1. Introduction

The general trend of system miniaturization has experienced extraordinary advances with the emergence of nanotechnologies. Optics has also followed this path with the investigation of light phenomena at subwavelength scale, also known as nano-optics [1]. These studies of the complex light-matter interactions at nanometer scale have led to the development of nano-optical devices such as photonic crystals [2–5] photon sieves [6, 7], plasmon circuitry [8–11] and more recently nanoantennas [12–14], which have brought new scientific and technologic solutions to many disciplines. Improved understanding of electric and magnetic light phenomena around nano-elements has also made it possible to design highly promising artificial materials, such as “left-handed” metamaterials [15–17] that show unprecedented optical properties.

Thus research in nano-optics is essentially relying upon the ability to control light at the nanoscale through the design of specific systems. However, given the significant complexity of the vectorial optical phenomena associated with subwavelength structures, nano-optical system design is starting to require a complete and precise knowledge of the electromagnetic optical fields in play, in order to perfect such designs. This means that the space-varying local amplitude and direction of the electric and magnetic fields, beyond just optical intensity, become particularly important information for predicting or analyzing nanodevice optical behaviors, especially for metamaterials [18, 19]. From this fundamental electromagnetic information, Poynting vector or energy density can be straightforwardly deduced, allowing for improved system understanding and design. Theoretical modeling has long been preparing solutions for this need, with the development of rigorous numerical methods such as the Finite Difference Time Domain [20], Green dyadic [21] and Multiple Multipole [22] methods, which allow complete and accurate light field simulation around arbitrarily complex nanodevices.

From an experimental point-of-view, access to the complete electromagnetic information associated with nanosystems remains both highly desirable and extremely challenging. Beyond the novel experimental insight of optical phenomena that occur at subwavelength scale, a sufficiently precise diagnostic of nanosystem optical properties would make it possible to assess reliability and predict achievable device performance. Because nano-optics stretches the very limits of currently available fabrication technologies, fabrication errors and defects are often unavoidable, leading to significant discrepancies between the desired and real optical properties and system performance. The ability to understand the non-obvious relationships between fabrication defects and system optical properties — by having a full comparative electromagnetic analysis between experimental systems and their model behavior — could have strong implications on scientific advances and commercialization of nano-optical systems.

Scanning Near-field Optical Microscopy (SNOM) is a highly capable method for probing subwavelength systems, providing high spatial-frequency information about even very fine sample details [8–11, 23]. Despite this fact, conventional SNOM does not quite provide the full set of required diagnostic tools, since the detected signal from most probes often represents a not completely detangled mixture of information from different field components at the sample. In collection-mode SNOM [24], it has been shown that metal-coated fiber probes can be sensitive to either the magnetic field [25, 26] or to the electric field [27, 28]. Probe-integrated annular nanoantennas have shown the ability to collect either the longitudinal magnetic or longitudinal electric fields [29]. It is known that dielectric bare probes are mostly sensitive to the electric field [30, 31], with almost no sensitivity to the magnetic field. However, they filter out the field component parallel to the tip axis [32, 33]. Recently, the local amplitude and polarization of the optical electric field has been retrieved experimentally with subwavelength resolution thanks to a novel SNOM architecture [34]. These unprecedented results represent highly promising advances in nano-inspection of optical structures and metamaterials. However, due to constraints in the detection process, this technique is limited to 1D surface samples. Moreover, information about the vector optical magnetic field is inaccessible.

Bethe and Bouwkamps’ conclusions on nano-apertures [35, 36] state that aperture tips cannot discriminate between the electric and magnetic optical fields. This is a serious problem for accurate interpretation of results, since in the presence of subwavelength spatial variations the electric and magnetic fields do not spatially overlap. They each carry complementary vectorial information, which has to be distinguished in order to fully analyze the electromagnetic response of such a nano-optical system. Recently, Burresi et. al. added an thin air gap to a metal-coated SNOM tip to create a “split-ring” nano-aperture capable of directly detecting the optical magnetic field [37]. Using this tip, they were able to show direct and simultaneous detection of the out-of-plane magnetic field and one transverse electrical field component above a ridge waveguide. Unfortunately, at least in these initial experiments, any signal from the other transverse electrical field component proved to be indistinguishable from the signal from the magnetic field [37]. Thus there is still a need to extend the capabilities of conventional SNOM, which currently provides either scalar or irretrievably mixed information about the vector optical field, in order to provide the accurate vector field diagnostics needed to unlock the full potential of nano-optics.

In this paper, we tackle the full 3-D electromagnetic reconstruction problem in near-field optical microscopy, developing a concept of near-field imaging of the amplitude and 3D local orientation of electric and magnetic fields around nanostructures. Our approach is based on a new SNOM configuration able to simultaneously and independently map the distributions (amplitude and phase) of two orthogonal electric-field components over a plane. From this single 2D acquisition right at the sample surface, the complementary electric and magnetic field lines and Poynting vector distribution can be reconstructed in the volume above the sample using rigorous numerical methods. The paper is organized as follows: In Section 2, the principle of our imaging concept is presented. In Sections 3 and 4, respectively, the SNOM architecture and field reconstruction technique are summarized. Then, the experimental SNOM set-up is detailed (Section 5) and the overall imaging concept is validated and discussed with two different complex yet analytically well-known vector-field distributions (Section 6). Our conclusions are presented in Section 7.

2. Basic principle

The full description of the structure of light near nanodevices requires the precise knowledge (with subwavelength resolution) of the spatial distributions of three electric and three magnetic field components in the volume surrounding the system. From this complete information, Poynting vector, energy density and electric and magnetic field lines can be deduced for a refined analysis of the optical nanostructures. However, based on the interplay between electric and magnetic fields described rigorously in Maxwell equations [38], it is obvious that not all six field components need to be measured. For example, the Maxwell-Faraday and Maxwell-Ampere equations allow retrieval of the vector magnetic field from the vector electric field, and vice versa, whereas the Maxwell-Gauss equation (and Maxwell-Thomson equation) shows that the electric-field vector components (and magnetic field components) are dependent upon each other. Following Maxwell equations in 3D space, the amplitude and direction of just two vector-field components need to be measured in order to fully describe the electromagnetic waves around samples. The other four components can be straightforwardly deduced from these two complex field acquisitions. For 2D problems that involve 1D samples, only one field component needs to be experimentally measured. For example, 1D subwavelength distributions of THz magnetic fields have been deduced from 2D electric field measurements (two components) by using the Maxwell-Faraday equation [39].

3. Concept of multichannel collection-mode SNOM

In this paper, we focus on extending this reconstruction approach to the complete characterization of 3D field distributions. To this end, we selectively map the spatial distributions of the two orthogonal components of the electric field parallel to a 2D measurement plane, namely, the E_x and E_y transverse electric field components. The SNOM architecture implemented for this purpose is depicted in Fig. 1. It works in collection mode, with the probe picking up light at the sample surface [40].

 

Fig. 1. (a) SNOM architecture for the simultaneous collection of E_x and E_y (amplitude and phase) at the sample surface. (b): polarization diagram of a pulled dielectric fiber tip.

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We have chosen to use an uncoated single-mode fiber probe as the nanocollector in this first demonstration of our SNOM set-up. Though dielectric probes exhibit modest resolution ability compared to aperture or apertureless probes, they are easier to produce in a highly reproducible manner and present adequate optical properties for demonstration of the concept. Several studies have shown that such probes are mostly sensitive to the optical electric field, with little or no sensitivity to the optical magnetic field [30, 31, 41, 42]. This is extremely important since the entire concept depends on being able to measure the two orthogonal transverse electric-field components with high accuracy and low crosstalk. Moreover, the use of sharp dielectric tips limits probe-to-sample optical coupling, which can be problematic in SNOM. It has been shown, for dielectric samples studied in photon tunneling SNOM configuration, that the tip-to-sample interaction is negligible, and thus a dielectric fiber tip can be considered to be a truly passive detector of the optical field [43]. Imaging of gratings illuminated with s-polarized light showed a good relationship between the SNOM images and intensities of the free space electric field [41, 42]. These works have shown that, despite some resolution limitations, dielectric fiber tips are good candidates to faithfully and non-invasively probe the electric field distributions at subwavelength scale.

However, imaging of 3D field distributions becomes more problematic than such intensity measurements, since the extraction properties of uncoated fiber probes are anisotropic with respect to the vector electric field. It has been shown that single mode tapered fibers filter out the electric field component parallel to the tip axis (longitudinal field) and collect only the transverse field [32, 44]. This selective collection behavior could be considered a considerable drawback for near-field imaging, since it represents the loss of potentially valuable information. In this work however, we take advantage of this polarization filtering property, since it leads to nanocollectors that are sensitive only to the transverse components of the electric field, with the longitudinal electric field filtered out by the single-mode fiber into which the probe feeds.

Given their rotationally symmetric geometry, dielectric fiber probes do not have any anisotropic sensitivity to the transverse electric field. In this context, we developed the multichannel optical collection scheme depicted in Fig.1(a), which uses polarization to discriminate the collected signals from the two transverse components E_x and E_y while the tip is scanned across the sample surface. First, the signal collected by the tip is split in two channels by means of a fiber coupler. One of the two output channels is connected to a detector (D₃) for the measurement of the overall intensity due to the two transverse fields. The other channel is connected to a second coupler for the separate detection of the signals from the E_x and E_y components. The simultaneous detections of E_x and E_y is achieved by inserting high quality polarizers (P₁ and P₂, each with 10⁴:1 polarization discrimination) between the end facets of the fiber pieces and the detectors (D₁ and D₂). The use of conventional polarizers coupled to fiber probes is typical in emission and reflection mode SNOM for polarization contrast and polarization modulation experiments [24, 45–47]. In our case, the intrinsic birefringence of the fibers is finely controlled with fiber polarization controllers (“babinet-soleil” or “three-paddle” type) to accurately define the polarization sensitivity of the overall probing system during calibration, and to maintain this sensitivity during measurements.

The full reconstruction of optical fields from the mapping of two electric field components requires the complete knowledge of the amplitude and phase of the probed waves. We achieve this by adding a heterodyne interferometer into the SNOM architecture shown in Fig. 1(a) [42, 48]. The “reference” branch of the interferometer is itself split in two channels, which are mixed into the two output channels of the polarizing fiber system through fiber couplers, enabling the simultaneous measurement of the amplitude and phase of E_x and E_y.

4. Full reconstruction of light structure beyond the sample

There are two possible ways to obtain the total electromagnetic light field in the volume surrounding the inspected nanostructures. The first way consists of reconstructing, with Maxwell’s equations, the missing electromagnetic information in the volume from experimental acquisition across the entire volume. This technique, previously carried out in THz near-field microscopy in order to define the magnetic field from the entire electric field of 1D samples [39], has the advantage of accurately describing electromagnetic field distributions independent of the particular finite scan window and the spatial properties of the fields being measured. However, this approach requires a multi-scan acquisition process at various tip-to-sample distances, making it difficult to implement with a heterodyne detection system that is prone to phase drifts and instabilities.

The alternative concept that we use here is to perform a single 2D acquisition right at the sample surface, and reconstruct numerically from these experimental results the entire optical field structure in the volume beneath the acquisition window. Various numerical methods of free space diffraction such as FDTD [20] or plane wave spectrum methods [49] can rigorously predict the complete structure of light in a 3D space from the knowledge of a limited number of field components across a single 2D face at the edge of the volume. We propose here to use 2D experimental acquisitions as experimental boundary conditions of these rigorous diffraction numerical methods. Note that no more-or-less realistic models of the samples need to be implemented in the numerical processes, so long as it is the fields above the sample which are to be reconstructed. All the sample optical properties and defects over the scan zone are faithfully described and stored in the 2D light field maps measured experimentally. The numerical methods are just used as optical field free space propagators in order to rebuild the entire light field distribution beyond the sample. The advantage is that such 3D light field distributions can be obtained from a single 2D near-field acquisition, thus limiting acquisition time and the amount of data required. However, the accurate description of the fields in the entire volume requires the imaging of square integrable field distributions that are contained in their entirety in the scan zone. This acquisition window becomes a diffracting aperture in the subsequent numerical processes, so that any non-negligible fields at the boundaries of the acquisition area will introduce parasitic and unphysical diffraction phenomena. Field apodization of the measured data before reconstruction helps reduce diffraction problems caused by the scan edges, at the expense of a reduction in both resolution and in the size of the volume over which 3D field distributions can be accurately reconstructed for a particular 2D scan zone. This problem will be discussed later.

5. Experimental SNOM set-up

The experimental SNOM set-up shown in Fig. 1(a) involves a stabilized laser source of high coherence length radiating at λ =660 nm. The linearly polarized laser beam is divided in two parts with a beam splitter. One part (the “reference” beam) is frequency shifted by 25 kHz with two acousto-optic modulators driven at 80.025 MHz and 80 MHz. This beam is then split with a 50/50 single-mode fiber coupler (Thorlabs) and coupled into the two output channels of the polarizing fiber device.

The rest of the original laser beam is directed to the sample. The SNOM probe is scanned across the sample surface with a home-made SNOM head that implements shear-force distance control. The dielectric fiber tips are fabricated by heating and pulling a single mode fiber with a micropipette puller from Sutter company (model P-2000). The multichannel device described above is constructed from standard 50/50 single mode optical couplers (Thorlabs), high quality polarizing films from Mlles Griot (10⁴:1 polarization ratio) and three-paddle manual polarization controllers (Fibercontrol). The signals from E_x and E_y are combined with the reference beam at the system output with unbalanced 10/90 single mode fiber couplers (Thorlabs), generating two separate beat signals at 25 kHz that are detected by standard silicon photodiodes. The use of unbalanced couplers helps compensate for the fact that the power level captured by the fiber tip is several times lower than the power of the reference beam. The simultaneous detection of the amplitude and phase of E_x and E_y is achieved by synchronous detection with two lock-in amplifiers (Stanford, model SR530) connected to a computer for data acquisition.

Figure 1(b) shows the typical polarization discrimination realized by our SNOM probe. This data was measured by projecting directly onto the tip a linearly polarized collimated laser beam, while rotating the polarization direction with a half-wave plate. The transmission diagram shows two sinusoids that are shifted by 90° to each other. The subwavelength collection system acts as a pair of crossed polarizers, able to probe locally and independently the information from two orthogonal transverse electric-field components. Due to the high-quality polarizers (10⁴:1) and calibration by manual polarization controllers, this polarization-sensitive SNOM probe realizes a polarization discrimination ratio better than 1:1500 (extinction ratio better than 32 dB). Note that this level of performance is achieved in both conventional and heterodyne SNOM configurations.

6. Validation of the imaging concept and related technique

We have chosen to validate our concept by reconstructing the full vectorial description of azimuthally and radially-polarized Bessel beams from a 2-D set of measurements with our polarization- and phase-sensitive near-field probe [50–53]. Given the axial symmetry of their electric and magnetic fields, radial and azimuthal polarizations are highly appropriate for focused-beam probing of the optical response of polarization-sensitive nanodevices such as single fluorophores [54] or nanoantennas [29]. The combination of these polarization states with propagative and evanescent Bessel beams offers the possibility for test field distributions of high aspect ratio that can be described by simple analytical expression (Bessel functions J₀ and J₁). The high aspect ratio relaxes the demands on probe positioning, while the simple field expressions allow obvious and straightforward comparisons between experimental and theoretical results. Since an arbitrary sample near-zone can contain both propagative and evanescent waves, we choose to validate our microscope system through the imaging of both an azimuthally-polarized propagative and a radially-polarized evanescent Bessel beam.

From an optical electromagnetic point of view, radially and azimuthally-polarized Bessel beams are very attractive electromagnetic test-objects, as they show basic electromagnetic functions that can be fully and unambiguously described with our microscope configuration. On the one hand, azimuthally-polarized Bessel beams exhibit the well-known configuration of a loop of optical displacement current density (ODCD) that induces a pure magnetic field along the loop axis. Such field properties are similar to those of magnetic moments (per unit length with Bessel beams). On the other hand, radially-polarized evanescent Bessel beam carry a longitudinal line of ODCD that originates a loop of magnetic field. Such a field behavior is analogous to that of single electric moments (per unit length).

6.1. Full description of an azimuthally-polarized propagative Bessel beam

The experimental acquisition of the azimuthally-polarized propagative Bessel beam is reported in Fig. 2. The test field-distribution is generated by projecting an azimuthally-polarized beam directly onto an axicon [55] of numerical aperture (NA) equal to 0.68 [Fig. 2(a)]. Azimuthal polarization is produced with a highly stable fiber device which minimizes phase drifts in the output doughnut beam [56]. The tip is immersed in the Bessel beam zone and is scanned to accumulate the electromagnetic field distribution in the (XY)-plane (or transverse plane) perpendicular to the beam’s propagation direction (Z). No distance control is required or available for this acquisition, so the phase accuracy achieved across the scan is a measure of the quality of our open-loop position control. The post detection system delivers both amplitude (A_x and A_y) and phase (ϕ_x and ϕ_y) of the orthogonal transverse electric field components (E_x and E_y). Figures 2(b) and 2(c) show the real parts of these field components, E_x = A_x cos ϕ_x and E_y = A_y cosϕ_y.

 

Fig. 2. (a): Scheme of the experimental set-up for the generation and full vectorial characterization of an azimuthally-polarized propagative Bessel beam. (b,c): Image of the orthogonal transverse electric field components E_x = A_x cosϕ_x and E_y = A_y cosϕ_y, respectively. A_x, A_y are the amplitude and ϕ_x, ϕ_y the phase distributions of the two field components, respectively (scale bar: 2 μm).

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The numerical reconstruction of the overall optical field from the 2D experimental results shown in Fig. 2(b) and 2(c), is realized with the FDTD method. Commercial software Fullwave (“R-Soft” company) is used for the numerical field propagation. The computation volume is limited laterally to 16 microns by 16 microns by the scan window and is 3 micron high in Z. All six boundaries are terminated with a Perfectly Matched Layer in order to avoid parasitic unphysical reflections. The resulting simulations are 512×512×150 grid cells in size. Because the lateral extent of Bessel beams produced in this study are several times larger than the scan zone, the experimental images must be apodized in order to avoid diffraction artifacts at the scan window edges. We have chosen a Gaussian apodization function with full width at 1/e of 8 microns. We find that the mismatch between the magnetic-field intensity distributions achieved with the apodized empirical and exact ideal Bessel beams do not exceed 1.6% of the intensity maximum across a volume 1 micron in height and 2.8 microns in diameter around the beam center. Note that while the propagation of empirically-measured and then apodized beam is simulated by FDTD, the propagation of the ideal Bessel beam is calculated analytically.

Figure 3 displays the reconstructed intensity distributions of (a,b) the electric field and (c,d) the magnetic field of the azimuthally-polarized propagative Bessel beam. The results are shown in (a,c) the longitudinal plane (y=0) and (b,d) the transverse plane (z=0), with fields expressed in Gaussian units. The non-diffracting nature of the Bessel beam can be observed with the almost Z-invariant intensity distributions of Figs. 3(a) and 3(c). We see at the beam center that the magnetic field reaches a maximum whereas the electric field is minimum. While in the ideal case the electric field should be null at this position, at the center of our reconstruction of the empirical Bessel beam the electrical field is 400 times smaller than the magnetic field. This “magnetic light” confinement is surrounded by a doughnut of electric field, indicating both a faithful production of the azimuthally-polarized Bessel beam by the axicon and highly accurate measurements by the phase- and polarization-sensitive SNOM probe.

 

Fig. 3. Intensity distributions, reconstructed from empirical data measured across the plane z=0 by our phase- and polarization-sensitive SNOM probe, of (a,b) the electric field and (c,d) the magnetic field of an azimuthally-polarized propagative Bessel beam, in (a,c) the longitudinal plane (y=0) and (b,d) the transverse plane (z=0). The fields are expressed in Gaussian units.

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Figure 4 and corresponding media file (Media 1) shows the complete submicron structure of the vector electromagnetic Bessel fields in (a) the longitudinal plane (x=0) and (b,c) the transverse plane (z=0). The background color image of Fig. 4(a) represents the x-component of the electric field distribution and black arrows show the magnetic field lines. The background color images of Fig. 4(b) and 4(c) displays the out-of-plane z-component of the electric and magnetic fields, respectively, whereas black arrows represent the in-plane transverse electric and magnetic field lines, respectively. Propagation of the beam wavefronts in the z-direction can be seen in the movie, as well as the phase quadrature between the longitudinal and transverse magnetic field components. The spatial concentration of pure longitudinally polarized magnetic field is confirmed.

From the full electric and magnetic vectorial information displayed in Fig. 4, the spatial distribution of the time-averaged Poynting vector can be straightforwardly deduced (see Fig. 5). We see a good relationship between (a) the energy flow of the experimental Bessel beam under consideration and (b) the flow of ideal azimuthally-polarized Bessel beam. Figs. 4 and 5 show that the longitudinal magnetic field acts as a reservoir of optical energy inside the beam since it is not associated with energy flow (there is no spatial overlap with the Poynting vector). This property was pointed out for the longitudinal component of the electric field of radially-polarized focused beams [54, 57].

Figure 6 displays a snapshot of the interplay between the magnetic field and the central loop of optical displacement current density (ODCD) (j⃗D = ∂D⃗/∂t) over half a beam wavefront. The toroïdal structure of the magnetic field that surrounds the central loop of ODCD is clearly visible. Given the size of the ODCD loop with respect to the wavelength, the magnetic field lines overlap and give rise to the single intense magnetic field confinement longitudinally polarized in the inner part of the loop, at the beam center.

 

Fig. 4. (Media 1) Full vectorial description of the reconstructed electromagnetic optical field for our empirical azimuthally-polarized propagative Bessel beam. (a): longitudinal cross-section (y=0): out-of-plane transverse electric field component E_x (background colored image) and in-plane magnetic field lines (black arrows). (b): electric field in the transverse plane (z=0): in-plane transverse electric-field lines (black arrows) and out-of-plane longitudinal electric field component E_z (background colored image). (c): magnetic field in the transverse plane (z=0): in-plane transverse magnetic-field lines (black arrows) and out-of-plane longitudinal magnetic field component H_z (background colored image).

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Fig. 5. Spatial distribution of the time-averaged Poynting vector in the longitudinal plane (x=0). (a): reconstructed from experimental SNOM data and (b) ideal Bessel beam

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Fig. 6. Interplay between the optical displacement current density (ODCD) and magnetic field over half a beam wavefront. (a):transverse plane: out-of-plane longitudinal magnetic-field component H_z (background colored image) and in-plane ODCD lines (black arrows). (b): longitudinal plane (x=0): out-of-plane x-component of the ODCD (background colored image) and in-plane magnetic-field lines (black arrows).

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6.2. Full description of an evanescent Bessel beam

While the detection and reconstruction of propagating waves by our phase- and polarization-sensitive SNOM probe is important, many nano-optical samples involve significant evanescent waves. To demonstrate the detection and reconstruction of such field components, we use the experimental set-up depicted in Fig. 7(a). The evanescent Bessel beam is achieved with a NA = 1.2 conical device composed of a solid immersion conical lens [58] and a microaxicon engineered directly at the end of the fiber radial polarizer introduced previously (see Ref. [59]). The tip is scanned over the flat output interface of this refractive system. During the raster scan, the tip-to-surface nanometer distance is maintained with a conventional shear-force distance control set-up. The real parts of the two orthogonal electric field components, E_x = A_x cos ϕ_x and E_y = A_y cos ϕ_y, in this 2D plane are shown in Figs. 7(b) and (c), respectively.

The plane wave spectrum method is a well-known and efficient technique for rigorously calculating the propagation of electromagnetic waves. Its efficiency lies in the ability to propagate waves from one plane to another using Fourier transforms. In a few words, general field distributions E⃗(x,y,z,t) are represented by the superposition of plane waves traveling in diverse directions:

where k⃗=(u,v,w) is the wavevector which satisfies the dispersion relation u² + v² + w² = k² where k is the propagation constant. Vector e⃗(u,v) is the “plane wave spectrum.” defined as:

E⃗(x,y,z = 0) represents the boundary conditions of the method. Note that the definition of the electric field component E_z(x,y,z,t) and the overall magnetic field H⃗(x,y,z,t) can be straightforwardly deduced from E_x(x,y) and E_y(x,y) by applying, in the inverse space (u,v), the Maxwell-Gauss equation - w(u, v)e_z(u, v) = ue_x(u,v) + ve_y(u,v) and the Maxwell-Faraday equationh h⃗(u,v) = k⃗(u,v) × e⃗(u,v)/ω, respectively.

 

Fig. 7. (a): Scheme of the experimental set-up for the generation and full vectorial characterization of a radially-polarized evanescent Bessel beam. (b,c): Image of the orthogonal transverse electric field components E_x = A_x cos ϕ_x and E_y = A_y cosϕ_y, respectively. A_x, A_y are the amplitude and ϕ_x, ϕ_y the phase distributions of the two field components, respectively (scale bar: 0.9 μm).

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Here, we use the experimental field distributions E_x(x,y) and E_y(x,y) of Figs. 7(b) and 7(c) as boundary conditions of the plane wave spectrum method to obtain in a rigorous way the overall electromagnetic optical field in the volume above the 2D experimental acquisition window, illustrating the evanescent decay in the air of the radially-polarized Bessel beam confined to the higher-index glass by total internal reflection. In this study, the acquisition window is 10 microns × 10 microns in size with 512 × 512 data points. The experimental results were apodized with a Gaussian function of full width at 1/e of 4 microns before reconstruction, in order to avoid parasitic unphysical diffraction caused by the edges of the acquisition window. We have verified that the differences between the electric-field intensity distributions achieved with the apodized empirical and exact ideal Bessel beams do not exceed 3.5 % of the intensity maximum in a volume 0.33 micron in depth beyond the glass-air interface and 1.6 microns in diameter around the beam center.

Figure 8 shows the reconstructed intensity distributions of (a,b) the electric field and (c,d) the magnetic field of our empirical radially-polarized evanescent Bessel beam in (a,c) the longitudinal plane (y=0) and (b,d) the transverse plane (z=0). The non-diffracting evanescent nature of the Bessel beam is evident in Figs. 8(a) and 8(c).

However, we see that the reconstructed intensity distribution is not rotationally symmetric. The visibility of the interference pattern vanishes along an angle of about 45° with respect to the x and y axis (referred to as the π-direction in Fig. 8 and in the following). Moreover, the magnetic field intensity is not negligible at the beam center, as would be expected in the ideal case. The origin of these defects can be observed in Fig. 9. This figure shows the plane wave spectrum of function $\sqrt{E_x{\left(x,y\right)}^2+E_y{\left(x,y\right)}^2},$ which is axis-symmetrical for the ideal Bessel beam [Fig. 9(a)] but obviously asymmetric for the experimental beam [Fig. 9(b)]. This non-uniformity in the experimental beam is due to an asymmetry of the conical beam diffracted by the micro-axicon of our experimental set-up (cf Section 6.1). The maximum discrepancy between two axis symmetrical spectral components is observed along the π-direction, which is in agreement with the minimum visibility observed in the reconstructed evanescent interference pattern.

 

Fig. 8. Intensity distributions, reconstructed from empirical data measured across the plane z=0 by our phase- and polarization-sensitive SNOM probe, of (a,b) the electric field and (c,d) the magnetic field of a radially-polarized evanescent Bessel beam, in (a,c) the longitudinal plane (y=0) and (b,d) the transverse plane (z=0). The fields are expressed in gaussian units. The π-line shown in (b) and (d) refers to the direction along which the visibility of the Bessel beam fringes vanish.

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Fig. 9. Plane wave spectrum of the transverse field amplitude $\sqrt{E_x{\left(x,y\right)}^2+E_y{\left(x,y\right)}^2}$ in (a) and (b) the ideal and experimental cases, respectively. (c): (a) modulated by a cosine function of the azimuthal angle: simulation of an unbalanced Bessel beam that approximate the experimental one.

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Fig. 10. (Media 2 and Media 3) (a,b,c): 3D electromagnetic distribution of the experimental radially-polarized evanescent Bessel beam. (d,e,f): 3D field distribution of the unbalanced theoretical Bessel beam. (a,d): longitudinal plane (y=0): out-of-plane transverse magnetic field component H_y (background colored image) and in-plane electric-field lines (black arrows). (b,e): transverse electric-field lines. (c,f): transverse magnetic-field lines.

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Figures 10(a)–10(c) display the complete electromagnetic vectorial structure of the experimental evanescent field distribution in (a) the longitudinal plane (y=0) and (b,c) the transverse plane (z=0). Figures 10(d)–10(f) compare the electromagnetic fields for an unbalanced theoretical Bessel beam whose plane wave spectrum more closely resembles the experimental spectrum [see Fig. 9(c)]. Figure 9(c) is achieved by modulating the ideal plane wave spectrum [Fig. 9(a)] with a cosine function of azimuthal angle whose maximum and minimum (10% of the maximum) are rotated to align with the observed π-direction. The background color image of Fig. 10(a), 10(d) represents the out-of-plane y-component of the magnetic field distribution and black arrows show the electric field lines. Figures 10(b), 10(e) and 10(c), 10(f) reports the electric and magnetic field lines in the transverse plane (z=0), respectively.

The good agreement between experimental and theoretical results shown both in Fig. 10 and in corresponding media files (Media 2 and Media 3) validates our imaging concept in the evanescent regime. The media files give the impression that the Bessel field distribution is translated along π-direction during field oscillation. Moreover, the magnetic loop and the related electric field distribution are spatially shifted along π-direction. These properties are due to the fact that we did not generate a pure stationary evanescent field. The asymmetric plane wave spectrum shown in Fig. 9(b) results, in the direct space, in interference between the pure ODCD line (achieved with an ideal stationary evanescent Bessel beam) and a p-polarized evanescent wave which propagates along the π-direction. The presence of the non stationary evanescent wave is responsible for the impression of a translation of the overall field. It also leads to a non-zero energy flow at the heart of the overall evanescent field distribution, in the near-field zone (Fig. 11). The energy flow shows a waist which may be seen as a 2D evanescent wave focusing property, analogous to the in-plane focusing of surface plasmons [11].

 

Fig. 11. Time-averaged Poynting vector of the evanescent field distribution in (a) the transverse plane (z=0) and (b) the longitudinal plane (y=0).

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These results imply that the discrepancies between the measured and ideal beams shown in Figs. 10 and 11 are reflective of the difficulty of synthesizing a perfectly ideal radially-polarized evanescent Bessel beam. The usefulness of the phase- and polarization-sensitive SNOM detector and the associated reconstruction procedures introduced here, for either further improvements in the generation of this particular Bessel beam or in any other nano-optical system, is highly evident.

7. Conclusion

We have introduced a new concept of near-field vectorial imaging that allows the full experimental characterization of the electric and magnetic field lines as well as the distribution of Poynting vector in the near-field and far-field zones surrounding a sample. This complete optical 3D information is achieved from a single 2D image acquisition of the amplitude and phase of two orthogonal transverse electric field components right at the sample surface. The overall optical field is rebuilt numerically using rigorous numerical methods such as FDTD or plane wave spectrum methods. This imaging technique has been validated with azimuthally-polarized propagative Bessel beams and radially-polarized evanescent Bessel beams. For the first of these two types of beams, electromagnetic structure similar to the radiation of magnetic moments has been fully characterized. We anticipate that this imaging principle will become an important diagnostic of nano-optical devices and especially of metamaterials, for which magnetic moments are the building blocks. Future advances in the technique will consist of improving the resolution ability of the probe without losing collection sensitivity to, and accurate discrimination of, the electric field. Recent advances in nanoantenna concepts may help fulfill this objective.