear process. Note that the FWM signals are normalized to those produced by a standard Si <1 0 0> wafer under the same incident power which is atomically flat and is very helpful for calibrating any deviation of collection efficiency from the experimental setup.

 

Fig. 2 (a) Spectral-temporal four wave mixing trace of the percolated Au film. (b) The intensity of the FWM signal as functions of the pump and probe beam intensity. Note that ω_FWM = 2ω_pump-ω_probe and $I_{FWM}\propto I_{pump}^2{I}_{probe}$are both held here which ensures a 3rd order nonlinear process.

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4. Polarization anisotropy

To verify whether the measured FWM signal was primarily from the plasmonic enhanced gap mode, polarization anisotropy measurement was carried out for films with 3 different filling fractions: 30.7%, 82.15%, and 92.25%, corresponding to the isolated, percolated, and continuous film, respectively. The experimental details were described in Appendix B. The statistical histograms were obtained from the FWM response of 250 different points sampled globally on each of the sample and the major features in Fig. 3 are summarized as follows: (1) The percolated film gives the largest FWM signal which is one order of magnitude higher as compared to the isolated and continuous films. (2) TE incidence gives higher intensity and broader distribution than that produced by TM incidence, in regardless of the filling fractions. (3) The separation between the TE and TM peak decreases with the increase of the filling fraction.

 

Fig. 3 Statistics of polarization dependent FWM from 250 sampled points on each of the nanostructured film with various filling fractions: (a) isolated, (b) percolated, and (c) continuous film. The result shows that the plasmon gap mode is the most possible cause for the enhanced signal.

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Feature (1) can be explained by the relatively high density of nanogaps formed at percolation threshold. By then, the gap plasmons give larger enhancement compared to those produced by particle plasmon resonance as well as the propagated surface plasmons. As for feature (2), since for TE incidence the electric field lies predominantly in the sample plane, the excitation of the plasmon gap mode is more efficient with respect to that done by TM incidence. This characteristic becomes more pronounced when the film is close to percolation, as can be seen from the difference between Figs. 3(a) and 3(b). The wide distribution of the FWM signal can be attributed to polydispersed nanogaps, where a variety of gap sizes were altogether sampled by the incident spot. For feature (3), when the film consists of isolated particles, localized surface plasmons contributed predominantly to the enhancement and thereby both polarizations can excite the required particle resonance. However for TM incidence, the presence of the substrate reduces EM field confinement, which results in a weaker peak intensity as compared to the case of TE incidence. Conversely, when the film becomes more and more interconnected, gap mode was reduced and the propagated surface plasmon mode excited by the TM wave increases. This brings the two peaks closer especially for the continuous film, where the signal from residual gap modes becomes less and is comparable to that contributed by the propagated surface plasmons, as shown in Fig. 3(c). These features made us to conclude that the significantly enhanced FWM was primarily originated from the plasmon mode supported by nanogaps.

5. Plasmon superlocalization and FWM hot spot steering

Having the filling fraction of the film and the incident polarization optimized for achieving the highest FWM signal, we then demonstrate the capability to localize and steer the plasmonic hot spot at various wavelengths. When the relative time delays were scanned from δt = −1.333 ps to δt = + 1.333 ps in step of 133 fs, the spectra show sequentially: a broad two photon luminescence (TPL), gradually increased FWM signal located from λ = 590nm to λ = 630 nm, and a TPL background once again. Figure 4 shows some selected snapshots of the far field FWM images and their corresponding spectra at different time delays. The movie file corresponds to continuous scan is available online (see Visualization 1). The first column depicts the raw images recorded by the EMCCD where the contour marked in blue (green) represents the region of interest (ROI) at the present (subsequent) time of delay for comparison purpose. The middle column shows the differentialized images between two consecutive time delays that is extremely useful in locating the emitting site [32] and tracking the movement of the hot spot. Essentially, dipolar-like intensity distributions can be easily observed in each image except for the first one which only subtracts the background and exhibits typical monopole characteristics. Nonetheless, it can be useful for the determination of the initial position of the FWM centroid by 2D Gaussian function fitting [33]. Since the decay of the dipolar field is faster, taking gradient of the differentialized image gives both the displacement and moving direction of the hot spot with high precision. These information are designated by the text and arrows in the figure and the corresponding spectra at each time delay are shown in the third column.

 

Fig. 4 Snapshots of the far field diffraction limited FWM images (the first column), differentialized FWM images (middle column) between consecutive time delays, and the corresponding spectra (the third column) at different time delays. The contour marked in blue (green) represents the ROI defined by 1/e² of the FWM intensity at present and (subsequent) time delay and the scale bar is 400 nm. Note that the first image in the middle column exhibits a monopole intensity distribution which is much smaller than the original far field pattern. Other differentialized intensities exhibit dipolar-like behavior and their gradient give the displacement of the hot spot which is indicated by the arrow and text in the figure. The movie file corresponds to continuous scan is available online (see Visualization 1).

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Figs. 5(a) and 5(b) show respectively the SEM image and the optical microscope (OM) image of the region under investigation. Regions produce FWM with S/N ratio larger than 900 were enclosed by the blue circles. It was observed that the far field FWM image consists of an intense central surrounded by 5 well-separated faint lobes within the diameter of the excitation spot. Since the excitation beam has a Gaussian profile whose intensity decreases rapidly away from the central, the presence of asymmetrically distributed satellite lobes indicates the existence of hot sites with high enhancement which can be effectively sieved and discriminated from the central, thanks to the nonlinear enhancement ($\propto E_{loc}^6$) of the FWM process that effectively broadens the intensity dynamic range and meanwhile reduces the overlapping between closely located emitters. To determine the coordinate of the FWM peak and centroid, a total incident power which ensures free of damage and meanwhile provides appropriate threshold was used. This allows us to consider only the central region where the FWM signal has an S/N ratio larger than 5000. Essentially, the excited plasmon mode manifested itself as a hot spot which was imaged onto the EMCCD via a lens to from the object PSF. The region of interest (ROI) was constructed by linking the points which define the width of a Gaussian profile in each line scan of the PSF in the x- and y direction iteratively. This rules out faint satellite lobes and reduces the area under investigation down to 0.9161 μm² (31 × 31 pixels). Next, the FWM peak position and the centroid obtained by weighting with local intensity at each time delay of the probe beam were determined from the recorded images. The results are in close agreement with that calculated by the maximum likelihood estimator [34], as illustrated in Appendix C1, wherein the centroid determination and the justification of the accuracy were presented in detail.

 

Fig. 5 (a) The SEM image of the region under study. Regions produce FWM with S/N ratio larger than 900 were enclosed by the blue circles. (b) The OM image of the same region under study. (c) The enlarged SEM image where the TPL peak positions, FWM peak positions, the trajectory of FWM centroid over λ = 580-640 nm, and the range of error are all designated on the same figure. Note that the color coded spot is the accumulated density of the superresolved FWM intensity, demonstrating superlocalization of plasmon emission within 60 nm alongside the identified metallic nano-interstice. The dynamic trajectory of a full cycle of scan is available online (see Visualization 2).

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To correlate the found hot spots with morphological features, we analyze the diffraction limited FWM images further combining superresolution techniques. Upon scanning the time of delays, the hot site can be switched on and off sequentially due to dispersion, which to our knowledge, is as-yet unused for discriminating spatially overlapped images of individual emitters. In combination with superresolution algorithms, the stream of the EMCCD images were deconvolved with the PSF of the imaging system to recover the superlocalized coordinates of the emitting sites at various time delays (or wavelengths). Next, the coordinates of the SEM and OM images were unified basing on 5 alignment marks, all were observable under OM and SEM. The detailed alignment process and parameter setting in the superresolution algorithm were presented in Appendix C2 and C3, respectively. As a result, the FWM peak positions, centroids, and superresolved plasmonic hot sites can be superimposed altogether on the same SEM image, establishing correlation with certain morphological features on the percolated film. It is identified that the peak positions of the FWM are distributed around the turning edge of a single metallic interstice which is about 130 nm long and 30 nm wide located at the center of the excitation spot. Among 21 time delays over λ = 580-640 nm, hot spots are distributed at 4 different positions, as shown by the pink dots in Fig. 5(c). This result indicates that certain hot sites may accommodate multiple resonances which rules out particle type plasmon resonance and was recognized as a signature of superlocalization [35]. Nevertheless, the explanation is still insufficient and there is more demanding to pinpoint the origin. In view of the calculation done by Kollmann H. et. al. [12], a 30 nm gap is unlikely to yield FWM with S/N as high as 5000. We therefore attribute these hot spots arise from even smaller nanostructures that can hardly be resolved by the SEM. Since nanogap of <1nm features a broadband resonance with extremely large EM enhancement, our result renders the grain-interstitial or the sharp turning edges associated with the identified interstice the most possible cause. In parallel to FWM peak positions, the centroids as a function of time delays are also plotted, which gives localization to subpixel resolution. Figure 5(c) shows the trajectory of the FWM signal where the orange curve links the centroid positions for positive time delays and the blue curve links the centroids for negative time delays. It is found that the trajectory lies well within a region bounded by the FWM peaks and most interestingly, it reveals the shape of the identified metallic interstice with high fidelity.

To validate our result, superresolved hot spot at different time delays were also calculated and localized on the SEM image by the common code: deconSTORM [36]. The movie clip (see Visualization 2) shows the trajectory of a full cycle of scan which is highly repeatable and coincides with the trace of centroids calculated previously by single emitter based Gaussian fitting method. The range of superlocalization, shown as a density spot in Fig. 5(c), was generated by accumulating the registered signals on each pixel of the EMCCD. It should be noted that at each time delay, the dwell time for integration is 400 ms which is much quicker compared to current single molecule Raman technique [37] and can be further improved to video rate in the future. The elliptical spot follows the shape of the identified metallic interstice, and the range of superlocalization is estimated to be ~60 nm (by the length of the major axis of the elliptical spot) which is very useful for single molecule sensing, spectroscopy, and manipulation combining plasmonic hot spot steering capability presented in this study.

6. Conclusion

We developed a practical technique to sieve, localize, and steer plasmonic hot spots along single morphological nano-interstice on percolated metallic films. Based on spatial-temporally superresolved FWM trace, the peak and centroid position of the FWM hot spot can be determined with accuracy as high as ~20 nm, far beyond the one photon diffraction limit. The femtosecond chirp-manipulated multi-wavelength FWM also allows one to steer the plasmonic hot spot, which provides an as-yet unused switching mechanism to separate overlapped images of closely located emitters. Combining with existing superresolution algorithms, the present method enables the realization and verification of label-free multi-color superlocalization of plasmons within the range of 60 nm. Overall, this result offers exotic advantages over existing superresolution techniques [38–41] in terms of the accessible wavelength and the acquisition time, and may be applied to investigate plasmonic nanostructured substrate for molecule monitoring, manipulation, sub-cellular bioscience, and quantum plasmonics.

Appendix A Setup for FWM spectroscopy and superlocalization imaging

The home-built setup for femtosecond chirp-manipulated four wave mixing (FWM) experiment was detailed in Fig. 6.

 

Fig. 6 Setup for multi-wavelength FWM spectroscopy and superlocalization imaging.

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Appendix B Polarization anisotropy

Polarization dependent FWM was achieved by tilting the sample with respect to the beam axis, as shown in Fig. 7. When the surface normal was parallel to the beam axis, the excitation is purely TE polarized. When the surface normal was tilted by 30°, TM component was introduced. The statistical result in Fig. 3 was obtained by sampling 250 different positions on each Au film, and the measured FWM signals were all normalized to that produced by Si <100> with the same experimental configurations.

 

Fig. 7 Experimental configuration for polarization dependent FWM.

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Appendix C FWM image processing

The analysis of the FWM image includes four steps: C1. peak and centroid finding, C2. correlation of hot spot and morphological features, C3. deconvolution super-localization, and C4. justification of accuracy.

C1. Peak and centroid finding

For each time delay, 10 FWM images were taken. The averaged intensity enclosed by the blue line (50×50 pixels) in Fig. 8 which contains no FWM signal was used as the background. This background signal was then subtracted from the measured FWM image.

 

Fig. 8 Region chosen for FWM image background subtraction.

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To find the FWM peak positions, we first outlined the region where pixels with S/N ratio larger than 5000 were selected, as shown by the blue circle in Fig. 9(a). Next, the region of interest (ROI) was defined by Gaussian fitting in both x- and y directions and connecting the full width at half maximum (FWHM) points, as shown by the contour lines marked in blue and green in Fig. 9(b). The ROI can be reduced further by zero the counts outside the ROI as shown in Figure 9(c), and Gaussian fitted again. This process was repeated twice which reduces the far field FWM plasmon emission to ~31×31 pixels. Finally, the pixel intensity counts were smoothed with 5×5 pixels and the peak position was determined by picking up the pixel coordinate with the maximum intensity counts.

 

Fig. 9 (a) Region of pixels with S/N ratio larger than 5000. (b) The ROI defined by the contour of the FWHM points of a 2D Gaussian function fitted in the x- and y diretions. (c) Zero the intensity counts outside the ROI and Gaussian fitted again may further reduce the ROI.

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To determine the centroid of the FWM hot spot, we first define the ROI as described previously. The centroid was then determined by finding the center of mass of the pixel coordinate within the ROI as follows:

(1)$$X_c=\frac{{\displaystyle \sum _i{X}_i\times I_i}}{{\displaystyle \sum _i{I}_i}}$$

(2)$$Y_c=\frac{{\displaystyle \sum _i{Y}_i\times I_i}}{{\displaystyle \sum _i{I}_i}}$$

Where X_c and Y_c are the central coordinates, X_i, Y_i, and I_i are the coordinate of the i^th pixel and the corresponding light intensity. To validate our result, we compare the peak and centroid position with that found by the maximum likelyhood estimator (MLE) [42] altogether in Fig. 10. It is found that the three results are in close agreement with one another, and the largest deviation between the centroid and that found by MLE is ~21 nm, ensuring the correctness of our method.

 

Fig. 10 Comparison of the FWM peak and centroid determined in this study and that by the MLE.

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C2. Correlation of hot spot and morphological features

To correlate the hot spot with morphological features, we first select two SEM images: one contains the alignment marks (feature a-e) covering a larger area and the other one with higher magnification containing only the features illuminated by the laser light. The latter image was then down-scaled to the former one, and the position and orientation were adjusted to overlap the morphological features on both images as close as possible. Once this step was done, the two images were merged into one as shown in Fig. 11. Figure 12 illustrates the accuracy of the alignment process. It can be seen that (a) is well-aligned as in contrast to the ill-aligned case (b). Due to aberration, regions outside of the dashed circle were ill-aligned in both cases. However, this caused negligible effect since the well-aligned region has sufficiently large area which already covers the spot of the excitation entirely.

 

Fig. 11 The alignment between two SEM images. a-e are alignment marks observed under low magnification. The inset in the middle was the image taken under higher magnification which was superimposed onto the low magnification image.

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Fig. 12 Illustration of the accuracy of alignment. (a) Well-aligned image. (b) Blurred image due to ill-alignment. The blurred image outside the dashed circle in (a) was due to aberration.

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To correlate the SEM image with the OM image, firstly, the FWM image labeled the peak and centroid position was stuck onto the OM image according to the coordinate of the CCD pixels, as shown in Fig. 13 (a). Next, Fig. 11 was superimposed onto Fig. 13(a) by appropriately shifting, tilting and scaling, making the five alignment marks a-e overlapped as close as possible. Fig. 13 (b) demonstrates the result and Fig. 13(c) zooms in each alignment mark individually for clarity. Finally, all layers of the figure were made transparent except for the SEM image with high magnification and the symbols marking the peak and centroid of the FWM on the OM image, which results in Fig. 5(c).

 

Fig. 13 The correlation of the OM and the SEM image. (a) FWM image correlates to the OM image with 5 alignment marks according to the coordinate of CCD pixels. (b) The correlation of the SEM image with (a) by overlapping the alignment marks. (c) The enlarged view of the alignment marks under OM, SEM and both.

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C3. Deconvolution super-localization

To compare and validate our results, we adopt Xiaowei Zhuang’s code [43] to recover the distribution and the spatial extend of plasmon superlocalization. Theoretically, the far field FWM image was formed by convolving the image of the localized emitters with the point spread function (PSF) of the optical system. In the reverse way, the image of the localized emitters can be obtained following deconvolution process, as shown schematically in Fig. 14.

 

Fig. 14 Illustration of the deconvolution method to recover localized emitters.

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The parameters setting in the algorithm are described as follows: d_samp represents the enhancement factor that increases image resolution after deconvolution. In the current case, we chose d_samp=4. Between two consecutive recording, α stands for the probability that the illuminated emitter remains to be active, and β stands for the probability that the illuminated emitter turns from inactive to be active. Since our experiment incorporated no fluorescent tags, it is assumed that there is no random process, i.e., as soon as the sample was illuminated, the emitters were activated. Therefore, we chose α=1 and β=0 in the simulation process. To estimate the width of the PSF of the optical system, we use the formula σ=1.22λ/N.A which gives a diffraction limited Gaussian spot with width σ=26.67 pixels (826.88 nm) at the wavelength of 610 nm. The background parameter b=10 is calculated as previously describe and the gain parameter g=1300 was chosen so as to optimize the maximum of signal counts.

C4. Justification of accuracy

According to reference [44], the position accuracy was defined by the following formula:

(3)$$〈{\left(\Delta x\right)}^2〉=\frac{s_x^2+a^2/12}{n}+\frac{8\pi s_x^4{b}^2}{a^2{n}^2}$$

(4)$$〈{\left(\Delta y\right)}^2〉=\frac{s_y^2+a^2/12}{n}+\frac{8\pi s_y^4{b}^2}{a^2{n}^2}$$

Where s_x and s_y are the standard deviation of the Gaussian-fitted PSF in both x- and y directions, $a$is the pixel size, n is the total number of collected photons, and b is the background noise. Parameter s_x and s_y were determined by fitting the measured spot (Fig. 10 at time delay δt = 0) within the ROI with a Gaussian function which gives s_x = 930.9 nm and s_y = 1075.2 nm, respectively. The pixel size, $a$ = 31 nm, obtained from the CCD specification, was confirmed by the SEM measurement. The background noise was measured by walking off the pump and probe beams and the statistical result of the 200 × 200 pixels yields b = 0.013 photon counts per pixel. The total collected photons n = 3734.059 was obtained by multiplying the total counts of the FWM image from the CCD with the calibrated conversion factor which was done by imaging a laser spot with known power. Having the abovementioned parameters determined, the error in position can be estimated to be ∆x = 15.9 nm and ∆y = 18.3 nm, which ensures that the correlation to the assigned morphological feature is reliable considering the size of the identified interstice (130 nm long and 30 nm wide) and the sparse distribution of the nano-interstices over the field of view. It should be noted that the biggest error among all alignment processes might arise from the OM to SEM images, which gives a potential error of ~100 nm. We therefore surround each identified hot spot a circle with such error as the radius (Fig. 15(a)) and select the outermost contour as the overestimated hot zone as shown in Fig. 15(b). It can be seen that this contour enclosed only very few morphological features as shown in Fig. 5(c), which again justified the capability to sieve and localize plasmonic hot spots with high accuracy using present method.

 

Fig. 15 The construction of the hot zone. (a) Each circle (~100 nm in radius) defines the range of error surrounding the central hot spot identified by the FWM. (b) The contour enclosed the FWM peak positions and the centroid which are basically alongside a major interstice as shown in Fig. 5(c).

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Acknowledgments

This work was sponsored by the Ministry of Science and Technology, Taiwan. The authors would like to thank for the grant support under contract number MOST 103-2112-M-008 −006 –MY3.

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