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Abstract
A three-dimensional (3D) hollow spot is of great interest for a wide variety of applications such as microscopy, lithography, data storage, optical manipulation, and optical manufacturing. Based on conventional high-numerical-aperture objective lenses, various methods have been proposed for the generation of 3D hollow spots for different polarizations. However, conventional optics are bulky, costly, and difficult to integrate. More importantly, they are diffraction-limited in nature. Owing to their unique properties of small size, light weight, and ease of integration, planar lenses have become attractive as components in the development of novel optical devices. Utilizing the concept of super-oscillation, planar lenses have already shown great potential in the generation of sub-diffraction, or even of super-oscillatory features, in propagating optical waves. In this paper, we propose a binary-phase planar lens with an ultra-long focal length (300λ) for the generation of a 3D hollow spot with a cylindrical vector wave. In addition, we experimentally demonstrate the formation of such a hollow spot with a sub-diffraction transverse size of 0.546λ (smaller than the diffraction limit of 0.5λ/NA, where NA denotes the lens numerical aperture) and a longitudinal size of 1.585λ. The ratio of central minimum intensity to the central ring peak intensity is less than 3.7%. Such a planar lens provides a promising way to achieve tight 3D optical confinement for different uses that might find applications in super-resolution microscopy, nano-lithography, high-density data storage, nano-particle optical manipulation, and nano-optical manufacturing.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, optical hollow spots have been found to have great potential in optical applications, such as stimulated emission depletion (STED) microscopy [1–3], super-resolution lithography [4], atom traps [5–7], optical tweezers [8], and optical manufacturing [9,10]. The size of the hollow spot is one of the key parameters determining the super-resolution in a STED microscope [11,12] and in lithography. A hollow spot of reduced size is also an elegant candidate for trapping a nanoparticle with a refractive index lower than its surroundings [13]. The common way to generate two-dimensional (2D) hollow spots is to focus azimuthally polarized waves [14] or circularly polarized waves [15,16] with high-numerical-aperture (NA) objective lenses. Metasurface lenses [17,18] and spiral phase microsphere [19] were also used to generate 2D hollow spots with angular momentum. Employing the concept of super-oscillation, sub-diffraction focusing of azimuthally polarized waves leads to the formation of 2D hollow spots with an inner full width at half-maximum (FWHM) less than the diffraction limit [20], or even super-oscillatory criteria [21,22]. However, a 2D hollow spot only provides optical confinement in the transverse plane, leaving open accesses along the optical axis. A complete optical confinement can be achieved with a three-dimensional (3D) hollow spot, which could enhance the axial resolution in STED microscopy [23] and super-resolution lithography, and improve the trapping stability for optical tweezers. A 4π focusing system has been proposed to generate 3D hollow spots, but it requires precisely overlapping two optical spots formed by focusing two counter-propagating radially polarized first-order Laguerre-Gaussian waves with two high-NA objective lenses. The theoretically predicted FWHMs of such hollow spots are approximately 0.35λ and 0.32λ in the transverse and axial directions, respectively [24]. 3D hollow spots can also be created by focusing a circularly polarized plane wave with the help of a circular π-phase-plate (πPP) [25]. Utilizing the destructive interference between the inner and outer rings of a double-ring-shaped radially polarized R-TEM11*-mode wave, the theoretical prediction shows that a hollow spot can be formed with FWHMs of 0.94λ and 0.68λ in the transverse and axial directions, respectively, by focusing with a high-NA objective lens (NA = 1.2) [26]. It was also suggested that a nearly spherical 3D super-resolution hollow spot can be generated by incoherent superposition of two radially polarized waves modulated by a circular πPP and a quadrant 0/π phase plate [27]. Cylindrical vector beams can also be used to produce multiple 3D hollow focal spots on the optical axis, and numerical results shows that FWHMs along the transverse and axial directions are 1.02λ and 0.70λ, respectively [23]. Although significant efforts have been carried out in investigating the generation of 3D hollow spots, to our knowledge no experimental demonstration of sub-diffraction 3D hollow spots has yet been reported. Recently, super-oscillation [28,29] has become an interesting topic in developing novel sub-diffraction focusing [30,31] and imaging [32–36] devices. Based on the concept of super-oscillation, 2D hollow spots [20], ultra-long hollow needles [21,22] have been experimentally demonstrated by shaping azimuthally polarized beams with planar lenses. Compared with conventional bulky lenses, planar lenses have unique properties of ultra-thin thickness, light weight, and ease of integration. More importantly, in contrast to conventional optics, a super-oscillation planar lens is not restricted to the diffraction limit. In this work, for cylindrical vector waves at a wavelength of λ = 632.8 nm, a planar lens based on binary-phase (0,π) modulation is proposed for generating the 3D hollow spot with a sub-diffraction transverse size. The lens has a NA value of 0.908, and its focal length is approximately 300λ. By optimizing the ratio of the radial and azimuthal polarizations in the cylindrical vector wave, a 3D hollow spot is experimentally demonstrated with a transverse FWHM of 346 nm (0.546λ) and a longitudinal FWHM of approximately 1.0 μm (1.585λ). The transverse size is smaller than the diffraction limit of 0.55λ (0.5λ/NA). The central intensity ratios (the ratio of the central minimum intensity and central ring peak intensity) are approximately 3.5% and 3.7% in the transverse and axial directions, respectively.
2. Theoretical consideration
2.1 Theoretical calculation of diffraction pattern
The cylindrical vector wave can be treated as a linear superposition of an azimuthally polarized wave and a radially polarized wave. In the following design, the electrical complex amplitude is described by a Laguerre-Gaussian profile for both polarizations, as given in Eq. (1), where E0 is the incident electrical field amplitude, w0 the beam waist, z0 = πw02/λ the Rayleigh range, R(z) = z[1 + (z0/z)2] the radius of curvature, w(z) = w0[1 + (z/z0)2]1/2 the beam width at z, and k = 2π/λ is the wavenumber:
For an azimuthally polarized wave and a radially polarized wave, the corresponding diffraction pattern is given by vectorial angular spectrum method as Eqs. (2) and (3), respectively:
where Eφ, Er, and Ez denote the azimuthal, radial, and longitudinal polarization components in the diffracted wave, respectively;; r and ρ are radial coordinates in the spatial and frequency domains, respectively; g(r) is the incident electrical field distribution described by Eq. (1) with w0 = 331μm and z = 276 mm; t(r) is the lens transmittance function; J1 is the first-order Bessel function, and q(ρ) = (1/λ2−ρ2)1/2.
2.2 Binary-phase lens design
Figure 1(a) depicts the structure of a binary-phase (0,π) planar lens, which consists of concentric Si3N4 ring belts on a glass substrate, where Rlens is the lens radius, Ri the central radius of the ith ring belt, w the minimum width of a ring belt, and t the thickness of the Si3N4 ring belt, which responds to a relative phase change of π compared to the phase change caused by a ring groove with a depth of t. Figure 1(b) illustrates the formation of a 3D hollow spot by focusing a cylindrical vector wave with the binary-phase planar lens, where the arrows indicate the polarizations of the cylindrical vector wave. Based on binary-phase modulation (0,π), the transmission function of the planar lens is optimized using the vectoral angular spectrum method and a particle swarm algorithm [37] for a wavelength of 632.8 nm. The radius of the lens Rlens is 650λ, the focal length f = 300λ, and the minimum ring belt width w = 400 nm. The optimized phase distribution of the planar lens is summarized in Table 1, which gives the phase value in the ith ring-belt area with a central radius of Ri and width of w. The binary-phase (0,π) distribution is coded in hexadecimal digits, the binary digital format of which gives the phase 0 for digital 0 and the phase π for digital 1 from the central to the outermost ring-belt areas. As some neighboring ring-belts might have the same phase value, and are combined into a larger ring-belt, therefore there are only 644 ring-belts in the final optimized binary-phase transmittance function.
Fig. 1 Binary-phase-modulation-based planar lens for 3D hollow-spot generation. (a) Structure of the binary-phase planar lens. (b) Generation of 3D hollow spot by focusing a cylindrical vector wave with the planar lens.
Table 1. Optimized phase distribution of the planar lens.
2.3 Numerical simulations
To investigate the generation of a 3D hollow spot, numerical simulations are conducted using COMSOL Multiphysics. In the simulation, the refractive index and thickness of the Si3N4 ring belt are set to be 1.9 and 352 nm, respectively, to achieve a relative phase change of π. According to the simulation data, for ratio of 0 between the azimuthal and radial polarizations, the optical intensity distributions of the transverse, longitudinal, and total electrical fields in the propagation plane are displayed in Figs. 2(a)–2(c), respectively.
Fig. 2 Numerical simulation results obtained with COMSOL Multiphysics. 2D intensity distributions of the (a) transverse, (b) longitudinal, and (c) total electrical fields in the propagation plane in the range between z = 290λ and 310λ for ratio of 0 between the azimuthal and radial polarizations, respectively. The intensity distribution (d) along the optical axis and (e) on the focal plane at z = 300.28λ for different ratios of azimuthal and radial polarizations, i.e., 0 (black-solid line), 0.6 (red-dotted line), and 0.8 (blue-dashed line), where AP and RP represent azimuthal and radial polarizations.
As shown in Fig. 2(a), the transverse component forms a hollow-needle-like structure with a length of approximately 5λ in the vicinity of the designed focal point at z = 300λ along the optical axis, while the longitudinal component creates two separated bright spots around the focal point on the optical axis, as shown in Fig. 2(b). The contributions from both polarization components result in the formation of a hollow spot at the designed focal point, as depicted in Fig. 2(c).
To further optimize the shape of the hollow spot, especially to achieve a small central intensity ratio, the incident cylindrical vector waves are applied to the numerical simulations with different ratios of the azimuthal and radial polarizations. Figures 2(d) and 2(e) plot the optical intensity distribution along the optical axis, and the optical intensity distribution on the focal plane as indicated by the dashed line in Fig. 2(c) for a polarization ratio of 0 (black-solid line), 0.6 (red-dotted line), and 0.8 (blue-dashed line), respectively. Since the two longitudinally polarized bright spots are mainly formed by the focused radial component, the on-axis optical intensity almost remains unchanged for all the three cases, as illustrated in Fig. 2(d). The minimum optical intensity is located at z = 300.28λ (190.0 μm), which is quite close to the designed focal point at z = 300λ. The peak-to-peak distance is 3.3λ, resulting in a longitudinal FWHM of approximately 1.65λ along the optical axis. The central intensity ratio is 2.6%. On the focal plane, as shown in Fig. 2(e), the central-ring peak-to-peak distance is 1.02λ. With the increase of the azimuthal component in the incident wave, the transverse size shows a slight increase. The corresponding transverse FWHMs are 0.490λ, 0.495λ, and 0.497λ for the above three cases, respectively. Meanwhile, the central intensity ratio is efficiently suppressed, as the azimuthal component gives a relatively greater contribution to the hollow-ring intensity. The value of the central intensity ratio decreases from 16.25% to 9.81%, as the ratio of the azimuthal and radial polarizations increases from 0 to 0.8. Therefore, a properly optimized ratio of azimuthal and radial polarizations helps to generate a 3D hollow spot with better optical confinement. Most of all, the transverse size of the 3D hollow spot is smaller than the diffraction limit of 0.551λ (0.5λ/NA) for all of the above three cases, and this sub-diffraction feature is created with a propagation wave, which is important for far-field optical applications.
3. Experimental results
3.1 Lens fabrication
Based on the optimized binary-phase distribution, a planar lens is fabricated by electron beam lithography (Vistec EBPG 5000plus ES). Using an inductively coupled plasma etch (Sentech PTSA SI 500), the Si3N4 concentric ring belts are formed by etching through a Si3N4 layer deposited on a glass substrate with inductively coupled plasma enhanced chemical vapor deposition (ICPECVD, Sentech SI 500D). During the etching process, a 70-nm-thick aluminum film is used as a hard mask to transfer the concentric ring-belt pattern into the Si3N4 layer. Figure 3 is the scanning electron microscopy (SEM) image of the binary-phase planar lens with a diameter of 822 μm, which consists of 644 Si3N4 ring belts. The thickness of the Si3N4 ring belts is 352 nm, corresponding to a relative phase change of π for a measured Si3N4 refractive index of 1.9.
3.2 Experimental setup
Although conventional optical microscopes with high NAs have been successfully used to obtain the sub-diffraction features in super-oscillatory propagation waves [31], it is only restricted to the measurement of the transverse polarization field [38]. Because the designed 3D hollow spot is formed by both transverse and longitudinal components, a custom testing system was built based on a tilted nano-fiber-probe mounted on a 3D nano-positioner. As shown in Fig. 4, the light source is a linearly polarized He-Ne laser (Thorlabs, HNL210L) emitting at a wavelength of 632.8 nm. The linearly polarized wave is then converted into a cylindrical vector wave using an s-wave plate (Workshop of Photonics, RPC-632.8-06-188). The ratio of azimuthal and radial polarization is controlled by rotating the s-wave plate around its optical axis. The cylindrical vector wave normally impinges on the binary-phase planar lens. After the lens, the optical intensity distribution of the generated 3D hollow spot is recorded by a tilted (30 degree with respect to the optical axis) nano-fiber-probe with a diameter of 100 nm (Nanonics Imaging Ltd., CFN-100), which is mounted on a 3D piezo-nano-positioner (Physik Instrumente, P-561.3CD). Finally, the optical signal collected by the nano-fiber-probe is sent to a single-photon-counter (Thorlabs, SPCM50A/M). By controlling the nano-positioner, 3D optical intensity in the far field can be recorded with a resolution better than 100 nm.
Fig. 4 Experimental setup for 3D hollow-spot characterization. The custom system consists of a linearly polarized He-Ne laser, optical isolator, linear polarizer, s-wave plate (SWP), mirror, binary-phase planar lens (BPPL), nano-fiber-probe, nano-positioner, and single-photon counter (SPC).
3.3 Experimental results
To achieve the smallest central intensity ratio, the polarization ratio is chosen as 0.8 between azimuthal and radial polarizations in the experiment. The experimentally obtained optical intensity distribution within the propagation distance between z = 295λ and 306λ after the lens is depicted in Fig. 5(a), while the theoretical simulation result is given in Fig. 5(b) for comparison. Good similarity is found between the two results: A clear 3D hollow spot is located around the designed focal point at z = 300λ; the hollow spot is surrounded by two bright spots on the optical axis and a weaker hollow ring between the two spots. To make a quantitative comparison, the normalized intensity distribution along the optical axis is plotted for the experimental (blue) and the theoretical (red) results in Fig. 5(c), which shows excellent agreement between the two. It is also noted that, in the experimental results, the peak-to-peak distance of the two spots is 2.845λ, the longitudinal FWHM is 1.585λ, and the central intensity ratio along the optical axis is 3.7%. All three parameters are identical to the theoretically predicted values, i.e., 3.3λ, 1.65λ, and 2.6%, respectively.
Fig. 5 Comparison of the optical intensity distribution on the propagation plane between experimental and theoretical results. 2D optical intensity distribution obtained with (a) experiment and (b) numerical simulation, respectively. (c) Normalized experimental (blue) and theoretical (red) intensity distributions along the optical axis, which show a longitudinal FWHM of 1.585λ and a peak-to-peak distance of 2.845λ in the experimentally generated hollow spot.
To achieve a better understanding of the 3D hollow-spot optical confinement around the optical axis, the optical intensity distribution on the focal plane is recoded, as presented in Fig. 6(a), which demonstrates an obvious dark spot surrounded by a bright circular ring at the center of the 3D hollow spot. The surrounding sidelobe rings have weaker intensities, and show reasonable circular symmetrical profiles. In the inset, the optical intensity distribution obtained with a high-NA microscope (NA = 95, 100 × ) within the same area is plotted for comparison. The corresponding intensity profile obtained with theoretical simulation is also presented in Fig. 6(b). Although there is some extent of non-symmetry and poor contrast in the experimental data obtained by nano-fiber-probe, due to the low signal-to-noise level caused by the low coupling efficiency of the nano-fiber-probe, it is seen that the experimental result are still consistent with the theory in the way both show a clear hollow spot with a bright central-ring lobe and weaker surrounding sidelobes. Owing to its non-symmetry, the intensity profile is obtained by averaging the intensity distribution in 10 different directions (0°, 18°, 36°, 54°, 72°, 90°, 108°, 126°, 144°, and 162°) crossing the hollow spot center with equal steps of 18°. This averaged intensity profile (blue-solid line) is plotted in Fig. 6(c) along with the theoretical result (red-dashed line). The two profiles show good consistency. The corresponding transverse FWHM is calculated by averaging the FWHMs taken in the above-mentioned 10 directions, and gives a value of 0.546λ (346 nm). The transverse size is larger than the theoretically predicted value of 0.497, while it is still slightly smaller than the diffraction limit 0.551λ (0.5λ/NA). Similarly, the experimentally obtained average central intensity ratio is only 3.5%, which is less than one-half of the theoretical one. The discrepancy between the experimental result and theoretical prediction is mainly caused by the optical alignment, as was pointed out in our previous work [20]. Actually, for a super-oscillatory hollow spot, the increase in spot size is usually accompanied by a reduction in the central intensity ratio. The curve (green-solid line) acquired by microscope is also plotted in Fig. 6(c) for comparison, and shows lower sidelobes, which are difference from the theoretical prediction and the data obtained with nano-fiber-probe. This discrepancy is caused by the missing of longitudinal polarized weak sidelobes, as seen in Fig. 2(b), because high-NA microscope only captures the transverse polarization components. This also demonstrates the advantage of using nano-fiber-probe over high-NA microscope for complex polarization measurement.
Fig. 6 Comparison of the optical intensity distribution on the focal plane between experimental and theoretical results. 2D optical intensity distribution obtained with (a) experiment and (b) numerical simulation, respectively, where the inset of (a) gives the intensity obtained by high-NA microscope within the same area. (c) Nano-fiber-probe obtained (blue-solid line), microscope obtained (green-solid line) and theoretical (red-dashed line) normalized intensity distributions with respect to the radial coordinate on the focal plane, which shows a peak-to-peak distance of 1.01λ and a transverse FWHM of 0.546λ in the experimentally generated 3D hollow spot. The transverse FWHM is smaller than the diffraction limit of 0.551λ (0.5λ/NA).
Here, it should be noted that, the optical coupling efficiency of the nano-fiber-probe has a strong dependency on field polarization, and the nano-fiber-probe only responses to the electrical field parallel to the tip front facet. Since the 3D hollow spot is formed by the two longitudinal spots and the transverse polarized hollow ring. To acquire the longitudinal field profile without significant distortion in the transverse field profile, the nano-fiber-probe was kept at a tilted angle of 30 degree with respect to the optical axis. Theoretically, the ratio between the minimum and maximum transverse coupling efficiencies is approximately 0.75 (cos2(30°)/1), while the ratio of maximum coupling efficiencies between longditudinal and transverse polarizations is approximately 0.25–0.33 (sin2(30°)/1–sin2(30°)/cos2(30°)). Therefore the transversely polarized field profile can be obtained with ignorable deformation, while the intensity deformation might happen when both polarizations are measured, and there is no way to distinguish each of them. Fortunately, the major parameters of the 3D hollow spot can be correctly obtained without ignorable distortion, as well as the intensity profile along the optical axis and the transverse intensity profile at the center of the hollow spot, as seen in Fig. 5(c) and Fig. 6(c), because each profile mainly consists of either transverse polarization or longitudinal polarization.
4. Conclusions
In present work, the binary-phase lens has to be optimized for both azimuthal and radial polarizations to the generate two longitudinally polarized focal spots and a azimuthally polarized hollow ring, which result in the formation of the desired 3D optical confinement. It should be noted that given independent phase control of the azimuthal and radial polarizations, a smaller hollow spot could be generated with tight optical confinement, e.g., by using a birefringent metasurface [39]. In such case, the transverse size of the 3D spot can be smaller than the super-oscillation criteria (0.38λ/NA). The size can also be further reduced by increasing the lens NA or by using medium-immersed approaches. Introducing more phase values between 0 and 2π in the lens design will also be helpful in improving the quality of the 3D hollow spot.
To summarize, we have proposed a binary-phase planar lens with a 300λ ultra-long focal length for generation of 3D hollow spots and experimentally demonstrated the formation of a 3D hollow spot with sub-diffraction transverse size by focusing of a cylindrical vector wave. The transverse hollow spot size is 0.546λ, which is smaller than the diffraction limit of 0.551λ (0.5λ/NA), while the axial size is 1.585λ. The central intensity ratio of the hollw spot is only 3.5% and 3.7% in the transverse and axial directions, respectively. Compared with conventional ways to generate hollow spots with bulky optics, planar lenses are light, ultra-thin, and easy to integrate. This provides a simple way to realize strong 3D optical confinement. Moreover, in our approach, the relative intensities of the transverse and longitudinal polarizations is controllable by rotating the s-wave plate to adjust the ratio of the incident radial and azimuthal polarizations. Such a 3D hollow spot might find applications in super-resolution microscopy, super-resolution lithography, high-density data storage, nano-particle optical manipulation, and nano-optical manufacturing.
Funding
National Key Basic Research and Development Program of China (Program 973) (2013CBA01700); China National Natural Science Foundation (61575031, 61177093); Open Fund of State Key Laboratory of Information Photonics and Optical Communications (University of Electronic Science & Technology of China), P. R. China; Fundamental Research Funds for the Central Universities (10611CDJXZ238826, 106112016CDJZR125503, 106112015CDJXY120007); National Key Research and Development Program of China (2016YFED0125200, 2016YFC0101100).
Acknowledgments
Authors also thank LetPub (www.letpub.com) for their linguistic assistance during the preparation of this manuscript.
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