On the impact of realistic point sources in spatial mode demultiplexing super resolution imaging

The desire to push beyond ‘Rayleigh’s curse’ has resulted in new techniques for super resolution imaging by deconstructing scattered light from point sources into several spatial modes, as coupling to higher order modes is exquisitely sensitive to lateral displacement. Here we implement such an approach for high numerical aperture objectives and demonstrate that for gold nanoparticles, their intrinsic asymmetry results in coupling to higher order modes without lateral displacement. This situation not only applies to practical nanoparticles but is applicable to any dipole emitter due to the asymmetry of the emission. However, with full polarization analysis we suggest that one may be able to apply such spatial mode demultiplexing techniques.

Resolving identical point-sources separated by distances less than the diffraction limit is a fundamental, yet complex problem in optics. This challenge has been addressed in many ways including approaches such as near field imaging (Kawata et al 2009), saturated scattering (Chu et al 2014, Wu et al 2016, localisation microscopy (Fernández-Suárez and Ting 2008, Huang et al 2010, Wegel et al 2016 and structured illumination (Gustafsson 2005, Chowdhury et al 2012. Inspired by quantum mechanics, there has recently been a paradigm shift to tackle this challenge arising from the realisation that collection and processing of the full electromagnetic field contains extra information that can be exploited. This concept has been demonstrated using spatial mode demultiplexing (SPADE) (Tsang et al 2016), in which the amplitude of each individual Hermite Gaussian mode scattered by point sources is collected, as conceptually illustrated in figure 1. Higher order modes are highly sensitive to displacement either side of the optical axis as a result of their spatial distribution. By considering the full modal make up of the collected signal, and therefore considering these modes with a higher sensitivity, it is predicted that the limitations imposed by diffraction can be overcome. This concept has been explored experimentally in a number of schemes using low numerical aperture (NA) lenses (Paúr et al 2016, Yang et al 2016, Tham et al 2017. However, to the best of our knowledge, this approach has not yet been implemented with the high NA objectives commonly used to image samples with high resolution. Intriguingly these high NA objectives, used in biological and fluorescence imaging, generate distinct spatial modes at the focus resulting from strong focusing of linear polarised light (Novotny and Hecht 2012). Conceptually, one of these spatial modes can be selected and utilised in a similar way to that of binary SPADE: a simplified version of SPADE that separates only the TEM 00 mode from all others (Tsang et al 2016).
Here we experimentally implement a variant of the binary SPADE approach for high NA objectives and image identical real point-sources separated by less than the diffraction limit. By comparing optical images of gold nanoparticles, obtained using an interferometric optical approach (Hong et al 2011), with those from atomic force microscopy (AFM) we demonstrate how the intrinsic asymmetric defects in real 'ideal' point sources affect the detected signal. We discuss how these alterations to the detected signal would affect the ability of SPADE to conduct super-resolution imaging with high NA objectives. A diagram of the optical system used is shown in figure 2(a). As a result of tightly focusing linearly polarised light with a high NA objective, such as the 1.45 NA objective used here, three different spatial modes are present in the focus, resembling TEM 00 , TEM 11 and TEM 01 spatial modes for the fields along the x, y and z axis, denoted by E x , E y and E z , respectively. The presence of a scattering object within the focal volume results in these components being scattered to the far field, which are collected by the second objective. The simulated in-phase component of these modes, based on the mathematical description by Novotny and Hecht (Novotny and Hecht 2012), is shown in figures 2(b)-(d). Each of these individual spatial modes has a distinct polarisation state. By selecting a frequency shifted local oscillator that matches the polarisation state of one of these modes, it is possible to image a sample using only the selected spatial mode, while collecting both the amplitude and phase of the light scattered by the point sources. Here the TEM 11 -like E y spatial mode is selected due to its orthogonal polarisation relative to the excitation beam enabling background free detection of scattering objects (Miles et al 2015b). The scattering signature of a ideal point source should therefore match that of the simulated signal in figure 2(c). Particles that are in close proximity to a second particle will contribute to higher order modes due to their placement relative to the optical axis. These extra modes should perturb the scattering signature detected. By characterising the deviation from the expected E y distribution it should therefore be possible to identify the particles separation, similar to the binary SPADE method.
In order to study the response of closely spaced point sources and characterise any resolution enhancements, gold nanoparticles with a diameter of 60 nm were imaged. The nanoparticles were selected due to their high scattering cross section and point-like particle nature. To convincingly demonstrate any resolution enhancement, the sample under study needs to be compatible with both the transmission based optical scheme presented and an imaging system with far greater resolution. While scanning electron microscopy (SEM) is often used for the imaging of gold nanoparticles, the requirement of conductive samples limits the potential substrate choice. AFM does not have this limitation and has a resolution significantly better than the optical diffraction limit. However to fully characterise the sample the same location needs to be imaged in both these imaging approaches.
Therefore a fiducial Cr/Au grid was fabricated through photolithography and thermal evaporation on a glass cover slip (Marienfeld 1.5H). To prepare the sample the fabricated gridded coverslip underwent a 10 min bath in a 2:1 solution of nitric and sulfuric acid followed by two 10 min baths in deionised water. Gold nanoparticles with a diameter of 60 nm were then drop deposited onto the surface and left for 2 min, after which any excess solution was removed through use of a nitrogen spray gun.
To demonstrate the strong dependence of the optical signal on particle separation figure 3 presents data for the instance of one, two and three particles, respectively. Figures 3(a)-(c) shows these situations under AFM with (d)-(f) displaying the collected in-phase scattering signal imaged with the E y polarised spatial mode. This result clearly shows that there is a distinct difference in the collected pattern for multiple closely spaced particles compared to the instance of a single particle. Showing the potential to be able to extract their position even at separations lower than the diffraction limit.
On closer inspection figure 3(d) reveals a deviation from the expected spatial distribution for a single point source. As demonstrated by the AFM image shown in figure 3(a) the collected signal comes from a single particle yet it does not produce the expected spatial mode seen in figure 2(c). It is particularly striking that the collected image does not present a central zero along the x and y axis, which is clearly present in the All of the example particles in figure 4 were selected from a single image collected from each imaging system, eliminating artifacts due to changes in the AFM tip shape that can occur during multiple imaging sessions. The variation between the optical scattering signatures may suggest changes in the alignment of the system over time as a result of drift. The drift in our optical system was however previously investigated (Miles et al 2015a); it was found that over 3.5 h there was only a 0.41 µm ± 0.04 µm displacement of the spatial structure. Given that here an individual image was collected in ∼8.5 min any drift is likely to be very minor. While the drift may only cause small displacements it would still result in an uneven overlap of the local oscillator and the collected signal, which could potentially generate asymmetric spatial signature. To demonstrate that the change in overlap would not be able to reproduce the structures depicted in figure 4, drift in the spatial distribution is simulated for small variations in the off axis angle of the signal beam, as depicted in figure 5(a). This was carried out by summing the local oscillator and signal fields with different offset of the central position, represented by d in figure 5(a), which yields the results shown in figures 5(b)-(e). This shows the enhancement of different lobes, but does not result in a signal between these lobes as seen throughout figure 4. (d)-(f) In-phase scattering signal of these same particles imaged with the generated Ey mode. The merging of the spatial patterns in the case of the multiple particles are clearly seen and the increasing complexity of the spatial structure when multiple particles are involved. Each image is 1.6 µm × 1.6 µm.

Figure 4.
A selection of individual nanoparticles selected from a single sample imaged both with AFM, in row one and optically in row two. This highlights how the variation of particle shape can affect the scattering signature produced to the extent that expected zeros are not present in the centre of the spatial distribution. Each image is 1.5 µm × 1.5 µm.
The obtained images can in fact only be understood by realizing that the E x and E z spatial modes, shown in figures 2(b) and (d), respectively, are also present in the final spatial structure. This is illustrated in figures 6(a)-(d) where varying degrees of the E x and E z spatial modes are mixed onto the expected isolated E y mode filling in the centre of the scattering signature as seen in figure 4. Varying the strength of these additional modes enables the detected spatial distribution for a single particle to be reproduced, as shown in figures 6(e)-(h). This implies that higher order modes are not exclusively generated by off axis particles in close proximity.
The AFM images shown in figure 4 shows that the shape of the individual particles varies throughout the single image. It is unlikely that the tip convolution differs at each particle, suggesting that particles differ in shape, as has also been identified in other work (Zoriniants et al 2017). The variation in the spatial distribution between collected signals is therefore likely to arise as a result of these real particles not being ideal point particles, in effect breaking the symmetry of the system for the single point source. Any such variation in shape would cause the particle to be asymmetric along different axes, which clearly has a significant effect on the resulting scattered light that is detected. This effect can be understood from the fact that asymmetry in the scattering structure projects polarised light onto other axes; in a similar manner to how a polariser at 45 • to an incident linearly x-polarised beam projecting equal amounts onto the x and y axis. In this optical interferometric scheme this results in a variation in the scattering signature due to the different polarisation modes contributing to the final scattering signature. This polarisation projection is caused by the symmetry breaking that is intrinsic to real nanoparticles compared to that of theoretical point particles, a property that has implications for other optical systems than the one used here as well. While the image in figure 6(f) was generated by mixing various combinations of the simulated E x , E y and E z and optimising the fit a full experimental implementation of binary SPADE could be achieved by changing the spatial mode of the local oscillator to one that enhances a selected mode, enabling each to be identified individually, as has been demonstrated with a low NA system (Pushkina et al 2021).
The imaging of gold nanoparticles with a diameter of 60 nm both optically and through AFM demonstrates that they cannot be treated as ideal point particles. While 60 nm gold nanoparticles are considered fairly large for the use in biological applications and are also more likely to be affected by surface irregularities, evidence of these effects has been seen with 30 nm particles with an asymmetry of only 0.5% and will also occur for smaller particles due to the crystal structure (Payne et al 2013, Zoriniants et al 2017. It should be noted that this effect of realistic point sources projecting into different spatial modes not only affects scattering. In fact this phenomena of asymmetry in the emission occurs for any dipole emitters, such as the single fluorophores often used in biological applications, as they have a different spatial emission pattern depending on their orientation (Gersen et al 2000).
This non-ideal nature of realistic point particles has a dramatic influence on the concept of using spatial demultiplexing of the generated modes to achieve super resolution imaging. The combination of multiple modes in the isolated signal from an individual asymmetric point source significantly influences the ability of the system to spatially resolve point particles in real applications. Cross-talk between spatial modes has previously been identified as an important factor, reducing the performance of determining the distance between two incoherent point-sources (Boucher et al 2020). The separation of closely spaced particles through systems such as SPADE relies on the asymmetric modes being generated from only the pair. Should the collected signal also contain additional modes generated from non-ideal point particles then separation becomes significantly more difficult. To highlight this complexity a structured sample of 80 nm gold disk pair was fabricated by creating a mask with e-beam lithography before evaporating gold into well defined positions. The layout is presented in figures 7(a) and (c) with (b) and (d) depicting the respective image collected when utilising the optical system described earlier, working in reflection due to the sample substrate. This structured sample was used to investigate the transition from a pair of disks to a single asymmetric disk, occurring at a separation of 75 nm, with a gradually reducing aspect ratio. The optical signatures seen in figure 7(b) at a separation of 25 nm and in (d) at 125 nm separation look surprisingly similar given the stark contrast in the sample being imaged. With both the asymmetry of an individual scattering structure and the close proximity of two individual scatters contributing additional modal contributions it is difficult to determine the source and therefore understand the full nature of the sample without additional information.
The separation and characterisation of the full situation therefore requires a method of determining the model contribution from each individual point source, not just the overall structure. Imaging with multiple polarisation angles has long been used to determine the orientation of various sources, from nanorods to single fluorophores (Axelrod 1979, Sönnichsen and Alivisatos 2005, Li et al 2012. In these methods the optical response of the targeted sources are dependent on sample orientation in respect to the imaging system, projecting respectively along the imaging axis. In the case of an imaging system dependent on additional modal components the rotation reveals additional information on the sample. To demonstrate  (d) show the corresponding optical image to (a) and (c) respectively. The incident polarisation is shown by the white arrow in the top right corner. The disk pair orientation difference of 45 • with respect to the incident polarisation between the two samples is equivalent to imaging with a polarisation rotated 45 • . Separation distance is determined from the centre of each disk and labelled above. this the optical signatures collected in figures 7(b) and (d) can be considered, specifically at the boundary between a separation of 100 nm and 75 nm. A significant change can be seen in figure 7(d) in the scattering signature between the case of separate particles and when they join to form a single asymmetric particle. This change however is not replicated in figure 7(b). Most notably the difference between the two orientations is the increase in the E x contribution to the scattering signature in figure 7(d). This should come as no surprise due to a significant increase in scattering cross section compared to two individual disks, but even so, is only detectable when imaged with a rotated frame.
This approach is able to pull additional information from the sample, including detail that sits below the diffraction limit. Combined with an understanding of potential sources for additional mode components, and their relative contribution, the technique would permit the sample to be understood to a greater degree. Integrating this approach into SPADE like systems could enable not only the full location information to be determined but also the characterisation of each source, recreating the full situation. This is currently the subject of further work.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).