Demonstration of longitudinally polarized optical needles

: Longitudinally polarized optical needles are beams that exhibit ultra-long depth of field, subwavelength transverse confinement, and polarization oriented along the longitudinal direction. Although several techniques have been proposed to generate such needles, their scarce experimental observations have been indirect and incomplete. Here, we demonstrate the creation and full three-dimensional verification of a longitudinally polarized optical needle. This needle is produced by generating a radially polarized Bessel-Gauss beam at the focus of a high numerical aperture microscope objective. Using three-dimensional spatial mapping of second-harmonic generation from a single vertically aligned GaAs nanowire, we directly verify such a longitudinally polarized optical needle’s properties, which are formed at the focus. The needle exhibits a dominant polarization, which is oriented along the longitudinal direction, an ultra-long depth of field (30 λ ), and high spatial homogeneity. These are in agreement with corresponding focal field calculations that use vector diffraction theory. Our findings open new opportunities for manipulation and utilization of longitudinally polarized optical needles.


Introduction
Vector Bessel beams represent the non-diffractive solutions to the full vectorial Helmholtz wave equation [1,2].They constitute the extension of the well-known Bessel beams, obtained from the scalar wave equation [3][4][5], to optical beams with various states of polarizations, like radially-(RP) or azimuthally-polarized (AP) beams.In principle, vector Bessel beams show an infinite depth of field similar to the well-known case of scalar Bessel beams.However, due to limitations in the physical size of utilizable apertures and available input energy, one can only create approximations of such beams in the laboratory, which are known as quasi-Bessel beams or Bessel-Gauss (BG) beams.Several methods exist to generate vector BG beams including spatial light modulators [6,7], axicons [8,9], or tuning the output mode of a laser [10].The vectorial behavior of such BG beams is particularly important under tight focusing conditions, where longitudinal electric field components naturally arise [11][12][13][14][15][16][17][18].
Longitudinally-polarized optical needle is a term used to represent longitudinal electric fields that show spot-like intensity distribution in the transverse plane with sub-wavelength confinement over a distance of several λ along the longitudinal axis (i.e., beam propagation axis), where λ is the excitation wavelength.Such optical fields are expected to be valuable in providing alternative illumination conditions in an optical microscope [19][20][21], increasing optical data storage density [22] or multi-plane micromanipulation of nanoparticles [23,24].Various methods, based on RP beams, have theoretically predicted the generation of optical needles using diffractive optical elements consisting of several belts [25][26][27][28][29] to promote constructive and destructive interferences at the focal region.The tight focusing of RPBG beams has been shown to be a practical way to generate longitudinally-polarized optical needles [30][31][32][33][34].To date, however, experimental verifications of the generation of longitudinally-polarized optical needles have been limited and mostly relied on the use of indirect measurement techniques [30,31,34,35] thereby substantially hindering the full characterization and subsequent tailoring of such needles in three dimensions.
Here, we experimentally demonstrate the creation and verification of a longitudinallypolarized optical needle generated by an RPBG beam at the focus of a high numerical aperture (NA) microscope objective.By forming a thin annular RP beam at the back focal plane of the microscope objective using polarization mode converter and an axicon in tandem, we create an RPBG beam at the focus of the microscope objective.We then probe directly the properties of the longitudinal electric field of such a beam by spatially mapping the secondharmonic generation (SHG) from a single vertically-aligned GaAs nanowire excited with the RPBG beam.By raster-scanning such nanowires, known to be mainly responsive to longitudinal electric fields [36][37][38][39][40], the three-dimensional spatial distribution of the optical needle is obtained unambiguously.The results are then compared with the SHG measurements from the same nanowire using RP and AP beams and an azimuthally-polarized Bessel-Gauss (APBG) beam.The experimental results are in good agreement with focal field calculations using vectorial diffraction theory.

Optical microscopy setup
The optical microscopy setup used to generate the vector beams and to probe the resulting optical needles at the focal region is shown in Fig. 1.The excitation beam originated from a mode-locked laser (excitation wavelength of 1060 nm, pulse length of 140 fs, repetition rate of 80 MHz).First, we modified the excitation path of our custom-built cylindrical-vectorbeam-equipped point-scanning nonlinear optical microscope [36][37][38][39]41,42] to incorporate the generation of the RPBG and related vector beams.This is done by converting the linearlypolarized Gaussian output of the laser into an RP or AP doughnut-shaped beam using a commercial polarization mode converter (MC, Arcoptix, S.A.).We then used this beam as input to an axicon (apex angle of 176°) to generate a vector BG beam [8].In order to generate a similar beam at the focus of the microscope objective, we focused our vector BG using a set of relay lenses (RFL) to generate a thin annular RP or AP beam with the proper size (around 6 mm in diameter) at the back focal plane of the microscope objective (MO) itself [15].The resulting thin annular RP and AP beams, located at the back focal plane of the microscope objective, were imaged with a beam profiler.Profiles were acquired by placing a beam splitter before the microscope objective to direct part of the beam to the beam profiler placed at the correct distance.m the polariza ular RP and AP g the appropria beam onto piezo stage-sca ).We then col same micros d using approp using a photo-m n a GaAs For this experiment, these nanowires are arranged in a periodic lattice with a pitch of 2.5 μm, have a diameter of 90 nm and a length of about 2.6 μm.We have demonstrated in previous works that SHG from such nanowires is driven by the longitudinal electric field and using pointscanning allows us to therefore map the longitudinal field distribution of a focused vector beam in three dimensions [36][37][38][39][40].The separation between each nanowire is large enough to avoid overlap of SHG signals and related SHG verification measurements (i.e., spectral and power dependence) were performed in previous works [36,37].

3D SHG microscopy parameters
In order to evaluate the three dimensional character of the generated optical needle, we performed systematic transverse (xy) and longitudinal (xz or yz) raster scans of the GaAs nanowire sample at the focal region.It is also important to note that, independently of the input polarization, we always compared SHG signals for the same GaAs nanowires.Transverse scans were performed over a 10 × 10 μm 2 area and longitudinal scans over a 7 × 15 μm 2 area for the data sets in Figs. 3 and 4. In Fig. 5, we used the maximum longitudinal scanning range of our system and scanned nanowires over a 7 × 50 μm 2 area.In all scanning measurements, a pixel resolution of 66.7 nm was used and maintained for each Cartesian axis.All scanning, i.e., pixel-by-pixel, maps were recorded using an average input power of 2 mW and a pixel-dwell time of 50 ms.Prior experiments showed that these exposure conditions are well below the damage thresholds of the used sample.All SHG intensity maps were plotted directly from the raw SHG counts recorded by the PMT using MATLAB, i.e., without image processing or related reconstruction.Each of these maps was independently normalized according to the minimum (Min) and maximum (Max) count values recorded for each scan.All maps also show a multiplication coefficient applied to all the pixel values in one image to show the relative strengths of the SHG signal levels with respect to the reference.

Focal field calculations
The focal field distributions were calculated using the formalism described by Richards and Wolf [11] for tight focusing of optical fields.Figure 6 shows the calculated transverse (E T ) and longitudinal (E Z ) electric field distributions in xy (z = 0) and xz (y = 0) planes for the different input vector beams and for settings that closely simulate our experimental conditions (i.e., excitation wavelength of 1060 nm,a focusing angle of 45°, the latter representing a NA below 0.8 due to under filling conditions in our experiments, environment of air).The angle between rays focused by the objective is taken to be 3° for the case of RPBG and APBG beams.

Results
We used 3D SHG microscopy of GaAs nanowires (See Methods) to verify the longitudinal electric fields at the focus of the microscope objective.We demonstrated in previous works that the SHG signal from a single vertically-aligned nanowire originates mainly from nanowire and not from the substrate, and showed that such nanowire is responsive mostly to longitudinal electric fields [36,37].In addition to this, and considering the rather short length of our nanowires (2.6 μm), by performing 3D SHG microscopy scans in a plane containing a nanowire and parallel to the longitudinal direction, one can get a good approximation of the depth of field of the longitudinal field created by a focused RP beam [40].For reference, we first used an RP beam at the input of the microscope objective to excite the nanowires.This was done by removing the axicon and relay-formation lenses (RFL) after the mode converter (MC) in the optical setup displayed in Fig. 1.As seen in Fig. 3(a), a SHG transverse raster scanning map reveals several point-like distributions that are centered with respect to the location of each GaAs nanowire.This is expected considering that the longitudinal electric field of a focused RP doughnut beam shows such a point-like distribution [13] when viewed in the transverse plane.This characteristic transverse distribution is also consistent with our previous works [36,37].The average count value from each nanowire (the substrate) is about 100 000 per 50 ms (2200 per 50 ms) in Fig. 3(a).We then acquired a longitudinal scanning map in a plane containing three arbitrarily chosen nanowires.The dashed yellow line in Fig. 3(a) shows the y coordinate along which we scanned these three nanowires in the xz plane.Figure 3(b) shows the corresponding longitudinal SHG intensity map from these three nanowires.From this scanning map, we clearly distinguish the location of the nanowires and can estimate the distance over which they were excited using the RP beam.The SHG counts recorded by the photo-multiplier tube (PMT) for this longitudinal map are similar to the ones recorded for the transverse scan for the brightest nanowire.Some variations in SHG signals from one nanowire to another can be seen in the longitudinal scan and are due to slight misalignment of the sample plane with respect to the scanning axes as seen in the transverse scan of Fig. 3(a).Such variations can also be observed in other longitudinal scans.From the SHG intensity maps of Figs.3a and 3b), we calculated the full width at half maximum (FWHM) value to estimate the transverse resolution.From both scans, we retrieved a consistent FWHM value of 0.42 λ, the reference being the excitation wavelength λ.We also calculated the FWHM over the longitudinal direction from Fig. 3(b) and obtained a value of 2.2 λ.
We then implemented the necessary optical components in our microscopy setup to generate an RPBG and acquired similar SHG measurements from the nanowires in the same regions of interest to determine whether we were able to create a sub-wavelength longitudinally-polarized optical field with extended depth of field at the focus of the objective.The input beam used for this was an RP ring in order to generate a tightly focused RPBG beam at the focus of the objective.The SHG signals from the GaAs nanowires of Fig. 3(c) show again a point-like distribution with a non-negligible surrounding ring, revealing the transverse spatial distribution of longitudinal electric field of an RPBG at the focus of a high NA objective.This result confirms earlier simulations as well as indirect observations of such longitudinal electric field distributions [34,44,45].The average SHG count value from each nanowire (the substrate) is about 6000 per 50 ms (110 per 50 ms) in Fig. 3(c), which is significantly lower compared to the count level obtained using an RP doughnut beam, this mostly due to the extended depth of field observed using an RPBG beam.The surrounding ring shows a count value of around 350 per 50 ms.dinal polarizati in Fig. 3, but from RP to AP ng the axicon iment, we inte he RPBG beam PBG beams, d refore, only, i. to scanning, w G), did not affe G intensity map ms.In Fig. 4(a mes lower comp from GaAs nanow udinal scanning m ellow lines in the t cans were perform al scan of the s seen in Fig. 3  show that the l of 15 μm.5(a) shows that the optical needle created at focus is not perfect and exhibits some inhomogeneities along the longitudinal direction.Such artifacts can be explained mainly by the high sensitivity of the BG beam to the axicon tip sharpness [49,50].An axicon tip parameter with roundness above 10 μm is enough to generate strong oscillations [49], explained by destructive interferences at focus.However, it is important to remember that in our nonlinear measurements, the SHG scattered signals were collected and not the linearly scattered signals.Therefore, the collected SHG intensities and the actual longitudinal electric field of the fundamental beam at focus are linked via the relation I(2ω) = E 4 (ω).For instance, the difference in SHG intensity between the two maxima of the needle (lower and central portion of maps in Fig. 4(a)) is below 20%.This corresponds to a difference of 5% in terms of electric field at the fundamental frequency.The differences in terms of SHG intensity between the maximum and minimum variations along the needle are contained below 35%, which corresponds to a difference of 10% in terms of electric field at the fundamental frequency.Overall, even though the SHG longitudinal scans show some imperfections, the optical needle possesses a great degree of homogeneity at the fundamental frequency ω.
The GaAs nanowires used in these experiments being sensitive only to longitudinal electric fields, it is hard to quantify experimentally the possible remaining proportion of transverse electric field at focus.However, considering the spatial distribution of the transverse component of an RPBG [1,45], and observable in Fig. 6(b), we can state that the optical needle created at focus is predominantly longitudinally-polarized.In more detail, based on our experimental conditions and according to Table 2 of [15], we can expect the longitudinal electric field component of the total beam at focus to be between 3 to 10 times higher than its transverse electric field component.On the other hand, our simulations suggest a longitudinal electric field only twice greater than the transverse electric field component.If the goal is to enhance even more the proportion of electric field polarized along the longitudinal direction, one can tune parameters such as the divergence angle or the focusing angle.
The physical properties of the nanowire influence the effective imaging performance of our SHG microscope.In our case, both height and diameter of the semiconductor nanowire need to be considered.In fact, we found before that the smallest nanowire probe that can be used to map the longitudinal electric fields via SHG exhibits a length of 100 nm and a diameter of 40 nm [36].The level of SHG signals acquired using this nanowire, however, is relatively low.It is also not practical to use very long nanowires to map the 3D spatial distribution of longitudinal electric fields.This is because in long nanowires, e.g., about 10 μm (assuming that the diameter is 40 nm), the electromagnetic modes aside from the inherent structure of the nonlinear tensor could also strongly influence the SHG efficiency.To date, the role of these electromagnetic modes in the SHG efficiency of the nanowire remains unknown.
For future work, it is valuable to explore the possibility of evaluating the 3D polarization purity of the optical needle.This task is possible but this will demand the development of new techniques and materials used as probes.It is important to remember that the full experimental measurement of all polarization components of an optical needle is complicated and not available to date.This is because both in-and out-of-plane electric fields of the optical needle should be measured directly at high reliability.So far, the probes used in farfield optical scanning microscopy to directly evaluate the focal fields are either sensitive to the total electric field or a particular electric field component only.Note that the implementation of full polarization (including amplitude and phase) measurements, which are based on either optical field reconstruction [51] or near-field nanoprobes, is not easy in practice due to the large spatial extent of the optical needle.
In conclusion, we have demonstrated the experimental creation of a longitudinallypolarized optical needle created by an RPBG beam at the focus of a high NA microscope objective.The needle was probed directly using three-dimensional SHG microscopy of single and vertically-aligned GaAs nanowires.We believe that the ability to generate and verify longitudinally-polarized optical needles will be useful for manipulating several nanoparticles on one line, optical microscopy, and optical data storage.
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