1. Introduction

In the last ten years, numerous techniques for far-field subdiffraction imaging have emerged, especially using fluorescent labels. For a densely labelled sample, there exist two main strategies that allow the superlocalization of isolated emitters: either by localized excitation of the fluorophores (e.g. in STED [1]) or stochastic switching (e.g. PALM [2] and STORM [3]). Both approaches finally obtain a wide-field image by accumulating points from a large number of measurements. Their success stems from their ability to image in liquid and biological environments. Using Brownian fluorescent particles, Cang et al. [4] were able to probe the intensity distribution of hot spots generated at a rough metallic surface. However, the extension of these methods to 3D imaging [5, 6] is still limited to a few micrometers in depth.

An interesting alternative to fluorophores is to use metallic nanoparticles (MNPs) as markers, since they are photostable and easy to chemically functionalize. Moreover, MNPs support localized surface plasmon resonances (LSPR), which can give them relatively large scattering cross sections. Recently, 2D superresolved images have been obtained using anisotropic MNPs as contrast agents [7]. Rotating the polarization of the incident light plays the same role as the random activation of the fluorophores, since LSPR of asymmetric nanoparticles are highly polarization-dependent. Besides, light scattering by MNPs preserves the coherence of light, thus allowing interferometric detection. Recent advances in digital holography now allow 3D localization and tracking of Brownian particles [8, 9, 10] provided that the typical interparticle distance allows their individual imaging.

Although all these methods rely on optics for detection, few of them have indeed been used to image electromagnetic fields, i.e. to retrieve an optical information with subwavelength resolution. Electromagnetic (EM) field mapping is nevertheless crucial, notably for plasmonics (e.g. for the characterization of nanoantennas) or SERS studies [11]. So far, imaging local EM fields has essentially been the realm of Near-field Scanning Optical Microscopy (NSOM) [12], since sub-λ features are filtered out by propagation. These microscopes use nano-apertures or scattering tips to reveal local fields, and can actually be seen as a special case of super-localization, the position of the tip being known with an excellent accuracy. However, NSOM microscopies are difficult to implement in liquids due to tip oscillation damping, although there exist some successful attempts [13]. An emerging alternative for the superresolved imaging of electromagnetic fields is based on nanostructurated media such as hyper- or meta-lenses [14]. Additionally, the recent extension to the optical range of time reversal imaging (initially developed in acoustics and in the microwave range [15]) now allows the retrieval of sub-diffraction near-field features. In this case, the prior analysis of the speckle patterns induced by a complex scattering medium in well-known conditions allows subwavelength focusing [16] as well as imaging [17, 18].

The technique we propose here is inspired by a combination of the concepts of superlocal-ization and NSOM microscopies. Like in NSOM, and as initially suggested by Synge [19], we propose the use of small objects to scatter light to the far-field. For monodisperse particles sets (i.e., at fixed scattering cross section σ_scatt), the power scattered by the particle provides an optical information on the local field intensity I(x, y, z) at the position of the particle, since I(x, y, z) = P_scat (x, y, z)/σ_scat. Although often overlooked in previous studies, this information gives a unique opportunity to map the amplitude and intensity of propagative and non-propagative optical fields with subwavelength accuracy.

Using a holographic setup, we acquire holograms of multiple gold nanoparticles undergoing Brownian motion in a liquid enclosed inside a microfluidic chamber, in the presence of spatially structured optical fields to be characterized. Warnasooriya et al. [20] and Verpillat et al. [10] recently showed that digital holography is able to reconstruct the 3D image of several metallic nanoparticles, resulting in multiple Point Spread Function (PSF) spots. Superlocalization can then be achieved in 3D on these PSFs with excellent accuracies. In this article, we do not only provide the subdiffraction position of multiple MNPs but also the locally scattered intensity. Due to random motion, particles visit most voxels within a few minutes, at a hologram frame rate of typically 20 Hz. A computer reconstruction of the scattered intensities at each position yields a 3D map of the optical field in the sample. The optical resolution of the technique is limited either by the size of the particle (60 nm here) or the localization accuracy (3 × 3 × 10 nm³), and is therefore well below the diffraction limit. The imaged volume is also unprecedentedly large: of the order of 40 × 40 × 30 μm³.

We first discuss the parameters that limit our localization precision. In order to illustrate the near-field imaging potential of our system, we then image the evanescent wave decay in a Total Internal Reflection (TIR) configuration [21]. Finally, highlighting the large 3D volume that can be imaged with our technique, we reconstruct the intensity profile of a focused Gaussian laser beam inside a microfluidic chamber.

2. Experimental setup

2.1. Digital holographic microscope (DHM)

The experimental setup is sketched in Fig. 1(a). The excitation laser beam is split with a polarizing beam splitter (PBS) into a reference and an illumination arm (fields E_R and E_I). A half-wave plate (HWP, λ/2) and neutral densities allow for the adjustment of the optical power in each arm. The reference beam passes through a beam expander (BE) to form a plane wave with a polarization angle adjusted with a HWP, in order to align it with the main polarization component of the scattered light and maximize the contrast of the holographic fringes. The object is illuminated under an angle θ, in glass, using a glass prism. The electric field scattered by the object under study E_S ≪ E_R is collected by a microscope objective with glass coverslip correction. This is essential in order to preserve the quality of the 3D PSF of the microscope when imaging inside microfluidic chambers. Off-axis optical interference between E_S and E_R results in a fringe pattern recorded with a EMCCD camera (Andor Ixon 885, cooled to 10°C, 512 × 512 square pixels of 8 μm).

 

Fig. 1 a) Schematic of the experimental setup. The polarizing beam splitter (PBS) and half wave plate (λ/2) allow balancing of the object and reference arms intensities, and the second half wave plate aligns the polarization of the reference arm to maximize interference contrast. b,c) Zoomed schematics of the microfluidic chamber containing gold nanoparticles in water. Above the glass/water total reflection angle (b), the particles are only illuminated by the evanescent wave, whereas below this critical incidence angle (c), propagative light directly illuminates the particles, and is then reflected at the glass chamber/air interface. In both cases, only the light scattered by the particles is collected by the objective.

Download Full Size | PDF

2.2. Microfluidic chamber

In order to enclose and protect the gold nanoparticle solutions, we fabricated microfluidic chambers using a glass slide/parafilm/coverslip stack : a double-layer ring of parafilm was heated to its melting point to serve as waterproof spacer of ∼ 150 μm thickness, and to seal tightly two plastic micropipettes which are used to fill the chamber after fabrication. In order to prevent particles from adhering to the walls, the glass chambers were treated with Bovine Serum Albumin (BSA) and washed prior to being filled with colloidal nanoparticle suspensions. In the following experiments, we used suspensions of citrate-coated gold nanoparticles purchased from BBI, with radii of r = 30 and 50 nm, and initial concentrations C = 2.6 × 10¹⁰ and 5.6 × 10⁹ particles/ml, respectively. When these solutions are illuminated by either a propagative or a non-propagative wave, a portion of the incident field is scattered, E_S, and collected by the detector.

In our system, the angle of incidence of the laser in the glass prism, θ, can easily be adjusted to switch between a non-propagative illumination of the colloidal particle solution, when θ is larger than the critical angle at the glass-water interface θ_c–gw = arcsin(n_water/n_glass) = 62.5° (Fig. 1(b)), or a propagative illumination for θ < θ_c–gw (Fig. 1(c)). In any case, total internal reflection conditions can be preserved at the last glass-air interface provided that θ > θ_c–ga = arcsin(n_glass/n_air) = 41.8°.

2.3. Imaging Brownian particles with DHM

Imaging Brownian nanoparticles is extremely challenging due to the very low number of available photons and the incessant motion of these nanoobjects. In order to obtain sharp images of moving objects, a short CCD integration time τ is essential. Over a time τ, Brownian particles travel a distance r(τ) = (6D_Bτ)^1/2, where D_B is the Stokes-Einstein diffusion coefficient. Any distance r(τ) substantially larger than the PSF of the microscope will therefore cause a blurring of the image of the particle, effectively reducing the accuracy of the localization.

Moreover, the scattering cross section of nanoparticles, which varies as the sixth power of their radius, is extremely low for small particles. In low light conditions, holography offers the possibility to increase the power of the reference beam E_R in order to increase the measured interference term E_SE_R and take maximum advantage of the CCD dynamic range. However, the detection of small moving particles remains a compromise between exposure time and particle size. With a diffusion coefficient D_B = 7, 2 μm²/s, a 30 nm radius particle undergoes a mean free displacement of 120 nm in each space direction within τ = 1ms. This is only marginally larger than the object pixel size 80 nm, and relatively small compared to the size of the lateral PSF of the microscope, ϕ_Airy = 1.22λ/NA = 763 nm for NA=0.85 and λ = 532 nm. Under such conditions, distortions are occasionally visible in the images of fast moving particles, however the vast majority of particles yield an image which is indistinguishable from the expected PSF of the microscope.

Different strategies for 3D localization, working either on the hologram itself [8, 9, 22, 23] or upon the hologram reconstruction have been proposed, with similar accuracies. We chose the second, which works in real space and makes the visual identification of features or artefacts, and the choice of parameters which avoid them, easier. In our method, the complex field E_scatt (x, y, z) in the full volume of the sample is numerically reconstructed from a single hologram at every time instant. This is achieved using the numerical angular spectrum method (see [10] for exhaustive details on the reconstruction algorithm and localization process).

A threshold is then applied to detect likely particles. For each region above the threshold, we determine the reconstruction plane along z and the xy lateral pixel for which the intensity is highest. Using these coarse xyz positions, we perform a Gaussian fit in the lateral direction and a second order polynomial fit along the axial direction in order to localize the particle position with subdiffraction resolution. In addition, the intensity of all the detected maxima is also recorded, yielding sets of I(x, y, z) values which reflect the intensity of the light scattered by each particle at a given instant.

The whole calculation is performed on a Graphical Processing Unit (Nvidia GeForce GTX560, 448 cores) using the CUDA language, which allows for massive parallelisation of the processing, accelerating it by almost two orders of magnitude as compared to classical CPU-based methods. In these conditions, the volume reconstruction and the localization of several particles is achieved in typically 200 ms per hologram.

3. Immobilized metallic nanoparticles

3.1. Experiment

To study the localization precision, we imaged particles immobilized on a substrate, in air, in order to obtain a lower limit on the achievable accuracy and to compare i) the scattered intensity and ii) the variations of the measured positions to models. Gold beads of 30 nm radius were fixed in a polyvinyl alcohol (PVA) matrix deposited by spin coating onto a glass slide. The bead concentration was chosen so as to obtain above-diffraction spacing between particles (average distance of the order of 10 μm). The spin coating conditions ensured a thin PVA film, with all particles lying in the plane of the glass substrate, as shown in an earlier study [24]. These slides were set onto a prism and illuminated at an angle θ > θ_c–ga in glass, to create TIR at the PVA-air interface. We used a single-mode frequency-doubled Nd:YAG laser at λ = 532 nm (CNILASER MXL-III, max. power 80 mW). The evanescent wave locally frustrated by the beads was therefore scattered into a propagative field E_S and collected by the objective (Olympus 100×, NA=0.85). Under these conditions, the bead diffraction spot spreads along a 40 × 40 pixel area, with an integrated intensity, expressed in arbitrary units (a.u.) as given by direct camera output, of 2.7 × 10¹⁸ a.u. As previously shown by Gross et al. [25], this intensity value can be converted to an absolute number of photo-electrons. For low light holographic images, the noise level is related to the shot noise of the reference beam and corresponds to 1 photo-electron per pixel. We thus determined the noise floor of our images by integrating the intensity of a 40 × 40 pixel area from the darkest region of the same axial plane containing the NPs. This relative value, measured at 4 × 10¹⁶ a.u., corresponds quantitatively to 40×40 = 1600 photo-electrons. Using this simple calibration step, we obtain an absolute value of 1.1×10⁵ photo-electrons scattered per particle in our reconstructed holograms. This method has the advantage of being independent of any direct optical power measurement, which is delicate for such weak signals.

Sets of 200 holograms of these immobilized beads were recorded. After reconstruction of the thin volume containing the NPs, the algorithm described above was applied to the detection the bright spots and to precisely fit each bead position in the lateral and axial directions. For each bead, we calculated the standard deviation over the successive 200 localizations in order to determinate our localization accuracy (see Fig. 2). As previously reported [10], the optimal localization precision is obtained for a hologram acquisition with particles slightly out of focus (typically ±10μm off the optimal focusing position), when the particle scattering is encoded over a larger portion of the hologram and recorded by more pixels. At this optimal position, we found that the standard deviation was 3 nm in the lateral xy-direction and 10 nm in the axial z-direction for a signal level of the order of 10⁵ scattered photons per particle.

 

Fig. 2 a) Position histograms for 200 successive localizations of the same immobilized NP. The values are centred around its mean position to highlight the standard deviation on the localization position. b) Sample defocusing during the acquisition time is corrected by subtracting a smoothed z position. c) 3D reconstructed intensity from a single snapshot of two NPs immobilized in the PVA film. The shadowed plane indicates the position of the glass substrate and the red ellipses are centred on the coordinates given by the localization algorithm.

Download Full Size | PDF

3.2. Analysis

The expected number of photons scattered by an individual immobilized NP under these conditions can also be expressed theoretically. For an incident laser power of 14 mW and an exposure time of τ_exp = 1 ms, we get 3.6 × 10¹³ photons impinging the plane of the NPs. The total illumination area is assimilated to a uniform disk of 20 μm radius (∼ 1250 μm²). We use the same analysis as Atlan et al. in [26] to calculate the expected signal and we obtain a value of 4.6 × 10⁵ photo-electrons scattered per bead, which is in good agreement with the measured value, 1.1 × 10⁵ photo-electrons.

The localization precision can also be compared to theoretical predictions. In light microscopy, despite the well-known diffraction limit that establishes a minimum size for an object image, the central position of a subdiffraction object can be determined with an arbitrary precision, given a sufficient number of photons in the spot. Owing to our off-axis filtering, our reconstructed images are mainly limited by shot noise. As shown by Thompson et al. in the case of fluorescent particles [27], each photon collected in the image gives a measurement of the position of the object and its position error is the same as the standard deviation of the PSF of the microscope. The best estimate of the position of the object is then given by the average of the positions of the individual photons, with a standard error of the mean position μ along the z direction of

With the signal level experimentally measured as discussed above, and using Eq. (1) we find an error in the localization position of 0.3 nm in the lateral xy-direction and 1.2 nm in the axial z-direction. This theoretical limit on the localization accuracy is almost one order of magnitude lower than the best experimental values, 3 nm and 10 nm along xy and z, respectively. Our experiments were conducted on standard vibration-isolated optical tables but residual vibrations, acoustical perturbation and thermal drifts, in addition to possible unaccounted for noise sources, are very likely to be the limiting factors here. However, our 10 nm accuracy value along the axial direction for 30 nm radius particles is still a noticeable improvement over previously reported performances, i.e. 70 nm axial accuracy for bigger (50 nm radius) particles [10]. We attribute this improvement to the higher NA of the microscope objective and to the improved PSF of refractive objectives as compared to Cassegrain objectives used in [10].

In the following sections, particles moving freely under Brownian motion in water were used as local scattering probes. In this case, the limited exposure times and/or the illumination conditions typically yield lower numbers of available photons and therefore lower localization accuracies.

4. Evanescent wave intensity distribution from TIR configuration

4.1. Experiment

A simple way to produce an evanescent wave from a propagative incident beam with our experimental setup is to induce total internal reflection at the interface between two mediums of different refractive index. For θ > θ_c–gw, as shown in Fig. 1(b), all the λ = 532 nm illumination light is reflected off the glass/water interface, and an exponentially decaying evanescent wave illuminates the nanoparticle-seeded water in the vicinity of the interface. In the weakly perturbative approximation, i.e. neglecting field disruption by the particles, the intensity in water is simply I(z) = I₀ e^−z/β, where $\beta =\lambda /4\pi {\left(n_s^2{\text{sin}}^2\theta -n_m^2\right)}^{-1/2}$ is the evanescent field penetration depth, n_s = 1.5 is the substrate refractive index, n_m = 1.33 is the liquid medium refractive index, and I₀ is the incident field intensity at the glass-water interface. In our experimental conditions, the angle of incidence θ = 72° corresponds to a penetration depth of β = 80 nm.

The glass chamber was filled with a solution of r = 30 nm particles with a concentration of 1.3 × 10⁹ particles/ml (1:20 dilution). Particle detection was achieved using a 100×, NA=0.85 objective, with adjustable correction to reduce the aberrations induced by the glass and water thickness, and imaged on the EMCCD at a frame rate f = 20 Hz and an exposure time τ = 1ms. A series of 2000 holograms was acquired, for a total duration of 100 s. From these holograms, after reconstruction and localization, we extracted a set of 1500 localization coordinates corresponding to isolated particles at different locations and instants, and we measured for each of them the scattered intensity in the plane of best focusing, I(x,y,z). Since the sample has no lateral structuring, the problem is x- and y- invariant, and the data can be reduced to a set of I(z) values. The 1500 measured values of I(z) are plotted in Fig. 3(a) (linear scale) and Fig. 3(b) (logarithmic scale), showing a general exponential decay well distributed around the expected exponential decay I(z) = I₀ e^−z/β, with β = 80 nm (solid black line, no adjustment).

 

Fig. 3 Normalized intensity I(z) of the light scattered by 30 nm radius nanoparticles represented in a) linear and b) logarithmic scale. 1500 individual localization events were extracted from a total of 2000 holograms (total acquisition time 100 s). The position of the glass/water interface, corresponding to z=0, was determined by localizing particles fixed to the interface. Detection events located in the z < 0 region result from noise in the localization. The solid black line corresponds to the theoretically expected exponential decay (β = 80 nm for θ = 72°, no adjustment). In red filled circles •, mean position values calculated from events inside the logarithmically distributed boxes indicated by thin grid lines. In (b), 95% confidence intervals are represented as horizontal error bars. The size of the experimental data symbols in a) is in real scale with the z axis, to give a pictorial flavour of a 30 nm radius nanoparticle probing an 80 nm decay length evanescent field. Alternatively, in b) symbols are intendedly smaller to emphasize the high number of localization events.

Download Full Size | PDF

4.2. Statistical analysis

As discussed in detail for immobilized particles, the localization accuracy is limited by the SNR of the recorded intensity. This explains the broadening, in Fig. 3, of the localization precision at low light level as particles that are further away from the glass surface (z = 0) scatter less light. More precisely, we normalised the scattered intensity of each localized particle by the intensity of the brightest particle, assuming that the error on the localization of the latter was minimal due to its higher SNR. In the following, we propose a statistical method to verify quantitatively that the stochastic I(z) scattered by the nanoparticles follows an exponential decay with the expected decay length, by taking into account the localization precision discussed above, which is our main source of error.

Let us call z_r the real particle distance to the surface (z_r = 0 corresponds to a particle in contact with the surface). Our hypothesis is that the scattered intensity, denoted I(z_r), follows the law:

Due to the uncertainty in the detected position of the particle, the measured distance to the interface is a random variable denoted Z₀. Accordingly to Eq. (1), we make the assumption that the error on the measured position can be written as:

The statistical validation for the relation in Eq. (2) with a specific value β relies on the following method:

In Fig. 3, we show that the hypothesis of Eq. (2) is validated by the experimental data, for our theoretical penetration length β = 80 nm. Grid lines in both figures show the N = 20 boxes, logarithmically distributed, used to compute the empirical mean values Ẑ_i. In the logarithmic plot (Fig. 3(b)) we have represented the confidence interval for each mean value: the only values not matching the theoretical exponential decay within a 95% confidence are those for very high and very low intensities. For very high intensities, this is due to the scarce number of events to compute the mean position; for low light intensities, the detected positions are not reliable any more and other weakly scattering artefacts may have been detected by our localization algorithm instead of nanoparticles.

5. Laser beam intensity mapping

The previous analysis neglected the x- and y-dependence of the intensity, since there was no lateral structuring in the geometry of the system. In order to illustrate the 3-dimensional abilities of the technique, we investigate here the case where the angle of incidence θ_c–ga < θ = 45° < θ_c–gw allows light propagation inside the water-filled chamber (see Fig. 1(c)). This light is then reflected off the last glass-air interface, preserving a dark-field illumination. The experiment was conducted on a similar setup, using an illumination wavelength of λ = 660 nm from a laser diode (Opnext HL6545MG, max. power 120 mW). The diode was expanded by a collimated telescope and focused inside the microfluidic chamber with a f′ = 50 mm converging lens.

Without nanoparticles, no light propagation occurred beyond the sample in these dark-field conditions and the laser beam was not observable. A colloidal solution of 50 nm radius particles with a concentration 5.6 × 10⁸ particles/ml (1:10 dilution) was used to scatter light to the microscope objective (Olympus 60×, NA 0.7, adjustable coverslip correction) and the camera. The holographic detection of the particles on a dark background was repeated over 6000 holograms acquired with an integration time τ = 1ms and a frame rate of 12 Hz, for a total recording time of 8.3 min. After processing with a detection threshold fixed at 45 times the background intensity (average signal measured in regions devoid of any particle) in order to avoid false detections, 36000 localization events were obtained. On average, 6 nanoparticles were therefore detected simultaneously in the volume reconstructed from each hologram. The corresponding (x, y, z) positions are shown in Fig. 4(a), with intensities I(x, y, z) coded by the size of the dots.

 

Fig. 4 a) Position of 36000 localization events detected when illuminating gold particles 50 nm in radius undergoing Brownian motion in water, with a λ = 660 nm diode laser beam. The intensity I(x, y, z) recorded at each location is represented by the size of the spheres. The red line ℒ corresponds to a 3D linear regression on the positions and indicates the direction of the laser beam. b) Projection of the values I(x, y, z) onto a plane perpendicular to the propagation axis, ℒ. The two axis of the laser diode beam are clearly identified (red and green lines). Fitted profiles along these axis allow the retrieval of the Gaussian characteristics of the beam (see text).

Download Full Size | PDF

Although the whole microchamber was filled with Brownian nanoparticles, only those which passed through the propagating laser beam were detected, and the scattered intensity was stronger when the particle was close to the center of the beam. The beam propagation axis was determined by a 3D linear fit on the whole set of coordinates (x, y, z), as indicated by the red line ℒ in Fig. 4(a).

The lateral properties of the beam can also be investigated. Due to our sparse experimental values, which reflect the stochastic character of our sampling methods, plotting a single cross-section perpendicular to propagation axis ℒ leads to a very noisy image, some regions containing several particles, others none. To avoid this, and since the focused Gaussian beam considered here has an almost constant profile along its Rayleigh length on each side of its best focusing region (2 × q_out,⊥ = 84 μm and 2 × q_out,‖ = 216 μm, see calculation below), we considered a subset of the 3D dataset along 60μm in the propagation direction to perform an average in the z direction. Laterally, the (x, y) data were binned in 200 × 200 nm boxes in order to obtain an image exempt of empty pixels, resulting in Fig. 4(b). This 2D image clearly reveals the elongated mode structure characteristic of laser diodes. Indeed, the laser diode manufacturer indicates a high degree of astigmatism of the fundamental mode, with values for the beam divergence angle of θ_‖ = 7.5° along its main axis and θ_⊥ = 15° in the perpendicular direction.

To quantify these values of beam asymmetry from our data, we fitted this 2D z-averaged cross-section to a 2D Gaussian asymmetric centroid by means of least-squares calculations, obtaining two profiles along its minor an major axis (red and green lines in Fig. 4(b)), which yield the experimental values of the waist of the focused beam along the main axis and perpendicular to it, $w_{\mathit{out},\perp }^{\mathit{exp}}=5.3\hspace{0.17em}\mu \text{m}$ and $w_{\mathit{out},\Vert }^{\mathit{exp}}=2.9\hspace{0.17em}\mu \text{m}$, respectively.

In order to estimate the size of the focused beam inside the microfluidic chamber, we first measured the size of the beam waist after the telescope, i.e. at the input of the focusing lens, using a beam analyser (Thorlabs BC106N-vis): w_in,‖ = 3.5 mm and w_in,⊥ = 2.2 mm, parallel and perpendicular to the long axis of the elliptic mode, respectively. After the focusing lens, the size of the emerging Gaussian beam can be calculated in the plane of best focusing as [28]:

Using Eqs. 5 and 6, and neglecting beam distortions induced by the thin microfluidic chamber and its water content, we obtain the expected geometrical parameters of the beam inside the chamber at the output of the f′ = 50 mm lens: w_out,⊥ = 4.8 μm, w_out,‖ = 3 μm and q_out,‖ = 108 μm, q_out,⊥ = 42 μm. These values are indeed in good agreement with the measured values, $w_{\mathit{out},\perp }^{\mathit{exp}}=5.3\hspace{0.17em}\mu \text{m}$ and $w_{\mathit{out},\Vert }^{\mathit{exp}}=2.9\hspace{0.17em}\mu \text{m}$, the main source of error being attributed to the determination of the best focusing region in the 3D plot.

6. Conclusions

In this paper, we show that digital holography is a powerful way to localize metallic nanoparti-cles in 3D with an unprecedented accuracy down to 3 × 3 × 10 nm³ for 30 nm radius particles. This accuracy strongly depends on the available illumination and could be further improved using higher laser power while staying under the particle or solvent damage thresholds. More importantly, the measurement of the scattered light offers a way to measure local optical information. Using an evanescent wave, we show that these particles behave as multiple near-field probes, revealing the field characteristics along its decay direction. In addition, we have illustrated the possibilities of the technique to deliver a full 3D image of a focused laser beam propagating in water. We believe that this technique has the potential to serve as an alternative to other near-field techniques such as NSOM, specially in liquid environments which are very difficult to probe with tip-based systems due to the strong damping of probe oscillations.

In addition to working with non-bleaching probes, our holographic setup offers access to 3D optical information over large volumes, whereas most fluorescence-based techniques have limited ranges along the z-direction (typically a few micrometers range). We observe that our localization accuracy is limited by the degradation of the PSF by aberrations when working far from the plane of best focus. However, localization accuracy remains acceptably small, i.e. lower than the particle diameter, within a depth of 50 to 100 μm depending on the objective and particle size. With the Olympus 100×, NA=0.85 air objective used here with 30 nm radius particles, this degradation only occurs at z = ±30 μm from the focusing plane, while the accuracy in the xy plane is still an excellent 10 nm. At lower NA, the best achievable accuracy will be decreased, but the acceptable range along z will increase. Therefore, the technique has the potential to image very large 3D fields with excellent resolutions.

Future steps will involve 3D imaging of complex near-field hot spots in a liquid environment, which will be crucial in order to characterize plasmonic structures. One major obstacle, however, is the light scattered to the far field by the plasmonic object itself. After holographic reconstruction, this parasitic light degrades the detection of the metallic nanoprobes and further studies are needed in order to preserve optimal localization accuracy. Finally, the technique developed in this paper could also be used to study the interaction potentials between a colloidal particle and a wall in microfluidics, as has already been been achieved using Total Internal Re-flection Microscopy (TIRM) [29], without requiring prior knowledge on the specific particle-wall hydrodynamic interactions in order to determine the intensity distribution I(z) with respect to the particle-wall distance.