The phenomenon of superoscillation produces oscillations that are faster than the fastest Fourier component of a system, potentially forming a local “hot spot” with a size below the diffraction limit. We show that a radially polarized Laguerre–Gaussian mode has the inherent ability to form superoscillation spots simply by controlling the incident beam size. We investigate this in detail, both numerically and experimentally. Our numerical simulations predict that lateral resolutions close to 100 nm are possible for practical confocal laser scanning microscopy with visible light. We demonstrate experimentally that superoscillation focusing can offer significant spatial resolution improvements for fluorescence imaging.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Concentrating light into a small spot is essential for many applications, including light microscopy, optical tweezers, laser microfabrication, and optical data storage, since, in most cases, the size of the focused spot directly impacts performance, such as spatial resolution, in these applications. However, due to the diffractive nature of light, the lateral spatial resolution of, for example, light microscopy is limited to about 200 nm when using an objective lens with a high numerical aperture (NA) for visible light. In recent years, substantial efforts have been made to greatly enhance the spatial resolution of light microscopy through a variety of techniques: structured illumination , stimulated emission depletion (STED) by a doughnut beam , and a localization method based on the measurement of single-molecule fluorescence . These methods have enabled us to achieve a spatial resolution of between several tens and a hundred nanometers in fluorescence imaging using visible light. Although these techniques are now commercially available, challenges still remain in terms of the usability and complexity of the system, the acquisition time, and photobleaching due to intense illumination when compared with widely used techniques like confocal laser scanning microscopy (LSM).
Another approach to improving spatial resolution in LSM is to produce a focal spot smaller than can be obtained with conventional laser beams by manipulating the characteristics of the beam itself . The longitudinal electric field of a focused radially polarized beam has been used for this purpose since it can generate a smaller focal spot in tight focusing conditions [5,6]. In this context, a radially polarized beam with an infinitely thin annulus is known to produce the smallest possible focal spot, with a full width at half-maximum (FWHM) size of , where is the wavelength of light in vacuum, corresponding to 125 nm, even in the practical conditions commonly used in LSM ( and ). While this is significantly smaller than the spot given by the well-known Airy pattern (), it cannot be achieved in practice due to the infeasibility of producing an infinitely thin annular beam.
Superoscillation, a concept that originates from an investigation by Torraldo di Francia , has the potential to further improve spatial resolution in far-field light microscopy [9,10] without relying on near-field probes, nonlinear processes, or specialized molecules. The idea behind superoscillation is that a band-limited function can locally oscillate faster than its fastest Fourier component when the superposition of its Fourier components is appropriately and precisely designed within a limited frequency range [11–14]. Such a well-designed superposition leads to the formation of a local “hot spot” with a size much smaller than the diffraction limit [15–21]. In general, this produces a very weak superoscillation spot, together with huge nearby side lobes, as the spot size decreases [22–24]. Despite these characteristics, recent progress in the design and implementation of a superoscillation filter or lens has demonstrated its effectiveness and potential for enhancing spatial resolution in light microscopy. The main limitation in most previous demonstrations has been that the superoscillation focusing was realized by planar, diffractive optical elements (DOEs) [9,17,19–21], which may restrict its applicability to conventional confocal LSM setups due to the presence of off-axis aberrations [25,26]. Nonetheless, these results have encouraged us to further investigate the possibility of producing much smaller focal spots by combining the polarization of a radially polarized beam with the concept of superoscillation, particularly in tight focusing conditions.
In this paper, we report, to the best of our knowledge, the first demonstration of superoscillation focusing of a radially polarized beam with a higher-order transverse mode in tight focusing conditions. We show that a higher-order radially polarized Laguerre–Gaussian mode beam inherently possesses superoscillation properties and can produce a much smaller focal spot with visible light, with a size close to 100 nm and moderate side-lobe intensity, beyond the lower limit for a radially polarized beam with an infinitely thin annulus. Since the superoscillation spot is achieved simply by focusing a higher-order radially polarized beam, it is, in principle, free from the undesirable aberration issue, except the aberrations associated with the focusing lens itself. The superoscillation spot is therefore readily applicable to many types of LSM for enhancing spatial resolution. Moreover, owing to the cylindrical symmetry of radial polarization, we obtain a perfectly circular spot at the focus of a high-NA lens. These characteristics offer significant advantages over the previous techniques for realizing superoscillation spots. The superoscillation behavior is first investigated via numerical simulations and then verified experimentally. We apply the superoscillation focusing of a higher-order radially polarized beam to fluorescent imaging including biological specimen to significantly enhance spatial resolution within the framework of a conventional fluorescence confocal LSM setup.
A. Superoscillation Criterion in Tight Focusing Conditions
By definition, superoscillation spots emerge as a result of oscillations that are faster than the fastest Fourier component in a band-limited function [11–14]. When focusing light with a lens, the fastest Fourier component corresponds to an annular component coming from the outer edge of the lens. In scalar theory, an annular mask of infinitely thin width produces a focal spot represented by , where is a Bessel function of the first kind of order zero, is the wavenumber of the light (in vacuum), and is the lateral displacement from the optical axis, resulting in the smallest possible focal spot of .
That said, however, the argument based on scalar theory is no longer precisely valid under high-NA focusing conditions due to the fact that the vector nature of light has a significant impact [28,29]: only the longitudinal electric field of an annular radially polarized beam can form a -Bessel pattern at the focus . Furthermore, even in the limit of an infinitely thin annulus, the residual radial component must be considered at the focus. Based on vector diffraction theory  for cylindrical vector beams, the rigorous intensity profile at the focus in the limit of an infinitely thin annular radially polarized beam can be written as 1) correspond to the longitudinal and radial components at the focus, respectively. Due to the doughnut-shaped profile of the -Bessel function, the radial component causes enlargement of the focal spot size unless . Hence, the superoscillation criterion used in the present study is the appearance of a spot smaller than can be obtained by Eq. (1). For with , the FWHM value for this criterion corresponds to , where is , which is larger than that produced purely by the longitudinal component of a radially polarized beam with an infinitely thin annulus ().
B. Superoscillation by Higher-Order Radially Polarized Beams
To realize superoscillation focusing by radially polarized beams, we consider the focusing properties of the higher-order transverse mode of a radially polarized Laguerre–Gaussian () beam . An beam has concentric rings, with the node spacing determined by its radial mode index . The characteristic amplitude and phase distributions of beams give rise to a variety of focusing behaviors, including the formation of a sharp (bright) spot  or a three-dimensional dark spot , when the incident beam width is appropriately adjusted in the pupil plane of the objective lens.
We now consider the focusing properties of an beam in more detail from the viewpoint of superoscillation focusing. The electric field of an beam in the pupil plane of an objective lens can be written as 
Figures 1(a) and 1(b) plot the calculated center intensity and FWHM values for the central focal spots at the focuses of , , and beams for and as a function of the incident beam size with respect to the pupil size of the objective lens. Here, we define the beam size parameter as the ratio of the pupil diameter to the second moment diameter measured for each beam. From the definition of the second moment diameter for a laser beam , for beams can be derived as . We assume that the input power of the focused beam at the focal plane is the same for all focusing conditions. In addition, the range of beam size parameters is such that all rings of the RP-LG beam are contained within the pupil of the objective lens. The ranges considered in our analysis correspond to , , and for , , and beams, respectively.
As shown in Fig. 1(a), the central intensity of the focal spot, which is composed only of the longitudinal component along the optical axis (-axis), strongly depends on the incident beam size. This can be attributed to phase flips of the multiring-shaped RP-LG beams changing the balance between constructive and destructive interference at the focus. Each mode therefore has a specific beam size parameter value that maximizes its central intensity (longitudinal component) at the focus, namely, , 1.056, and 1.061 for the , , and beams, respectively. Since the central spot sizes of RP-LG beams in such situations are already smaller than those of conventional linearly or circularly polarized Gaussian beams , applying an RP-LG beam to confocal LSM  and two-photon excitation LSM  can enhance spatial resolution. In addition, we have reported that the spot size of an RP-LG beam can be decreased for larger mode indexes , approaching the focusing ability of a radially polarized beam with an infinitely thin annulus . Hence, the achievable spot size is still larger than that obtained by the superoscillation focusing criterion when the beam size parameter is chosen for maximizing the focal spot intensity.
However, as shown in Figs. 1(a) and 1(b), decreasing the value from the value that maximizes the central spot intensity causes the central spot’s FWHM value to decrease rapidly and the central spot intensity also to decrease to zero, resulting in the formation of a dark spot . In Fig. 1(b), the superoscillation focusing criterion () for and is indicated by the green dashed line. The spot sizes below this point, shown as the light-green region in Fig. 1(b), imply that superoscillation focusing can be achieved in a limited range of beam size parameter values. As an example, Fig. 2 plots the intensity profiles at the focus of an beam for the conditions marked as A to D in Fig. 1(b). Decreasing the beam size parameter from the maximum condition (A) [Fig. 2(a)] causes the central peak intensity to decrease but the outer side lobes to be enhanced [Figs. 2(b)–2(d)]. The FWHM values of the central spots, measured across the total intensity profiles in Figs. 2(a)–2(d), are , , , and , respectively. These results suggest that RP-LG beams inherently possess superoscillation characteristics that can be readily manifested by adjusting the incident beam size with respect to the pupil size of a high-NA objective lens.
In addition, Fig. 1(b) shows that the difference in the FWHM value between the total intensity (solid line) and its longitudinal component (dashed line) becomes smaller as the beam size parameter changes from the maximum condition (A) to the superoscillation focusing condition. In particular, for condition D in Fig. 1(b), which corresponds to Fig. 2(d), the FWHM values for the total and longitudinal components are identical, indicating that the radial component within the first dark ring of the focal spot is almost zero and, accordingly, that the central spot predominantly consists of the longitudinal component. This property greatly contributes to shrinking the central spot in superoscillation focusing. Note that the discontinuities in the solid lines (e.g., at for ) in Fig. 1(b) mean that the focal spot for the total intensity is no longer formed as a single peak due to the reappearance of an increased radial component instead of the weakened longitudinal component.
A significant problem, commonly observed in superoscillation focusing, is prominent side lobes appearing near the central peak [22,23]. Superoscillation with an RP-LG beam also exhibits intense side lobes in the outermost part of the focal pattern, as can be seen in Figs. 2(c) and 2(d). However, because the peak intensity of the central spot is sufficiently strong compared with those of the outer side lobes in superoscillation focusing, the effect of these side lobes, especially away from the central spot, can be simply suppressed using a pinhole in confocal LSM, as shown later. However, the first side lobe (the nearest ring to the central spot) may significantly affect image quality, even with a small confocal pinhole. We evaluated the side lobes generated by RP-LG beams by considering the side-lobe ratio, defined as the ratio () of the first (innermost) side lobe to the central spot intensity ; see the inset in Fig. 1(c). Figure 1(c) plots the side-lobe ratio as a function of the beam size parameter. The side-lobe ratios for , , and beams for the maximum condition are 0.24, 0.18, and 0.17, respectively. When the focal spot size is larger than the superoscillation limit, the variation in the side-lobe ratio is relatively small with respect to the change in input beam size. By contrast, as the focal spot size decreases below the superoscillation limit, the side-lobe ratio increases significantly, as expected. However, it is worth noting that the side-lobe ratio of the superoscillation spot at D in Fig. 1(b) is still smaller than 1 (). This point is a distinctive feature that differs from previous superoscillation implementations, where the first side-lobe intensity substantially exceeds the central intensity of the superoscillation spot (see, e.g., [9,16,17]). Superoscillation focusing of RP-LG beams thus exhibits a superior ability to practically suppress undesirable side lobes.
These calculations also reveal that, for the same superoscillation spot size, higher-order RP-LG beams have lower side-lobe ratios, although their peak intensities are weaker. Therefore, both the mode order and the beam size parameter should be carefully chosen to achieve higher spatial resolution with a sufficiently strong signal in LSM, accounting for the actual experimental conditions, for example, the quantum yield or the photobleaching resistance of fluorescent specimens. However, it is worth mentioning that such an optimization can be done in the straightforward and controllable fashion by only changing the incident beam size of an RP-LG beam with an appropriate mode order. This is a distinct difference from the previous superoscillation approaches based on DOEs or phase masks since the superoscillation characteristic of those approaches are designed mostly using exhaustive computational searching.
C. Simulation of Confocal LSM Using a Superoscillation Spot
To confirm the practical imaging ability of superoscillation spots for confocal LSM, we calculated the point spread function (PSF) for confocal LSM and the images expected for line objects, as shown in Fig. 3. In the numerical simulations, we assumed an excitation wavelength of 488 nm and a fluorescent signal with a peak wavelength 1.05 times longer than that of the excitation. The confocal PSF intensity distributions were calculated based on vector diffraction theory [39,40]. We assumed an objective lens of () and a confocal pinhole of 1.0 or 0.5 Airy units (AU) with a magnification of 100. The images expected for fluorescent, thin line objects with spacings from 110 to 160 nm [Fig. 3(a)] were simulated by two-dimensional convolution of the PSF and object.
For comparison, we begin by considering conventional confocal LSM using a circularly polarized Gaussian beam as an excitation beam because circularly polarized beams can produce circular focal spots without polarization-dependent elongation in high-NA focusing . In this case, the lateral FWHM values of the confocal PSFs were 190 and 160 nm for confocal pinholes of 1.0 and 0.5 AU, respectively. Accordingly, the line objects could not be distinguished by the conventional PSF for 1.0 AU [Fig. 3(b)]. Although a smaller 0.5 AU pinhole [Fig. 3(c)] led to better spatial resolution, the line objects with spacings of 140 nm and below were still blurred.
The lateral size of the PSF decreased further when the superoscillation spot of an beam was used, as shown in Figs. 3(d) and 3(e), where we have assumed beam size parameters of and and a 0.5 AU confocal pinhole. The confocal pinhole effectively suppressed the outer side lobes of the superoscillation spot [see Figs. 2(c) and 2(d)], although the first side lobe (the closest to the center spot) remained, as shown in Fig. 3(e). Nonetheless, the FWHM values of these PSFs were 109 nm [Fig. 3(d)] and 96 nm [Fig. 3(e)], meaning that we expect a significant enhancement in lateral spatial resolution in practice for confocal LSM. The line objects with spacings of 140 and 120 nm could be clearly visualized using a superoscillation spot. Notably, even the structure of the 110 nm line object could be distinguished using the PSF, as shown in Fig. 3(e), although weak fringes, caused by the residual side lobes, are present in the image. These numerical simulations suggest that superoscillation with an RP-LG beam has the potential to substantially enhance spatial resolution in confocal LSM.
3. EXPERIMENTAL DEMONSTRATION
To realize superoscillation focusing, we built a confocal laser microscope, illustrated in Fig. 4(a), by modifying our previous setup . We chose an beam to produce the superoscillation spot, as this would give both an acceptable side-lobe intensity and a sufficiently strong signal in our setup with a small confocal pinhole. An excitation laser beam with a wavelength of 532 nm was converted from an input linearly polarized (LP) Gaussian beam (polarized along the -axis) to an beam by utilizing a mode converter consisting of two types of transmissive liquid crystal devices [35,36]. One device was a concentric binary phase shifter that produced multiple phase flips at specific radii (, 2, 3), corresponding to the three nodes of an beam. The other operated as a 12-segment half-wave plate to change the incident polarization from linear to radial. The mode converter was projected onto the pupil plane of an oil-immersion objective lens of (UPLSAPO 100XO, Olympus) using relay optics with a magnification that was appropriate for controlling the incident beam size of the beam. We adopted a focusing condition that produced a superoscillation spot with suitable spatial resolution, as shown in Figs. 4(b) and 4(c). In our setup, the node radii in the pupil plane were, with respect to the pupil radius , , and . The input Gaussian beam was expanded such that the ratio of the pupil diameter to the Gaussian beam width was 0.65. Under this focusing condition, the intensity distribution at the focus consisted of a small central spot with large side lobes [Fig. 4(b)], and the expected FWHM value of the central spot was 138 nm, which meets the superoscillation criterion (). In addition, the intensity distribution in the -plane [Fig. 4(c)] showed a clear constriction with two peaks along the axial direction (-axis), which is characteristic of superoscillation focusing with an RP-LG beam, as discussed later.
To verify these features experimentally, we measured the intensity distribution of the superoscillation spot by imaging an isolated fluorescent orange bead with a diameter of 100 nm without a confocal pinhole. The sample was scanned using a piezo stage (Nano-PDQ350HS, Mad City Labs). The fluorescent signal was measured using a photodetector (R10467U-40, Hamamatsu) equipped with a photon-counting unit (C9744, Hamamatsu). The measured intensity distributions [Figs. 4(d) and 4(e)] were in good agreement with those predicted by the numerical simulations, and this agreement was further verified by comparing the measured and calculated profiles along the - and -axes, as shown in Figs. 4(f) and 4(g). These results indicate that the beam in our setup was able to generate a superoscillation spot experimentally, as expected.
We then put a confocal pinhole, equivalent to 0.5 AU, in the image plane to evaluate the confocal PSF for the superoscillation spot. Figures 5(a) and 5(b) show the intensity distributions of the measured confocal PSFs obtained as images of the 100 nm fluorescent bead using LP and beams, respectively. Figures 5(c) and 5(d) plot the corresponding intensity profiles along the and -axes for each PSF. These results demonstrate that the side lobes of the superoscillation spot [Fig. 4(d)] were reduced sufficiently by the confocal pinhole to produce a single-peak pattern in the focal plane [Fig. 5(b)]. The FWHM values of the measured PSF for the superoscillation spot along the - and -axes were estimated to be 130 nm and 125 nm, respectively, compared with 166 nm and 189 nm for the conventional LP beam, indicating that the LP beam had an elongated focal pattern under the tight focusing condition.
Figures 5(e) and 5(f) show images of clusters of fluorescent beads with diameters of 170 nm, mounted on a coverslip, taken by the LP and beams. Compared with the image obtained by the LP beam [Fig. 5(e)], the individual fluorescent beads in Fig. 5(f) can be more clearly distinguished, reflecting the fact that the spatial resolution of the image acquired by the superoscillation spot was substantially finer. As shown in Fig. 5(g), this improvement can also be seen by plotting the intensity profiles along the dashed lines in Figs. 5(e) and 5(f).
Finally, to demonstrate the applicability of superoscillation spots for the imaging of biological samples, we made images of microtubules in fixed HeLa cells labeled by anti--tubulin antibodies and AlexaFluor546-conjugated secondary antibodies (Thermo Fisher Scientific). The coverslip was mounted on a glass slide with antifade reagents (ProLong Gold, Thermo Fisher Scientific) to reduce photobleaching. Figures 6(a)–6(c) show the images obtained by the conventional LP beam with confocal pinholes of 1.0 AU and 0.5 AU and by the superoscillation spot produced by an beam (0.5 AU), respectively. The images shown in Figs. 6(a) and 6(b), as well as their magnifications in Figs. 6(d) and 6(e), demonstrate that the spatial resolution of conventional images made using an LP beam can be slightly improved by using a smaller confocal pinhole, as discussed for Fig. 3. Meanwhile, when the beam was changed to the superoscillation spot produced by the beam, the fine network structures of the microtubules could be seen more sharply [Figs. 6(c) and 6(f)], and the adjacent microtubules were clearly resolved, as plotted in Fig. 6(g). These improvements can be attributed to the smaller PSF size realized by the superoscillation spot. The experimental results thus demonstrate that the superoscillation spot produced by the RP-LG beam was able to significantly enhance the lateral spatial resolution of confocal LSM images of biological samples.
4. DISCUSSION AND CONCLUSION
The method introduced in this paper can dramatically improve lateral spatial resolution in confocal LSM, although this comes at the cost of degraded axial spatial resolution, as illustrated in Figs. 4(c) and 4(e). This is because the smaller lateral spot size in superoscillation focusing results in axial elongation due to the strong constriction of the focal spot, in contrast to other superoscillation techniques using DOEs [9,17,19–21], inevitably causing peak splitting of the focal spot along the axial direction, as plotted in Fig. 4(g). This implies that the superoscillation spot produced by an RP-LG beam for confocal LSM will be most suitable for observing thin specimens. However, by introducing a small pinhole for detection, this axial elongation can be effectively suppressed: the axial extent of the resultant confocal PSF was reduced to about 1 μm by a confocal pinhole of 0.5 AU, sustaining the elongation rate of the axial PSF below two compared to the conventional LP beam (data not shown). Superoscillation spots can thus be used to visualize a variety of specimens in confocal LSM.
The other limitation of superoscillation focusing using an RP-LG beam is the weak intensity of the central spot compared with that for conventional beam focusing, as has been generally observed with other superoscillation techniques [15–21]. In fact, the peak intensity of the superoscillation spot used in our setup (Fig. 4) was calculated to be 25 times lower at the focus than that of an LP Gaussian beam for the same laser power. However, in our experiment, increasing the input laser power by a factor of was enough to obtain images with acceptable signal-to-noise ratios using superoscillation focusing. This partly depends on the fluorescence quantum yield, the photostability of the fluorophores, and the sensitivity of the photodetector used in our experiment. In other words, the selection of appropriate fluorophores and recent technological advances in low-noise and high-sensitivity photodetectors will greatly facilitate and encourage the use of superoscillation spots in LSM.
Importantly, despite these present constraints, lateral spatial resolution in confocal LSM can be enhanced significantly within the framework of conventional LSM. The only modification introduced here is the use of an RP-LG beam with properly designed phase and polarization distributions as the excitation beam. Extremely small focal spots, for example, close to 100 nm [ in Fig. 2(d) for a 488 nm beam], will potentially be available to enhance the spatial resolution in LSM, provided that a sufficiently strong fluorescent signal can be detected using a small confocal pinhole. Another important point in this context is that the spot size produced by superoscillation with an RP-LG beam can be explicitly and predictably varied by adjusting the beam size parameter, as shown in Fig. 1(b). This means that superoscillation focusing has a controllability and versatility that is not, in general, readily attainable with DOE-based superoscillation spots. In addition, since the present method is highly compatible with conventional LSM designs, further improvements in spatial resolution may be possible by introducing a deconvolution technique and also by the combined use of nonlinear absorption and emission processes, such as multiphoton excitation, second-harmonic generation, sum-frequency generation, and STED.
In summary, the superoscillation characteristics of a tightly focused, higher-order RP-LG beam have been revealed and investigated in detail. Precise control of the incident beam width in the pupil plane for the focusing lens enables an extremely small spot to be produced, exceeding the superoscillation criterion. Numerical simulations have proved that lateral spatial resolutions close to 100 nm are possible in practical confocal LSM using visible light. In addition, this extremely small spot size can be obtained with weak but acceptable peak intensity and moderate side-lobe intensity, demonstrating that the present method has significant practical advantages. We have also demonstrated superoscillation focusing experimentally with an beam, obtaining significant spatial resolution improvements in confocal imaging for fluorescent specimens, including biological samples, using superoscillation spots. In addition, the present method has advantages in versatility due to needlessness of specially designed fluorescent probes, potential capability of fast acquisition required for live cell imaging, and simplicity that can be achieved without any skilled techniques to operate microscopes. These characteristics of the present method with the significantly enhanced spatial resolution are quite desirable for many situations in practical biological imaging.
Japan Society for the Promotion of Science (JSPS) (16H05985, 15H05953); Core Research for Evolutional Science and Technology (CREST); Japan Science and Technology Agency (JST).
We thank Dr. T. Hibi of Nagoya University and Prof. T. Nemoto of Hokkaido University for the biological sample preparation. We also thank Citizen Watch Co., Ltd., for the provision of the liquid crystal device.
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