Abstract

Polarization is a significant factor in a great variety of optical phenomena, playing an important role in determining the focusing properties of lenses, in the resolution of optical systems, and in the performance during laser processing. Knowing the polarization distribution in focused light is critical to understanding and designing relevant optical devices and systems. However, it remains challenging to characterize the vectorial polarization distribution in optical fields. We develop a polarization-conversion-based optical microscope for directly acquiring the distribution of three orthogonal polarizations in focused light and theoretically prove and experimentally demonstrate its validity by characterizing super-resolution focused light with different incident polarizations.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

The vectorial characteristic is a fundamental nature of electromagnetic waves. The vibration of optical electrical vectors, or polarization, has great effects on optical focusing and imaging. In tightly focused scalar optical fields including linearly, circularly, and elliptically polarized fields, there is a strong longitudinally polarized optical field pointing in the direction of the optical axis, which has a larger size and stronger intensity compared to the corresponding transverse components and thus leads to significant extension in spot size and degradation in the spatial resolution of optical systems. In addition to scalar optical fields, there has been an increasing interest in generating and engineering vectorial optical fields. As an important form of vectorial optical fields, radially polarized beams are characterized by an optical field vector vibrating along the radial direction with respect to the beam center. Due to the unique property of the strong longitudinal and non-propagating optical field in the focal region [1], the focusing of a radially polarized beam is of special interest for particle manipulation [24], particle acceleration [5], high-resolution imaging [6], tip-enhanced Raman spectroscopy [7], and laser processing of materials [8,9], where a small focal spot and longitudinally polarized optical field are of critical importance. Theoretical studies have predicted that focusing radially polarized light can create a longitudinally polarized focal spot with a size much smaller than that generated by focusing linearly polarized or circularly polarized light [1015]. Various methods have been proposed to generate longitudinally polarized sub-wavelength focal spots, including a parabolic mirror [10], optical filter [1117], negative-index grating lens [18], and ${4}\pi$ focusing system [19]. Recently, optical super-oscillation has been proposed as a promising way to realize far-field sub-diffraction focusing and imaging without evanescent waves [20,21], showing great potential in label-free far-field super-resolution optical microscopy [2224]. A sub-diffraction solid focal spot of longitudinal polarization has been realized by focusing radially polarized light through a properly designed binary-phase planar lens [25]. Two-dimensional hollow spots were demonstrated by focusing radially polarized light with an engineered microsphere [26]. Three-dimensional hollow focal spots with a sub-diffraction feature have also been reported experimentally [27] and theoretically [28] by focusing vectorial optical fields. A longitudinally polarized non-diffraction beam can be successfully generated by focusing a radially polarized beam with a binary-phase planar lens [29]. As a promising application, a longitudinally polarized focal spot has been applied to confocal microscopy [3032] and super-resolution fluorescent microscopy [33].

Knowing the vectorial polarization distribution in focused light is critical to understanding and designing relevant optical devices and systems. The intensity distribution of transversely polarized optical fields can be directly acquired by a high-numerical-aperture optical microscope equipped with a high-resolution digital camera (DC) [3440]. However, longitudinally polarized optical fields cannot be obtained in the same way, as both theoretical and experimental studies have identified an unavoidable significant attenuation of the longitudinal components on the image plane due to the optical lever effect of the microscope, especially when the magnification is large [41]. This absence of longitudinal polarization in the imaged field has been verified in experimental investigations of focused super-oscillation optical fields [27,42], where a clear deviation was observed in the imaged field from its theoretically predicted entire field, and a good agreement was found between the imaged field and corresponding transversely polarized field obtained with numerical simulation. A scanning aperture optical microscope, such as a scanning near-field optical microscope, can be used to detect the spatial distribution of a longitudinally polarized field [25,27,43]. However, it has strong polarization selectivity [41], which makes it impossible to characterize the distribution of the three orthogonal components in optical fields. In addition, scanning aperture probing is time consuming and depends on the interaction with the field under test, which unavoidably results in the deformation of the realistic optical field. Although a polarization-conversion-based confocal microscope has been reported for far-field mapping of longitudinal magnetic and electric optical fields, it depends on point-by-point scanning by moving the optical field under test, which is slow and not applicable to fast optical field acquisition [44].

It remains a challenge to directly image the vectorial polarization distribution in optical fields. In the present work, a polarization-conversion microscopy (PCM) is proposed. Based on the method, a polarization-conversion-based optical microscope (PCBOM) of large magnification is developed for characterizing the distribution of transverse and longitudinal polarizations simultaneously and independently. Both theoretical and experimental investigations have been conducted to verify the concept, and show good agreements between the theoretical predictions and experimental results. The proposed method provides a promising tool for easy, fast, and noninvasive characterization and study of focused optical fields. It also provides a powerful tool to understand the formation of optical super-resolution and super-resolution related optical phenomena.

2. PCM THEORY

An optical field can be decomposed into three orthogonal components, namely, radial, azimuthal, and longitudinal polarizations, but they are not independent. The longitudinal component is a result of oblique propagation of the radially polarized optical field. In the following, theoretical analyses are conducted to demonstrate that the transversely and longitudinally polarized optical fields in focused light can be independently obtained by the PCM method.

A. Longitudinally Polarized Optical Field in Focused Light

As shown in Fig. 1, the radially polarized incident field, i.e., ${\mathop{E}\limits^\rightharpoonup } _{{\rm inc}} = {E_{{\rm inc}}}{{{{{\mathop{n}\limits^\rightharpoonup }}} } _\rho}$, is focused by a lens with focal length of $f$, which is represented by a reference sphere $S$ (dashed curve), and ${\mathop{n}\limits^\rightharpoonup }_{\rho}$ and ${\mathop{n}\limits^\rightharpoonup }_{\theta}$ are unit vectors in the radial and polar directions, respectively. During propagation, the entire electrical vector is perpendicular to the corresponding light ray.

 figure: Fig. 1.

Fig. 1. Focusing of radially polarized optical field by a lens with a focal length of $f$, which is represented by a reference sphere $S$ (dashed curve).

Download Full Size | PPT Slide | PDF

The focused optical field consists of longitudinal and transverse polarizations, which can be calculated by using the vectorial angular spectrum method (VASM) [45]. Equation (1) gives the spatial distribution of the longitudinally polarized optical field on the focal plane for the incident wavelength of $\lambda$, where $k = {2}\pi /\lambda$, $P(\theta)$ describes the amplitude distribution of the incident wave, ${J_0}$ is the zero-order Bessel function, ${\mathop{n}\limits^\rightharpoonup }_{z}$ is the unit vector in $z$ direction, and $r$ is the radial coordinate on the focal plane at $z = F$. Here, we have assumed that the refractive index is equal to one on both sides of the reference sphere:

$$\begin{split}{\mathop{E}\limits^{\rightharpoonup}}_{z}(r) &= \left\{{ikf\exp\! \left({ikf} \right)\int_0^{{\theta _{{\max}}}} {P(\theta){{\left({\cos \theta} \right)}^{1/2}}{{\left({\sin \theta} \right)}^2}{J_0}(kr\sin \theta){\rm d}\theta}} \right\}\\&\quad \times {E_{{\rm inc}}}{\mathop{n}\limits^\rightharpoonup }_{z}.\end{split}$$

B. PCM for Measuring Longitudinally Polarized Optical Field

The far-field pattern associated with longitudinal polarization can be treated as the result of radiation generated by a dipole orientated in the $z$ direction [46,47]. To directly observe the longitudinally polarized optical field, a radial-to-linear PCM (RLPCM) is proposed, where an ${ S}$-wave-plate (SWP) and a linear polarizer (LP) are put in the light path between the objective lens (OL) and tube lens (TL) in a conventional optical microscope (COM). The SWP [48] is a kind of polarization convertor that converts radial polarization into linear polarization, and vice versa. Figure 2 presents the working principle of the proposed PCM scheme for directly acquiring the longitudinally polarized optical field on the front focal plane of the OL (represented by the reference sphere ${S_1}$).

 figure: Fig. 2.

Fig. 2. Radial-to-linear polarization conversion microscopy for characterizing a longitudinally polarized optical field. The objective lens and tube lens are represented by two reference spheres, i.e., ${S_1}$ and ${S_2}$, respectively. The ${S}$-wave plate (SWP) and linear polarizer (LP) are set in the light path between the objective lens and the tube lens. The dipole is an analog of the longitudinally polarized optical field.

Download Full Size | PPT Slide | PDF

As shown in Fig. 2, an electrical dipole, i.e., ${\mathop{\mu}\limits^\rightharpoonup }=\mu_{z}{\mathop{n}\limits^\rightharpoonup }_{z}$, is set at the focal point of the OL (${S_1}$) with a focal length of $f$, and the dipole orientation is in the $z$ direction, or the longitudinal direction. The emitted optical field of the dipole can be expressed as ${{\mathop{E}\limits^\rightharpoonup }_1} = {{({\omega ^2}} / {{\varepsilon _0}{c^2}}}){{\mathop{G}\limits^\leftrightarrow }} \cdot {\mathop{\mu}\limits^\rightharpoonup }$, where $\omega$ is the time-domain angular frequency of the incident field, ${\varepsilon _0}$ is the permittivity of vacuum, $c$ is the light speed in vacuum, and ${\mathop{G}\limits^\leftrightarrow }$ is the dyadic Green’s function [45]. The radiated optical field is collected and collimated by the OL (${S_1}$), and then passes through the SWP and LP. The SWP converts the corresponding radial component into a $y$ linearly polarized one, and the LP allows only the $y$ linearly polarized component to pass through. Finally, the optical field is converged on the focal plane of the TL (represented by the reference sphere ${S_2}$) with a focal length of $f^\prime$.

Equations (2.1)–(2.4) give the electrical field of ${\mathop{E}\limits^\rightharpoonup}_{1}$, ${\mathop{E}\limits^\rightharpoonup}_{2}$, ${\mathop{E}\limits^\rightharpoonup}_{3}$, and ${\mathop{E}\limits^\rightharpoonup}_{4}$, as indicated in Fig. 2, where we have assumed that the refractive index on both sides of the two reference spheres is equal to one, and approximations are made for ${\cos}\,\theta ^\prime = {1}$ and ${\sin}\,\theta ^\prime = {0}$ for the case of a small angel of $\theta ^\prime$, or large magnification, i.e., $M = f^\prime /\;f\; \gg \;{1}$. The large magnification is critical to filter out the contribution from transverse components in the focused field on the image plane of the TL, which can be explained as the optical lever effect in RLPCM:

$${\mathop{E}\limits^\rightharpoonup}_{1}\def\LDeqtab{} = {\mu _z}\left({\frac{{{\omega ^2}}}{{{\varepsilon _0}{c^2}}}} \right)\frac{{\exp \left({ikf} \right)}}{{4\pi f}}\left[{\begin{array}{*{20}{c}}{- \cos \phi \sin \theta \cos \theta}\\{- \sin \phi \sin \theta \cos \theta}\\{{{\sin}^2}\theta}\end{array}} \right]\!,$$
$$\begin{split}{\mathop{E}\limits^\rightharpoonup}_{2} &= {\left({\cos \theta} \right)^{- 1/2}}\left[{\left({{\mathop{E}\limits^\rightharpoonup}_{1} \cdot {{\mathop{n}\limits^\rightharpoonup}_{\theta}}} \right){{\mathop{n}\limits^\rightharpoonup}_{\rho}} +{({{\mathop{E}\limits^\rightharpoonup}_{1} \cdot {{\mathop{n}\limits^\rightharpoonup}_{\phi}}})}{\mathop{n}\limits^\rightharpoonup}_{\theta}} \right]\\& = \left\{{- {\mu _z}\left({\frac{{{\omega ^2}}}{{{\varepsilon _0}{c^2}}}} \right)\frac{{\exp \left({ikf} \right)}}{{4\pi f}}{{\left({\cos \theta} \right)}^{- 1/2}}\sin \theta} \right\}{{\mathop{n}\limits^\rightharpoonup}_{\rho}},\end{split}$$
$$\begin{split}{\mathop{E}\limits^{\rightharpoonup}}_3& = \left({\mathop{E}\limits^{\rightharpoonup}}_2 \cdot {\mathop{n}\limits^{\rightharpoonup}}_\rho\right){\mathop{n}\limits^{\rightharpoonup}}_y\\&= \left\{{- {\mu _z}\left({\frac{{{\omega ^2}}}{{{\varepsilon _0}{c^2}}}} \right)\frac{{\exp \left({ikf} \right)}}{{4\pi f}}{{\left({\cos \theta} \right)}^{- 1/2}}\sin \theta} \right\}{\mathop{n}\limits^{\rightharpoonup}}_y,\end{split}$$
$$\begin{split}{\mathop{E}\limits^\rightharpoonup}_{4} &= \left[{\left({\mathop{E}\limits^\rightharpoonup}_{3}{ \cdot {\mathop{n}\limits^\rightharpoonup}_{\rho}} \right){\mathop{n}\limits^\rightharpoonup}_{\theta^{\prime}}+( {\mathop{E}\limits^\rightharpoonup}_{3}{ \cdot {\mathop{n}\limits^\rightharpoonup}_{\phi}}){\mathop{n}\limits^\rightharpoonup}_{\phi}} \right]{\left({\cos \theta ^\prime} \right)^{1/2}}\\ &= \left\{{- {\mu _z}\left({\frac{{{\omega ^2}}}{{{\varepsilon _0}{c^2}}}} \right)\frac{{\exp \left({ikf} \right)}}{{4\pi f}}{{\left({\cos \theta} \right)}^{- 1/2}}\sin \theta} \right\}{\mathop{n}\limits^\rightharpoonup}_{y}.\end{split}$$

As shown in Eq. (2.2), it is interesting to notice that the collimated radiation right after the OL (${S_1}$) is purely radially polarized, which is consistent with the previous claim that a longitudinally polarized optical field can be treated as a dipole lying along the $z$ direction [46,47]. Due to the lever effect [41], the final optical field after the TL (${S_2}$) consists of only a $y$ linearly polarized field, as shown in Eq. (2.4). Again, using the VASM, the optical field on the focal plane of the TL (${S_2}$) can be expressed as Eq. (3), which consists of only $y$ polarization, and where $C = ({\frac{{{\omega ^2}}}{{{\varepsilon _0}{c^2}}}})\frac{k}{{4\pi}}\frac{{f^\prime}}{f}\exp [{ik({f - f^\prime})}]$:

$${\mathop{E}\limits^{{\rightharpoonup}}} _{\rm image}(\rho) = \left\{{- iC\int_0^{{\theta _{{\max}}}} {{{\left({\cos \theta} \right)}^{1/2}}{{\left({\sin \theta} \right)}^{2}}{J_0}(k\sin \theta {\rho / M}){\rm d}\theta}} \right\}{\mu _z}{\mathop{E}\limits^{{\rightharpoonup}}} _{y},$$
$${\mathop{E}\limits^{{\rightharpoonup}}} _{\rm image}(r) = \left\{{- iC\int_0^{{\theta _{{\max}}}} {{{\left({\cos \theta} \right)}^{1/2}}{{\left({\sin \theta} \right)}^{2}}{J_0}(k\sin \theta ){\rm d}\theta}} \right\}{\mathop{n}\limits^{{\rightharpoonup}}} _{y},$$

By comparing Eq. (1) and Eq. (3), it is interesting to note that they share the same format of the integral, indicating the same spatial distribution in the two optical fields. The only difference between the two is the constant coefficient outside the integral. Furthermore, $r = \rho /M$ is, in fact, the radial coordinate on the objective plane, and then Eq. (3) can be rewritten as Eq. (4) (we have let ${\mu _z} = {1}$). This indicates that the spatial distribution of the optical field obtained on the image plane of the RLPCM represents exactly that of the longitudinally polarized optical field in the focused light on the objective plane of the RLPCM. It also indicates that the RLPCM cannot obtain the transversely polarized optical field. Therefore, the scheme illustrated in Fig. 2 can be used to characterize the longitudinally polarized optical field in focused light. In fact, Eq. (4) gives the point spread function (PSF) of the RLPCM, as shown in Fig. 2, for the longitudinally polarized field.

C. COM for Measuring Transversely Polarized Optical Field

Although COM has been successfully applied to the characterization of super-resolution focused optical fields of transverse polarizations [20], there have been no corresponding theoretical analyses conducted to prove its validity. Here, we use above method to verify its validity.

Under the illumination of a linearly polarized optical field, for example, a polarization in $x$ direction, the focused optical field consists of transverse polarization in $x$ direction and longitudinal polarization in $z$ direction. The corresponding transversely polarized field is given by Eq. (5):

$$\begin{split}{\mathop{E}\limits^{{\rightharpoonup}}} _{x}(r) &= \left\{ikf\exp \left({ikf} \right)\int_0^{{\theta _{{\max}}}} P(\theta){{\left({\cos \theta} \right)}^{1/2}}\right.\\&\quad \times \left.\frac{1}{2}\left({1 + \cos \theta} \right)\sin \theta {J_0}(kr\sin \theta)d\theta \right\}{E_{{\rm inc}}}{\mathop{n}\limits^\rightharpoonup}_{x}.\end{split}$$

For a COM with a large magnification, according to the VASM, the PSFs for $x$-polarized optical fields can be written as Eq. (6), which consists of only $x$ polarization. The large magnification is critical to filter out the contribution from longitudinal components in the focused field on the image plane of the TL in COM, which is called the optical lever effect [41]. Similarly, Eqs. (5) and (6) show the same spatial distribution, except for the constant coefficients outside the integrals, indicating that the image field obtained by a COM represents exactly the transversely polarized optical field in focused light. It also reveals that a COM cannot acquire the longitudinally polarized optical field. Therefore, in contrast to the RLPCM, the COM can be used to obtain only the transversely polarized optical field in focused light:

$$\begin{split}&{\mathop{E}\limits^{{\rightharpoonup}}} _{\rm image}(r, \varphi)\\&\quad = \left\{{- iC\int_0^{{\theta _{{\max}}}} {{{\left({\cos \theta} \right)}^{1/2}}\frac{1}{2}\left({{1 +}\cos \theta} \right)\sin \theta {J_0}(k\sin \theta r){\rm d}\theta}} \right\}{\mathop{n}\limits^\rightharpoonup}_{x}.\end{split}$$

According to Eqs. (5) and (6), the PSF of transverse polarization given by Eq. (6) is approximately 1.24 times greater than that of longitudinal polarization given by Eq. (4) for an OL of NA = 0.95, i.e., $\theta_{\max}=71.8^{\circ}$, which is later used to calibrate the experimentally obtained intensities of the transverse and longitudinal polarizations.

3. EXPERIMENTAL SETUP OF PCBOM

A PCBOM system for simultaneously measuring both transversely and longitudinally polarized fields is built to characterize the distribution of the three orthogonal optical fields, i.e., ${E_x}$, ${E_y}$, and ${E_z}$, in focused light. The system consists of two microscopes sharing a common OL, as shown in Fig. 3. Each microscope includes an OL and TL. The vertical arm and the OL form a RLPCM for directly observing the longitudinal polarization, and the horizontal arm and the OL form a COM for observing the transverse polarizations. In the RLPCM, there are SWP and LP between the OL and TL for radial-to-linear polarization conversion. In the COM, there is a LP between the OL and TL to select desired transverse polarizations, i.e., ${E_x}$ and ${E_y}$. In Fig. 3, the lens is used to generate a focused optical field to be characterized by the PCBOM system.

 figure: Fig. 3.

Fig. 3. Polarization-conversion-based optical microscope for independently measuring both transversely and longitudinally polarized fields. Vertical arm and objective lens form RLPCM, and horizontal arm and objective lens form COM. COM and RLPCM are for observing transverse polarizations, i.e., ${E_x}$ and ${E_y}$, and longitudinal polarization, i.e., ${E_z}$, respectively. The lens is used to generate the focused optical field for the polarization imaging experiment.

Download Full Size | PPT Slide | PDF

As illustrated in Fig. 3, a He–Ne laser (HNL210L, Thorlabs Inc.) emitting at the wavelength of $\lambda = {632.8}\;{\rm nm}$ is used as a coherent source for illumination. A lens is used to generate the focused optical field for the polarization imaging experiment. The optical field on the front focal plane of the OL is collected by the OL (CF Plan ${100}\! \times \;{/0.95}$, Nikon) and then split into two parts by a cube beam splitter (BS) (BS013 50:50, Thorlabs Inc.). One is sent to the COM arm equipped with a LP (WP25M-VIS, Thorlabs Inc.), which is used to obtain the $x$-polarized and $y$-polarized fields by rotating to the targeted polarization direction; the other is sent to the RLPCM arm equipped with a SWP (RPC-632.8-06-188, Workshop of Photonics) and a LP (WP25M-VIS, Thorlabs Inc.), which are used to obtain the longitudinally polarized fields. Finally, the transversely polarized field and longitudinally polarized field are imaged on the back focal plane of the TLs (ITL200, Thorlabs Inc.) and captured by DCs (acA3800-14 µm, Basler Inc.) in the COM arm and RLPCM arm, respectively. The OL is mounted on a nano-positioner (NP) (EO-S1047, Edmund Optics), which can be used to obtain the polarization distribution in three-dimensional space.

4. RESULTS AND DISCUSSION

To verify the proposed method, we used the PCBOM system to characterize the tightly focused super-resolution optical fields. Experiments have been conducted to obtain the intensity distribution of three orthogonal optical fields, i.e., ${E_x}$, ${E_y}$, and ${E_z}$, generated by super-resolution metalenses under the illumination of three typical polarizations, i.e., linear, circular, and radial polarizations, respectively. Theoretical simulations have also been carried out to verify the corresponding experimental results.

A. Super-Resolution Optical Field Generated by Focusing Linearly Polarized Light

Linearly polarized light has optical fields oscillating in a fixed direction. In the circularly symmetric case, the propagation of linearly polarized light can be described by the angular spectrum diffraction formulas in cylindrical coordinate ($r,\varphi ,z$). Equation (7) give the transverse and longitudinal optical fields, i.e., ${E_{{{\hat n}_{\phi ^\prime}}}}(r,z)$ and ${E_z}(r,\varphi ,z)$, in the diffraction pattern of an optical device under the illumination of linearly polarized light with a polarization angle of $\varphi^{\prime}$:

$$\left\{\begin{array}{l}{E_{{{\hat n}_{\phi ^\prime}}}}(r,z) = \int_0^\infty {{A_0}({f_{\textit{xy}}})\exp (i2\pi {f_z}z){J_0}(2\pi {f_{\textit{xy}}}r)2\pi {f_{\textit{xy}}}{\rm d}{f_{\textit{xy}}}}, \\[5pt]{E_z}(r,\phi ,z) = - i\cos (\phi - \phi ^\prime)\int_0^\infty {A_0}({f_{\textit{xy}}})({{{f_{\textit{xy}}}} / {{f_z}}})\\[5pt]\exp (i2\pi {f_z}z){J_1}(2\pi {f_{\textit{xy}}}r)2\pi {f_{\textit{xy}}}{\rm d}{f_{\textit{xy}}}, \\[5pt]{A_0}({f_{\textit{xy}}}) = \int_0^\infty {{E_{{{\hat n}_{\phi ^\prime}}}}(r,z = 0)T(r){J_0}(2\pi {f_{\textit{xy}}}r)2\pi r{\rm d}r}, \end{array} \right.$$
where ${\hat n_{\phi ^\prime}}$ is the unit vector of the polarization direction, ${A_0}({f_{\textit{xy}}})$ is the angular spectrum, ${f_z}$ is the spatial frequency in $z$ direction, ${f_{\textit{xy}}} = {({1/}{\lambda ^2} - f_z^2)^{1/2}}$ is the in-plane spatial frequency, ${J_0}$ and ${J_1}$ are the zero-order and first-order Bessel functions, respectively, and $T(r)$ is the transmission function of the optical device. Therefore, the diffracted optical field includes a transverse component ${E_{{{\hat n}_{\phi ^\prime}}}}(r,z)$ with the same polarization of the incident field and a longitudinal component ${E_z}(r,\;\varphi ,\;z)$ with polarization in the axial direction, i.e., $z$ direction. For a positive focusing lens, ${E_{{{\hat n}_{\phi ^\prime}}}}(r,z)$ contributes to the formation of a solid circular focal spot, while ${E_z}(r,\;\varphi ,\;z)$ results in a dumbbell-shaped spot lying in the direction of ${\hat n_{\phi ^\prime}}$ because of the orientation-dependent factor of ${\cos}(\phi - \phi ^{\prime})$.

To observe the spatial distribution of ${E_{{{\hat n}_{\phi ^\prime}}}}(r,z)$ and ${E_z}(r,\;\phi ,\;z)$, a previously reported super-resolution metalens [29] is used to generate a non-diffraction optical field with sub-diffraction features. Based on binary phase (zero and $\pi$) modulation, the lens is designed with a radius of ${638}\lambda$ and working distance of ${230}\lambda$ for a wavelength of $\lambda = {632.8}\;{\rm nm}$. It can generate a circularly polarized non-diffraction beam with propagation distance of approximately ${100}\lambda$, whose transverse size is smaller than the Abbe diffraction limit (ADL) within the range of ${z} = {240}\lambda$ and 330λ. Figures 4(a)–4(d) present the theoretical predictions of the intensity distributions of the three orthogonal optical fields and the entire optical field, i.e., ${E_x}$, ${E_y}$, ${E_z}$, and $E$, on the focal plane under the illumination of linearly polarized light with a polarization angle of 45º with respect to the $x$ axis, or $\phi^{\prime}=45^{\circ}$. The intensity of the entire optical field is obtained by summing up the intensities of the three orthogonal optical fields, i.e., $|E{|^2} = |{E_x}{|^2} + |{E_y}{|^2} + |{E_z}{|^2}$. The corresponding experimental results are depicted in Figs. 4(e)–4(h), respectively. The measurements are taken on the plane ${280}\lambda$ away from the metalens. To make comparisons, all intensity distributions are normalized with respect to the peak intensity of the corresponding entire optical field $E$. As expected, the transversely polarized fields, i.e., ${E_x}$ and ${E_y}$, form a solid spot with a good rotational symmetry, while the longitudinally polarized field creates two spots orientated in the linear polarization direction, leading to a significant broadening of the entire optical field.

 figure: Fig. 4.

Fig. 4. Intensity distribution of optical fields created by focusing linearly polarized light. Theoretical results for (a) $|{E_x}{|^2}$, (b) $|{E_y}{|^2}$, (c) $|{E_z}{|^2}$, and (d) $|E{|^2}$. Experimental results for (e) $|{E_x}{|^2}$, (f) $|{E_y}{|^2}$, (g) $|{E_z}{|^2}$, and (h) $|E{|^2}$. Comparison of intensity distributions between theoretical and experimental results for (i) $|{E_x}{|^2}$ and (j) $|{E_y}{|^2}$ in the $x$ direction, and (k) $|{E_z}{|^2}$ and (l) $|E{|^2}$ in the incident polarization direction, as indicated by the white lines in (c) and (g), where red curves are VASM results, and blue curves are experimental results. The scalar bar represents a length of ${1}\lambda$.

Download Full Size | PPT Slide | PDF

Figures 4(i) and 4(j) give the intensity curves of ${E_x}$ and ${E_y}$ on the $x$ axis, while results for ${E_z}$ and $E$ along the linear polarization direction are presented in Figs. 4(k) and 4(l), respectively. Again, the experimental results show excellent fits of the theoretical ones. It is noted that the intensity maxima in ${E_x}$ and ${E_y}$ appear at different locations compared with their counterparts in ${E_z}$. For example, in the ${E_x}$ and ${E_y}$ intensity curves, the second maxima are located at approximately ${x} = \pm {0.75}\lambda$, while in the ${E_z}$ intensity curve, the second maxima are located at $L = \;\pm {1}\lambda$, where $L$ is the coordinate along the polarization direction, as indicated by the white lines in Figs. 4(c) and 4(g). This indicates that the observed longitudinal components $|{E_z}{|^2}$ and the transverse components ($|{E_x}{|^2}$ or $|{E_y}{|^2}$) come from different polarization components in the focused optical field.

Tables Icon

Table 1. Major Parameters of Optical Intensity Created by Focusing Linearly Polarized Light

 figure: Fig. 5.

Fig. 5. Intensity distribution of optical fields created by focusing circularly polarized light. Theoretical results for (a) $|{E_x}{|^2}$, (b) $|{E_y}{|^2}$, (c) $|{E_z}{|^2}$, and (d) $|E{|^2}$. Experimental results for (e) $|{E_x}{|^2}$, (f) $|{E_y}{|^2}$, (g) $|{E_z}{|^2}$, and (h) $|E{|^2}$. Comparison of intensity distributions between theoretical and experimental results for (i) $|{E_x}{|^2}$, (j) $|{E_y}{|^2}$, (k) $|{E_z}{|^2}$, and (l) $|E{|^2}$ on the $x$ axis, where red curves are VASM results, and blue curves are experimental results. The scalar bar represents a length of ${1}\lambda$.

Download Full Size | PPT Slide | PDF

The major parameters of the focused fields are listed in Table 1, including the size of the spot (FWHM, full-width-at-half-maximum) and the corresponding sidelobe ratio (SR, ratio of maximum sidelobe intensity to the central peak intensity). According to the experimental results, the FWHMs of $|{E_x}{|^2}$ and $|{E_y}{|^2}$ are ${0.430}\lambda$ and ${0.426}\lambda$, respectively, which are smaller than ADL (${0.5}\lambda {\rm /NA}$) [20]; the inner FWHM of $|{E_z}{|^2}$ is ${0.333}\lambda$, which is smaller than the super-oscillation criteria (${0.38}\lambda {\rm /NA}$) [20]; and the FWHM of the entire field $|E{|^2}$ is ${1.043}\lambda$, which is much greater than ADL. The corresponding theoretical results are also presented in the table for comparison. The experimental results show good agreements with the corresponding theoretical predictions. The difference between the experimental and theoretical results is less than 3% in FWHM. The SR varies between 19% and 36% for different components, which is also consistent with the theoretical values between 16% and 35%.

To test the response of the RLPCM to the weakly focused optical field, an experiment is conducted to obtain the polarization distribution of the focused optical field generated by an OL with a small NA of 0.25. With such a small NA, for linearly polarized incoming light, the focused field consists of a comparatively strong linearly polarized component and a weak longitudinally polarized component, which agrees with our experimental observation. The detailed information can be found in Supplement 1. This indicates that the pattern obtained by the RLPCM comes solely from the contribution of the longitudinally polarized optical field on the front focal plane of the OL.

B. Super-Resolution Optical Field Generated by Focusing Circularly Polarized Light

Circularly polarized light has a rotating electrical vector, which can be treated as a superposition of two orthogonal linearly polarized optical fields with a phase difference of $\pi /{2}$. According to Eq. (7), for a positive lens, the focused transverse optical fields ${E_x}$ and ${E_y}$ have the shape of a solid spot with rotational symmetry, while the intensity of the longitudinal optical field forms a donut-shaped hollow spot because ${{\rm \cos}^2}(\varphi - \varphi ^{\prime}) + {{\rm \cos}^2}(\varphi - \varphi^{\prime} + \pi /{2}) = {1}$ and ${J_1}({2}\pi {f_{\textit{xy}}}r)$ is zero at $r = {0}$.

To investigate the spatial distribution of the focused fields, the same metalens [29] is used to generate a super-resolution optical field under the illumination of circularly polarized light. Figures 5(a)–5(d) present the intensity distributions of ${E_x}$, ${E_y}$, ${E_z}$, and $E$ on the plane at ${z} = {280}\lambda$ obtained with VASM. Figures 5(e)–5(h) illustrate the corresponding experimental results for the three orthogonal polarization components and the entire optical field, respectively. The predicted donut-shaped intensity distribution of the longitudinally polarized field is directly observed by the RLPCM arm, which results in a clear broadening of the spot size of the entire field. Figures 5(i)–5(l) plot normalized intensity distribution curves of ${E_x}$, ${E_y}$, ${E_z}$, and $E$ on the $x$ axis for both theoretical and experimental results. Again, the experimental and theoretical results show an excellent agreement in all three orthogonally polarized optical fields and the entire optical field. Similar to the case of incident linearly polarized light, it is noted that the intensity maxima in ${E_x}$ and ${E_y}$ appear at different locations compared with their counterparts in ${E_z}$. Again, this indicates that the observed longitudinal components $|{E_z}{|^2}$ and the transverse components ($|{E_x}{|^2}$ or $|{E_y}{|^2}$) come from different polarization components in the focused optical field.

Tables Icon

Table 2. Major Parameters of Optical Intensity Created by Focusing Circularly Polarized Light

By comparing Fig. 4 and Fig. 5, it is noted that the normalized intensity distribution of transverse components is almost the same for the cases of linear and circular polarization illuminations, while the focused longitudinal field becomes weaker due to the continuous rotation of the polarization in the circularly polarized optical field. Therefore, circularly polarized light can achieve a smaller focal spot compared to linearly polarized light.

The corresponding focusing parameters are presented in Table 2 for both experimental and theoretical results, including FWHM and SR. To give a better evaluation of the focusing parameters, FWHM and SR are calculated by averaging their value in 10 different directions between 0° and 162° (including 0° and 162°) with an angular interval of 18º in the cases of circular symmetrical intensity distribution. According to the experimental results, the FWHMs of $|{E_x}{|^2}$ and $|{E_y}{|^2}$ are ${0.418}\lambda$ and ${0.424}\lambda$, respectively, which are also smaller than ADL and close to the values obtained in the case of linear polarization illumination; the inner FWHM of $|{E_z}{|^2}$ is ${0.347}\lambda$, which is super-oscillatory; and the FWHM of the entire field $|E{|^2}$ is ${0.743}\lambda$, which is much smaller than that obtained under the illumination of linearly polarized light due to the comparatively weaker $|{E_z}{|^2}$ in the present case. However, the spot size of the entire optical field is greater than ADL. The deviation between the experimental and theoretical results is approximately 2% in FWHM. The SR varies between 21% and 31% for different components, which is consistent with the theoretical values between 16% and 35%.

C. Super-Resolution Optical Field Generated by Focusing Radially Polarized Light

Radially polarized light has optical fields oscillating in the radial direction with respect to the beam center. For an optical device with transmission function of $T(r)$, under the illumination of radially polarized light, the diffracted optical field consists of the radially polarized optical field ${E_\rho}$ and longitudinally polarized field ${E_z}$. The two orthogonal optical fields have rotational symmetry and can be described by Eq. (8):

$$\left\{\begin{array}{l}{E_\rho}(r,z) = \int_0^\infty {{A_1}({f_{\textit{xy}}})\exp (i2\pi {f_z}z){J_1}(2\pi {f_{\textit{xy}}}r)2\pi {f_{\textit{xy}}}{\rm d}{f_{\textit{xy}}}}, \\[5pt]{E_z}(r,z) = i\int_0^\infty {A_1}({f_{\textit{xy}}})({{{f_{\textit{xy}}}} / {{f_z}}})\exp (i2\pi {f_z}z){J_0}(2\pi {f_{\textit{xy}}}r)\\[5pt]2\pi {f_{\textit{xy}}}{\rm d}{f_{\textit{xy}}}, \\[5pt]{A_1}({f_{\textit{xy}}}) = \int_0^\infty {{E_\rho}\left({r,z = 0} \right)T\left(r \right){J_1}(2\pi {f_{\textit{xy}}}r)2\pi r{\rm d}r}, \end{array} \right.$$
where ${A_1}({f_{\textit{xy}}})$ is the angular spectrum of the radially polarized optical field right behind the optical device. Unlike the cases of linear and circular polarizations, in the present case, the diffracted radially polarized field has a donut-shaped profile, which is a superposition of dumbbell-shaped ${E_x}$ and ${E_y}$; however, the longitudinally polarized field forms a circular solid spot.

To characterize the polarization distribution created by focusing radially polarized light, a previously reported vectorial-optical-field super-resolution metalens [49] was adopted to generate a longitudinally polarized focal spot under the illumination of a linearly polarized light. The metalens integrates the functions of polarization conversion (linear-to-azimuthal and linear-to-radial) and super-resolution focusing. The phase profile of the lens is designed based on binary phase (zero and $\pi$) with a radius of ${250}\lambda$ and focal length of ${100}\lambda$ for a wavelength of $\lambda = {632.8}\;{\rm nm}$. When the incident linear polarization aligns with the fast axis of the metalens, the incident beam is first converted to radial polarization and then focused into a longitudinally polarized spot. On the focal plane, except for the tightly focused longitudinally polarized component, a weaker donut-shaped radially polarized component is also expected. Figures 6(a)–6(d) and Figs. 6(e)–6(h) present the normalized intensity of ${E_x}$, ${E_y}$, ${E_z}$, and $E$ on the focal plane obtained by VASM and the finite element method (FEM) (COMSOL Multiphysics), respectively. The corresponding experimental results are depicted in Figs. 6(i)–6(l). The dumbbell-shaped intensity distributions of ${E_x}$ and ${E_y}$ are the result of the donut-shaped optical intensity of the radial component ${E_\rho}$. For comparison, the normalized intensity distribution of each orthogonal polarization is plotted for theoretical and experimental results in Figs. 6(m)–6(p). Again, it is found that the intensity maxima in ${E_x}$ and ${E_y}$ appear at different locations compared with their counterparts in ${E_z}$, indicating that the observed longitudinal components $|{E_z}{|^2}$ and the transverse components ($|{E_x}{|^2}$ or $|{E_y}{|^2}$) are the results of different polarization components in the focused optical field.

 figure: Fig. 6.

Fig. 6. Intensity distribution of optical fields created by focusing radially polarized light. Theoretical results for (a) $|{E_x}{|^2}$, (b) $|{E_y}{|^2}$, (c) $|{E_z}{|^2}$, and (d) $|E{|^2}$. FEM simulation results for (e) $|{E_x}{|^2}$, (f) $|{E_y}{|^2}$, (g) $|{E_z}{|^2}$, and (h) $|E{|^2}$. Experimental results for (i) $|{E_x}{|^2}$, (j) $|{E_y}{|^2}$, (k) $|{E_z}{|^2}$, and (l) $|E{|^2}$. Comparison of intensity distributions between theoretical and experimental results for (m) $|{E_x}{|^2}$, (n) $|{E_y}{|^2}$, (o) $|{E_z}{|^2}$, and (p) $|E{|^2}$, where red curves are VASM results, green curves are FEM results, and blue curves are experimental results. The scalar bar represents a length of ${1}\lambda$.

Download Full Size | PPT Slide | PDF

In all cases, the field distribution shows an excellent agreement between the theoretical and experimental results in ${E_x}$, ${E_y}$, ${E_z}$, and $E$. The intensity profiles of ${E_x}$ and ${E_y}$ show dumbbell-shaped spots along the polarization direction, which are typical distributions of the $x$ and $y$ components of a radially polarized optical field, indicating a successful linear-to-radial polarization conversion. It is interesting to note that the transversely polarized donut-shaped structure (not shown in the figure) formed by the superposition of ${E_x}{|^2}$ and $|{E_y}{|^2}$ has much less intensity compared to its counterpart formed by the longitudinally polarized optical field as shown in Figs. 5(c), 5(g), and 5(k). Compared with the focusing of linearly and circularly polarized light, focusing of radially polarized light results in a comparatively smaller spot size of the entire optical field. Table 3 gives FWHM and SR of the focused field for each polarization. According to the experimental results, the inner FWHMs of $|{E_x}{|^2}$ and $|{E_y}{|^2}$ are ${0.352}\lambda$ and ${0.380}\lambda$, respectively, which are super-oscillatory and close to their theoretical value of ${0.34}\lambda$; the FWHM of $|{E_z}{|^2}$ is ${0.421}\lambda$; and the FWHM of the entire field $|E{|^2}$ is ${0.467}\lambda$. The SR varies between 17% and 46% for different components, which is consistent with the theoretical values between 14% and 37%. Unlike the cases of linear and circular polarization, in the present case, the FWHM is sub-diffracted for all three orthogonal optical fields, indicating the advantage of using the radially polarized optical field for super-resolution applications.

Tables Icon

Table 3. Major Parameters of Optical Intensity Created by Focusing Radially Polarized Light

 figure: Fig. 7.

Fig. 7. Intensity distribution of optical fields on the propagation plane created by focusing radially polarized light. VASM results for (a) $|{E_x}{|^2}$, (b) $|{E_y}{|^2}$, (c) $|{E_z}{|^2}$, and (d) $|E{|^2}$. FEM results for (e) $|{E_x}{|^2}$, (f) $|{E_y}{|^2}$, (g) $|{E_z}{|^2}$, and (h) $|E{|^2}$. Experimental results for (i) $|{E_x}{|^2}$, (j) $|{E_y}{|^2}$, (k) $|{E_z}{|^2}$, and (l) $|E{|^2}$. Comparison of the major parameters (intensity, FWHM, and SR) between theoretical and experimental results for (m) $|{E_x}{|^2}$, (n) $|{E_y}{|^2}$, (o) $|{E_z}{|^2}$, and (p) $|E{|^2}$ along the optical axis, where red curves are VASM results, green curves are FEM results, blue curves are experimental results, magenta dashed lines are ADL, and black dashed lines are super-oscillation criterion.

Download Full Size | PPT Slide | PDF

By conducting the $z$-direction scan, the three-dimensional distribution of the three orthogonal polarizations is acquired. Figure 7 gives the intensity distributions of ${E_x}$, ${E_y}$, ${E_z}$, and $E$ on the propagation plane, or $xz$ plane, for both theoretical and experimental results. Figs. 7(a)–7(d) and Figs. 7(e)–7(h) give the theoretical results obtained with VASM and FEM, respectively. There is only a small difference in the intensity distribution and the focusing parameters. Due to the large computation load, in both theoretical simulations, the influence caused by the meta-atom structure is ignored. The entire lens is treated as consisting of concentric ring belts, and the phase is assumed as constant within each ring belt. Figures 7(i)–7(l) present the experimentally acquired results of ${E_x}$, ${E_y}$, ${E_z}$, and $E$. There is a clear extension in the intensity distribution along the $z$ axis for transversely and longitudinally polarized fields, and this deviation from the theoretical prediction is attributed to two factors. One is the approximation made in the simulations mentioned above; the other is the depth of focus of the optical microscope. The focusing parameters are plotted against the $z$ coordinate for ${E_x}$, ${E_y}$, ${E_z}$, and $E$ in Figs. 7(m)–7(p), respectively. Except for the spatial extension in the intensity, the major focusing parameters, i.e., FWHM and SR, have a good agreement between experimental and theoretical results, especially in the area around the peak intensity. For the transverse components, the inner FWHM is smaller than the super-oscillation criterion as indicated by the black dashed line. The FWHM of the longitudinal component is below ADL and right above the super-oscillation criterion. Due to the comparatively small intensity of the transverse components, the lateral size of the entire field remains smaller than ADL, as shown in Fig. 7(p).

To test the response of the RLPCM to the pure transversely polarized field, an experiment is also conducted to obtain the polarization distribution of the azimuthally polarized optical field generated with the same metalens by aligning its slow axis to the incident linear polarization. As expected, the RLPCM has no any response to the focused field. The detailed information can be found in Supplement 1. Again, this indicates that the RLPCM responds solely to the longitudinally polarized optical field on the front focal plane of the OL.

5. CONCLUSION

Knowing the polarization distribution in focused light is critical to understanding and designing optical devices and systems for applications including focusing, imaging, laser processing, and studying optical polarization singularity [50]. In the present work, a PCM for directly imaging the distribution of a longitudinally polarized optical field in focused light is proposed. Based on the method, a PCBOM has been developed to characterize the distributions of three orthogonal polarizations of the focused optical field in three-dimensional space. Theoretical analyses and experimental demonstrations have verified the validity of the proposed method. The transverse and longitudinal optical fields are simultaneously observed in tightly focused light with sub-diffraction features under different incident polarizations, including linear, circular, and radial polarizations. An excellent agreement has been found between the experimental results and theoretical predictions. Using an additional quarter-wave plate in the COM arm, one can further obtain the Stokes parameters and determine the polarization state in the $xy$ plane. Our approach provides a promising tool for easy, fast, and noninvasive characterization and study of focused optical fields. It also provides a powerful tool to understand the formation of optical super-resolution and super-resolution related optical phenomena. It could find its application in super-resolution imaging, super-resolution microscopy, optical data storage, and laser processing.

Funding

National Natural Science Foundation of China (61927818); Natural Science Foundation of Chongqing (cstc2020jcyj-cxttX0005); Fundamental Research Funds for the Central Universities (10611CDJXZ238826).

Acknowledgment

We thank Mr. Hongchun Liu for his help in English polishing.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

REFERENCES

1. H. Kang, B. Jia, and M. Gu, “Polarization characterization in the focal volume of high numerical aperture objectives,” Opt. Express 18, 10813–10821 (2010). [CrossRef]  

2. Y. Liu, D. Cline, and P. He, “Vacuum laser acceleration using a radially polarized CO2 laser beam,” Nucl. Instrum. Methods Phys. Res. 424, 296–303 (1999). [CrossRef]  

3. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004). [CrossRef]  

4. W. Liu, D. S. Dong, H. Yang, Q. H. Gong, and K. B. Shi, “Robust and high-speed rotation control in optical tweezers by using polarization synthesis based on heterodyne interference,” Opto-Electron. Adv. 3, 200022 (2020). [CrossRef]  

5. D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368, 402–407 (2007). [CrossRef]  

6. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001). [CrossRef]  

7. N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6241 (2004). [CrossRef]  

8. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999). [CrossRef]  

9. M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007). [CrossRef]  

10. H. Dehez, A. April, and M. Piché, “Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent,” Opt. Express 20, 14891–14905 (2012). [CrossRef]  

11. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008). [CrossRef]  

12. Z. Nie, G. Shi, X. Zhang, Y. Wang, and Y. Song, “Generation of super-resolution longitudinally polarized beam with ultra-long depth of focus using radially polarized hollow Gaussian beam,” Opt. Commun. 331, 87–93 (2014). [CrossRef]  

13. V. Ravi, P. Suresh, K. B. Rajesh, Z. Jaroszewicz, P. M. Anbarasan, and T. V. S. Pillai, “Generation of sub-wavelength longitudinal magnetic probe using high numerical aperture lens axicon and binary phase plate,”J. Opt. 14, 055704 (2012). [CrossRef]  

14. Y. Zha, J. Wei, H. Wang, and F. Gan, “Creation of an ultra-long depth of focus super-resolution longitudinally polarized beam with a ternary optical element,” J. Opt. 15, 075703 (2013). [CrossRef]  

15. R. Zuo, W. Liu, H. Cheng, S. Chen, and J. Tian, “Breaking the diffraction limit with radially polarized light based on dielectric metalenses,” Adv. Opt. Mater. 6, 1800795 (2018). [CrossRef]  

16. K. B. Rajesh, Z. Jaroszewicz, and P. M. Anbarasan, “Improvement of lens axicon’s performance for longitudinally polarized beam generation by adding a dedicated phase transmittance,” Opt. Express 18, 26799 (2010). [CrossRef]  

17. Q. Li, X. Zhao, B. Zhang, Y. Zheng, L. Zhou, L. Wang, Y. Wu, and Z. Fang, “Nanofocusing of longitudinally polarized light using absorbance modulation,” Appl. Phys. Lett. 104, 061103 (2014). [CrossRef]  

18. S. Wang, J. Xu, Y. Zhong, R. Ren, Y. Lu, H. Wan, J. Wang, and J. Ding, “Focus modulation of cylindrical vector beams through negative-index grating lenses,” Opt. Commun. 372, 245–249 (2016). [CrossRef]  

19. Z. Nie, W. Ding, D. Li, X. Zhang, Y. Wang, and Y. Song, “Spherical and sub-wavelength longitudinal magnetization generated by 4π tightly focusing radially polarized vortex beams,” Opt. Express 23, 690–701 (2015). [CrossRef]  

20. G. Chen, Z. Q. Wen, and C. W. Qiu, “Superoscillation: from physics to optical applications,” Light: Sci. Appl. 8, 56 (2019). [CrossRef]  

21. M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019). [CrossRef]  

22. E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012). [CrossRef]  

23. F. Qin, K. Huang, J. F. Wu, J. H. Teng, C. W. Qiu, and M. H. Hong, “A supercritical lens optical label-free microscopy: sub-diffraction resolution and ultra-long working distance,” Adv. Mater. 29, 1602721 (2017). [CrossRef]  

24. G. H. Yuan, K. S. Rogers, E. T. F. Rogers, and N. I. Zheludev, “Far-field superoscillatory metamaterial superlens,” Phys. Rev. Appl. 11, 064016 (2019). [CrossRef]  

25. A. Yu, G. Chen, Z. Zhang, Z. Wen, L. Dai, K. Zhang, S. Jiang, Z. Wu, Y. Li, C. Wang, and X. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6, 38859 (2016). [CrossRef]  

26. Y. Zhou and M. H. Hong, “Formation of polarization-dependent optical vortex beams via an engineered microsphere,” Opt. Express 29, 11121–11131 (2021). [CrossRef]  

27. Z. Wu, Q. Jin, S. Zhang, K. Zhang, L. Wang, L. Dai, Z. Wen, Z. Zhang, G. Liang, Y. Liu, and G. Chen, “Generating a three-dimensional hollow spot with sub-diffraction transverse size by a focused cylindrical vector wave,” Opt. Express 26, 7866–7875 (2018). [CrossRef]  

28. Z. Wu, Q. Zhang, X. Jiang, Z. Wen, G. Liang, Z. Zhang, Z. Shang, and G. Chen, “Broadband integrated metalens for creating superoscillation 3D hollow spot by independent control of azimuthally and radially polarized waves,” J. Phys. D 52, 415103 (2019). [CrossRef]  

29. Z. Wu, K. Zhang, S. Zhang, Q. Jin, Z. Wen, L. Wang, L. Dai, Z. Zhang, H. Chen, G. Liang, Y. Liu, and G. Chen, “Optimization-free approach for generating sub-diffraction quasi-non-diffracting beams,” Opt. Express 26, 16585–16599 (2018). [CrossRef]  

30. W. T. Tang, E. Y. S. Yew, and J. R. Sheppard, “Polarization conversion in confocal microscopy with radially polarized illumination,” Opt. Lett. 34, 2147–2149 (2009). [CrossRef]  

31. P. W. Meng, S. Pereira, and P. Urbach, “Confocal microscopy with a radially polarized focused beam,” Opt. Express 26, 29600–29613 (2018). [CrossRef]  

32. Y. Kozawa, R. Sakashita, Y. Uesugi, and S. Sato, “Imaging with a longitudinal electric field in confocal laser scanning microscopy to enhance spatial resolution,” Opt. Express 28, 18418 (2020). [CrossRef]  

33. Y. Kozawa, D. Matsunaga, and S. Sato, “Superresolution imaging via superoscillation focusing of a radially polarized beam,” Optica 5, 86–92 (2018). [CrossRef]  

34. T. Wang, X. Wang, C. Kuang, X. Hao, and X. Liu, “Experimental verification of the far-field subwavelength focusing with multiple concentric nanorings,” Appl. Phys. Lett. 97, 231105 (2010). [CrossRef]  

35. Y. Liu, H. Xu, F. Stief, N. Zhitenev, and M. Yu, “Far-field superfocusing with an optical fiber based surface plasmonic lens made of nanoscale concentric annular slits,” Opt. Express 19, 20233–20243 (2011). [CrossRef]  

36. Z. Wu, Q. Jin, K. Zhang, Z. Zhang, F. Liang, Z. Wen, A. Yu, and G. Chen, “Binary-amplitude modulation based super-oscillatory focusing planar lens for azimuthally polarized wave,” Opto-Electron. Eng. 45, 170660 (2018). [CrossRef]  

37. G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2015). [CrossRef]  

38. F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015). [CrossRef]  

39. G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017). [CrossRef]  

40. S. Zhang, H. Chen, Z. Wu, K. Zhang, Y. Li, G. Chen, Z. Zhang, Z. Wen, L. Dai, and L. Wang, “Synthesis of sub-diffraction quasi-non-diffracting beams by angular spectrum compression,” Opt. Express 25, 27104–27118 (2017). [CrossRef]  

41. T. Grosjean and D. Courjon, “Polarization filtering induced by imaging systems: effect on image structure,” Phys. Rev. E 67, 046611 (2003). [CrossRef]  

42. T. Liu, S. Yang, and Z. Jiang, “Electromagnetic exploration of far-field superfocusing nanostructured metasurfaces,” Opt. Express 24, 16297–16308 (2016). [CrossRef]  

43. F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007). [CrossRef]  

44. C. Ecoffey and T. Grosjean, “Far-field mapping of the longitudinal magnetic and electric optical fields,” Opt. Lett. 38, 4974–4977 (2013). [CrossRef]  

45. L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

46. J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express 18, 21965–21972 (2010). [CrossRef]  

47. Y. Yu and Q. Zhan, “Optimization-free optical focal field engineering through reversing the radiation pattern from a uniform line source,” Opt. Express 23, 7527–7534 (2015). [CrossRef]  

48. M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011). [CrossRef]  

49. Z. Wu, F. Dong, S. Zhang, S. Yan, G. Liang, Z. Zhang, Z. Wen, G. Chen, L. Dai, and W. Chu, “Broadband dielectric metalens for polarization manipulating and superoscillation focusing of visible light,” ACS Photon. 7, 180–189 (2020). [CrossRef]  

50. F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. H. Kang, B. Jia, and M. Gu, “Polarization characterization in the focal volume of high numerical aperture objectives,” Opt. Express 18, 10813–10821 (2010).
    [Crossref]
  2. Y. Liu, D. Cline, and P. He, “Vacuum laser acceleration using a radially polarized CO2 laser beam,” Nucl. Instrum. Methods Phys. Res. 424, 296–303 (1999).
    [Crossref]
  3. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
    [Crossref]
  4. W. Liu, D. S. Dong, H. Yang, Q. H. Gong, and K. B. Shi, “Robust and high-speed rotation control in optical tweezers by using polarization synthesis based on heterodyne interference,” Opto-Electron. Adv. 3, 200022 (2020).
    [Crossref]
  5. D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368, 402–407 (2007).
    [Crossref]
  6. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [Crossref]
  7. N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6241 (2004).
    [Crossref]
  8. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
    [Crossref]
  9. M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
    [Crossref]
  10. H. Dehez, A. April, and M. Piché, “Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent,” Opt. Express 20, 14891–14905 (2012).
    [Crossref]
  11. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
    [Crossref]
  12. Z. Nie, G. Shi, X. Zhang, Y. Wang, and Y. Song, “Generation of super-resolution longitudinally polarized beam with ultra-long depth of focus using radially polarized hollow Gaussian beam,” Opt. Commun. 331, 87–93 (2014).
    [Crossref]
  13. V. Ravi, P. Suresh, K. B. Rajesh, Z. Jaroszewicz, P. M. Anbarasan, and T. V. S. Pillai, “Generation of sub-wavelength longitudinal magnetic probe using high numerical aperture lens axicon and binary phase plate,”J. Opt. 14, 055704 (2012).
    [Crossref]
  14. Y. Zha, J. Wei, H. Wang, and F. Gan, “Creation of an ultra-long depth of focus super-resolution longitudinally polarized beam with a ternary optical element,” J. Opt. 15, 075703 (2013).
    [Crossref]
  15. R. Zuo, W. Liu, H. Cheng, S. Chen, and J. Tian, “Breaking the diffraction limit with radially polarized light based on dielectric metalenses,” Adv. Opt. Mater. 6, 1800795 (2018).
    [Crossref]
  16. K. B. Rajesh, Z. Jaroszewicz, and P. M. Anbarasan, “Improvement of lens axicon’s performance for longitudinally polarized beam generation by adding a dedicated phase transmittance,” Opt. Express 18, 26799 (2010).
    [Crossref]
  17. Q. Li, X. Zhao, B. Zhang, Y. Zheng, L. Zhou, L. Wang, Y. Wu, and Z. Fang, “Nanofocusing of longitudinally polarized light using absorbance modulation,” Appl. Phys. Lett. 104, 061103 (2014).
    [Crossref]
  18. S. Wang, J. Xu, Y. Zhong, R. Ren, Y. Lu, H. Wan, J. Wang, and J. Ding, “Focus modulation of cylindrical vector beams through negative-index grating lenses,” Opt. Commun. 372, 245–249 (2016).
    [Crossref]
  19. Z. Nie, W. Ding, D. Li, X. Zhang, Y. Wang, and Y. Song, “Spherical and sub-wavelength longitudinal magnetization generated by 4π tightly focusing radially polarized vortex beams,” Opt. Express 23, 690–701 (2015).
    [Crossref]
  20. G. Chen, Z. Q. Wen, and C. W. Qiu, “Superoscillation: from physics to optical applications,” Light: Sci. Appl. 8, 56 (2019).
    [Crossref]
  21. M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
    [Crossref]
  22. E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012).
    [Crossref]
  23. F. Qin, K. Huang, J. F. Wu, J. H. Teng, C. W. Qiu, and M. H. Hong, “A supercritical lens optical label-free microscopy: sub-diffraction resolution and ultra-long working distance,” Adv. Mater. 29, 1602721 (2017).
    [Crossref]
  24. G. H. Yuan, K. S. Rogers, E. T. F. Rogers, and N. I. Zheludev, “Far-field superoscillatory metamaterial superlens,” Phys. Rev. Appl. 11, 064016 (2019).
    [Crossref]
  25. A. Yu, G. Chen, Z. Zhang, Z. Wen, L. Dai, K. Zhang, S. Jiang, Z. Wu, Y. Li, C. Wang, and X. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6, 38859 (2016).
    [Crossref]
  26. Y. Zhou and M. H. Hong, “Formation of polarization-dependent optical vortex beams via an engineered microsphere,” Opt. Express 29, 11121–11131 (2021).
    [Crossref]
  27. Z. Wu, Q. Jin, S. Zhang, K. Zhang, L. Wang, L. Dai, Z. Wen, Z. Zhang, G. Liang, Y. Liu, and G. Chen, “Generating a three-dimensional hollow spot with sub-diffraction transverse size by a focused cylindrical vector wave,” Opt. Express 26, 7866–7875 (2018).
    [Crossref]
  28. Z. Wu, Q. Zhang, X. Jiang, Z. Wen, G. Liang, Z. Zhang, Z. Shang, and G. Chen, “Broadband integrated metalens for creating superoscillation 3D hollow spot by independent control of azimuthally and radially polarized waves,” J. Phys. D 52, 415103 (2019).
    [Crossref]
  29. Z. Wu, K. Zhang, S. Zhang, Q. Jin, Z. Wen, L. Wang, L. Dai, Z. Zhang, H. Chen, G. Liang, Y. Liu, and G. Chen, “Optimization-free approach for generating sub-diffraction quasi-non-diffracting beams,” Opt. Express 26, 16585–16599 (2018).
    [Crossref]
  30. W. T. Tang, E. Y. S. Yew, and J. R. Sheppard, “Polarization conversion in confocal microscopy with radially polarized illumination,” Opt. Lett. 34, 2147–2149 (2009).
    [Crossref]
  31. P. W. Meng, S. Pereira, and P. Urbach, “Confocal microscopy with a radially polarized focused beam,” Opt. Express 26, 29600–29613 (2018).
    [Crossref]
  32. Y. Kozawa, R. Sakashita, Y. Uesugi, and S. Sato, “Imaging with a longitudinal electric field in confocal laser scanning microscopy to enhance spatial resolution,” Opt. Express 28, 18418 (2020).
    [Crossref]
  33. Y. Kozawa, D. Matsunaga, and S. Sato, “Superresolution imaging via superoscillation focusing of a radially polarized beam,” Optica 5, 86–92 (2018).
    [Crossref]
  34. T. Wang, X. Wang, C. Kuang, X. Hao, and X. Liu, “Experimental verification of the far-field subwavelength focusing with multiple concentric nanorings,” Appl. Phys. Lett. 97, 231105 (2010).
    [Crossref]
  35. Y. Liu, H. Xu, F. Stief, N. Zhitenev, and M. Yu, “Far-field superfocusing with an optical fiber based surface plasmonic lens made of nanoscale concentric annular slits,” Opt. Express 19, 20233–20243 (2011).
    [Crossref]
  36. Z. Wu, Q. Jin, K. Zhang, Z. Zhang, F. Liang, Z. Wen, A. Yu, and G. Chen, “Binary-amplitude modulation based super-oscillatory focusing planar lens for azimuthally polarized wave,” Opto-Electron. Eng. 45, 170660 (2018).
    [Crossref]
  37. G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2015).
    [Crossref]
  38. F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
    [Crossref]
  39. G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017).
    [Crossref]
  40. S. Zhang, H. Chen, Z. Wu, K. Zhang, Y. Li, G. Chen, Z. Zhang, Z. Wen, L. Dai, and L. Wang, “Synthesis of sub-diffraction quasi-non-diffracting beams by angular spectrum compression,” Opt. Express 25, 27104–27118 (2017).
    [Crossref]
  41. T. Grosjean and D. Courjon, “Polarization filtering induced by imaging systems: effect on image structure,” Phys. Rev. E 67, 046611 (2003).
    [Crossref]
  42. T. Liu, S. Yang, and Z. Jiang, “Electromagnetic exploration of far-field superfocusing nanostructured metasurfaces,” Opt. Express 24, 16297–16308 (2016).
    [Crossref]
  43. F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
    [Crossref]
  44. C. Ecoffey and T. Grosjean, “Far-field mapping of the longitudinal magnetic and electric optical fields,” Opt. Lett. 38, 4974–4977 (2013).
    [Crossref]
  45. L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).
  46. J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express 18, 21965–21972 (2010).
    [Crossref]
  47. Y. Yu and Q. Zhan, “Optimization-free optical focal field engineering through reversing the radiation pattern from a uniform line source,” Opt. Express 23, 7527–7534 (2015).
    [Crossref]
  48. M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
    [Crossref]
  49. Z. Wu, F. Dong, S. Zhang, S. Yan, G. Liang, Z. Zhang, Z. Wen, G. Chen, L. Dai, and W. Chu, “Broadband dielectric metalens for polarization manipulating and superoscillation focusing of visible light,” ACS Photon. 7, 180–189 (2020).
    [Crossref]
  50. F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
    [Crossref]

2021 (1)

2020 (3)

Y. Kozawa, R. Sakashita, Y. Uesugi, and S. Sato, “Imaging with a longitudinal electric field in confocal laser scanning microscopy to enhance spatial resolution,” Opt. Express 28, 18418 (2020).
[Crossref]

W. Liu, D. S. Dong, H. Yang, Q. H. Gong, and K. B. Shi, “Robust and high-speed rotation control in optical tweezers by using polarization synthesis based on heterodyne interference,” Opto-Electron. Adv. 3, 200022 (2020).
[Crossref]

Z. Wu, F. Dong, S. Zhang, S. Yan, G. Liang, Z. Zhang, Z. Wen, G. Chen, L. Dai, and W. Chu, “Broadband dielectric metalens for polarization manipulating and superoscillation focusing of visible light,” ACS Photon. 7, 180–189 (2020).
[Crossref]

2019 (4)

Z. Wu, Q. Zhang, X. Jiang, Z. Wen, G. Liang, Z. Zhang, Z. Shang, and G. Chen, “Broadband integrated metalens for creating superoscillation 3D hollow spot by independent control of azimuthally and radially polarized waves,” J. Phys. D 52, 415103 (2019).
[Crossref]

G. H. Yuan, K. S. Rogers, E. T. F. Rogers, and N. I. Zheludev, “Far-field superoscillatory metamaterial superlens,” Phys. Rev. Appl. 11, 064016 (2019).
[Crossref]

G. Chen, Z. Q. Wen, and C. W. Qiu, “Superoscillation: from physics to optical applications,” Light: Sci. Appl. 8, 56 (2019).
[Crossref]

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

2018 (6)

2017 (3)

F. Qin, K. Huang, J. F. Wu, J. H. Teng, C. W. Qiu, and M. H. Hong, “A supercritical lens optical label-free microscopy: sub-diffraction resolution and ultra-long working distance,” Adv. Mater. 29, 1602721 (2017).
[Crossref]

G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017).
[Crossref]

S. Zhang, H. Chen, Z. Wu, K. Zhang, Y. Li, G. Chen, Z. Zhang, Z. Wen, L. Dai, and L. Wang, “Synthesis of sub-diffraction quasi-non-diffracting beams by angular spectrum compression,” Opt. Express 25, 27104–27118 (2017).
[Crossref]

2016 (3)

T. Liu, S. Yang, and Z. Jiang, “Electromagnetic exploration of far-field superfocusing nanostructured metasurfaces,” Opt. Express 24, 16297–16308 (2016).
[Crossref]

A. Yu, G. Chen, Z. Zhang, Z. Wen, L. Dai, K. Zhang, S. Jiang, Z. Wu, Y. Li, C. Wang, and X. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6, 38859 (2016).
[Crossref]

S. Wang, J. Xu, Y. Zhong, R. Ren, Y. Lu, H. Wan, J. Wang, and J. Ding, “Focus modulation of cylindrical vector beams through negative-index grating lenses,” Opt. Commun. 372, 245–249 (2016).
[Crossref]

2015 (4)

Z. Nie, W. Ding, D. Li, X. Zhang, Y. Wang, and Y. Song, “Spherical and sub-wavelength longitudinal magnetization generated by 4π tightly focusing radially polarized vortex beams,” Opt. Express 23, 690–701 (2015).
[Crossref]

G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2015).
[Crossref]

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref]

Y. Yu and Q. Zhan, “Optimization-free optical focal field engineering through reversing the radiation pattern from a uniform line source,” Opt. Express 23, 7527–7534 (2015).
[Crossref]

2014 (2)

Q. Li, X. Zhao, B. Zhang, Y. Zheng, L. Zhou, L. Wang, Y. Wu, and Z. Fang, “Nanofocusing of longitudinally polarized light using absorbance modulation,” Appl. Phys. Lett. 104, 061103 (2014).
[Crossref]

Z. Nie, G. Shi, X. Zhang, Y. Wang, and Y. Song, “Generation of super-resolution longitudinally polarized beam with ultra-long depth of focus using radially polarized hollow Gaussian beam,” Opt. Commun. 331, 87–93 (2014).
[Crossref]

2013 (2)

Y. Zha, J. Wei, H. Wang, and F. Gan, “Creation of an ultra-long depth of focus super-resolution longitudinally polarized beam with a ternary optical element,” J. Opt. 15, 075703 (2013).
[Crossref]

C. Ecoffey and T. Grosjean, “Far-field mapping of the longitudinal magnetic and electric optical fields,” Opt. Lett. 38, 4974–4977 (2013).
[Crossref]

2012 (3)

V. Ravi, P. Suresh, K. B. Rajesh, Z. Jaroszewicz, P. M. Anbarasan, and T. V. S. Pillai, “Generation of sub-wavelength longitudinal magnetic probe using high numerical aperture lens axicon and binary phase plate,”J. Opt. 14, 055704 (2012).
[Crossref]

H. Dehez, A. April, and M. Piché, “Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent,” Opt. Express 20, 14891–14905 (2012).
[Crossref]

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012).
[Crossref]

2011 (2)

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[Crossref]

Y. Liu, H. Xu, F. Stief, N. Zhitenev, and M. Yu, “Far-field superfocusing with an optical fiber based surface plasmonic lens made of nanoscale concentric annular slits,” Opt. Express 19, 20233–20243 (2011).
[Crossref]

2010 (4)

2009 (1)

2008 (2)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

2007 (3)

F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368, 402–407 (2007).
[Crossref]

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
[Crossref]

2004 (2)

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
[Crossref]

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6241 (2004).
[Crossref]

2003 (1)

T. Grosjean and D. Courjon, “Polarization filtering induced by imaging systems: effect on image structure,” Phys. Rev. E 67, 046611 (2003).
[Crossref]

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

1999 (2)

Y. Liu, D. Cline, and P. He, “Vacuum laser acceleration using a radially polarized CO2 laser beam,” Nucl. Instrum. Methods Phys. Res. 424, 296–303 (1999).
[Crossref]

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[Crossref]

Adamo, G.

G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2015).
[Crossref]

Aharonov, Y.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Anbarasan, P. M.

V. Ravi, P. Suresh, K. B. Rajesh, Z. Jaroszewicz, P. M. Anbarasan, and T. V. S. Pillai, “Generation of sub-wavelength longitudinal magnetic probe using high numerical aperture lens axicon and binary phase plate,”J. Opt. 14, 055704 (2012).
[Crossref]

K. B. Rajesh, Z. Jaroszewicz, and P. M. Anbarasan, “Improvement of lens axicon’s performance for longitudinally polarized beam generation by adding a dedicated phase transmittance,” Opt. Express 18, 26799 (2010).
[Crossref]

April, A.

Arie, A.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Bahabad, A.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Beresna, M.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[Crossref]

Berry, M.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

Chad, J. E.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012).
[Crossref]

Chen, G.

Z. Wu, F. Dong, S. Zhang, S. Yan, G. Liang, Z. Zhang, Z. Wen, G. Chen, L. Dai, and W. Chu, “Broadband dielectric metalens for polarization manipulating and superoscillation focusing of visible light,” ACS Photon. 7, 180–189 (2020).
[Crossref]

Z. Wu, Q. Zhang, X. Jiang, Z. Wen, G. Liang, Z. Zhang, Z. Shang, and G. Chen, “Broadband integrated metalens for creating superoscillation 3D hollow spot by independent control of azimuthally and radially polarized waves,” J. Phys. D 52, 415103 (2019).
[Crossref]

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

G. Chen, Z. Q. Wen, and C. W. Qiu, “Superoscillation: from physics to optical applications,” Light: Sci. Appl. 8, 56 (2019).
[Crossref]

Z. Wu, K. Zhang, S. Zhang, Q. Jin, Z. Wen, L. Wang, L. Dai, Z. Zhang, H. Chen, G. Liang, Y. Liu, and G. Chen, “Optimization-free approach for generating sub-diffraction quasi-non-diffracting beams,” Opt. Express 26, 16585–16599 (2018).
[Crossref]

Z. Wu, Q. Jin, S. Zhang, K. Zhang, L. Wang, L. Dai, Z. Wen, Z. Zhang, G. Liang, Y. Liu, and G. Chen, “Generating a three-dimensional hollow spot with sub-diffraction transverse size by a focused cylindrical vector wave,” Opt. Express 26, 7866–7875 (2018).
[Crossref]

Z. Wu, Q. Jin, K. Zhang, Z. Zhang, F. Liang, Z. Wen, A. Yu, and G. Chen, “Binary-amplitude modulation based super-oscillatory focusing planar lens for azimuthally polarized wave,” Opto-Electron. Eng. 45, 170660 (2018).
[Crossref]

G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017).
[Crossref]

S. Zhang, H. Chen, Z. Wu, K. Zhang, Y. Li, G. Chen, Z. Zhang, Z. Wen, L. Dai, and L. Wang, “Synthesis of sub-diffraction quasi-non-diffracting beams by angular spectrum compression,” Opt. Express 25, 27104–27118 (2017).
[Crossref]

A. Yu, G. Chen, Z. Zhang, Z. Wen, L. Dai, K. Zhang, S. Jiang, Z. Wu, Y. Li, C. Wang, and X. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6, 38859 (2016).
[Crossref]

Chen, H.

Chen, S.

R. Zuo, W. Liu, H. Cheng, S. Chen, and J. Tian, “Breaking the diffraction limit with radially polarized light based on dielectric metalenses,” Adv. Opt. Mater. 6, 1800795 (2018).
[Crossref]

Chen, W.

Chen, Y.

F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

Cheng, H.

R. Zuo, W. Liu, H. Cheng, S. Chen, and J. Tian, “Breaking the diffraction limit with radially polarized light based on dielectric metalenses,” Adv. Opt. Mater. 6, 1800795 (2018).
[Crossref]

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Chu, W.

Z. Wu, F. Dong, S. Zhang, S. Yan, G. Liang, Z. Zhang, Z. Wen, G. Chen, L. Dai, and W. Chu, “Broadband dielectric metalens for polarization manipulating and superoscillation focusing of visible light,” ACS Photon. 7, 180–189 (2020).
[Crossref]

Cline, D.

Y. Liu, D. Cline, and P. He, “Vacuum laser acceleration using a radially polarized CO2 laser beam,” Nucl. Instrum. Methods Phys. Res. 424, 296–303 (1999).
[Crossref]

Colombo, F.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Courjon, D.

T. Grosjean and D. Courjon, “Polarization filtering induced by imaging systems: effect on image structure,” Phys. Rev. E 67, 046611 (2003).
[Crossref]

Dai, L.

Z. Wu, F. Dong, S. Zhang, S. Yan, G. Liang, Z. Zhang, Z. Wen, G. Chen, L. Dai, and W. Chu, “Broadband dielectric metalens for polarization manipulating and superoscillation focusing of visible light,” ACS Photon. 7, 180–189 (2020).
[Crossref]

Z. Wu, K. Zhang, S. Zhang, Q. Jin, Z. Wen, L. Wang, L. Dai, Z. Zhang, H. Chen, G. Liang, Y. Liu, and G. Chen, “Optimization-free approach for generating sub-diffraction quasi-non-diffracting beams,” Opt. Express 26, 16585–16599 (2018).
[Crossref]

Z. Wu, Q. Jin, S. Zhang, K. Zhang, L. Wang, L. Dai, Z. Wen, Z. Zhang, G. Liang, Y. Liu, and G. Chen, “Generating a three-dimensional hollow spot with sub-diffraction transverse size by a focused cylindrical vector wave,” Opt. Express 26, 7866–7875 (2018).
[Crossref]

S. Zhang, H. Chen, Z. Wu, K. Zhang, Y. Li, G. Chen, Z. Zhang, Z. Wen, L. Dai, and L. Wang, “Synthesis of sub-diffraction quasi-non-diffracting beams by angular spectrum compression,” Opt. Express 25, 27104–27118 (2017).
[Crossref]

G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017).
[Crossref]

A. Yu, G. Chen, Z. Zhang, Z. Wen, L. Dai, K. Zhang, S. Jiang, Z. Wu, Y. Li, C. Wang, and X. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6, 38859 (2016).
[Crossref]

Dehez, H.

Dennis, M. R.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012).
[Crossref]

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

Ding, J.

S. Wang, J. Xu, Y. Zhong, R. Ren, Y. Lu, H. Wan, J. Wang, and J. Ding, “Focus modulation of cylindrical vector beams through negative-index grating lenses,” Opt. Commun. 372, 245–249 (2016).
[Crossref]

Ding, W.

Dong, D. S.

W. Liu, D. S. Dong, H. Yang, Q. H. Gong, and K. B. Shi, “Robust and high-speed rotation control in optical tweezers by using polarization synthesis based on heterodyne interference,” Opto-Electron. Adv. 3, 200022 (2020).
[Crossref]

Dong, F.

Z. Wu, F. Dong, S. Zhang, S. Yan, G. Liang, Z. Zhang, Z. Wen, G. Chen, L. Dai, and W. Chu, “Broadband dielectric metalens for polarization manipulating and superoscillation focusing of visible light,” ACS Photon. 7, 180–189 (2020).
[Crossref]

Ecoffey, C.

Eleftheriades, G. V.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Eliezer, Y.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Fang, Z.

Q. Li, X. Zhao, B. Zhang, Y. Zheng, L. Zhou, L. Wang, Y. Wu, and Z. Fang, “Nanofocusing of longitudinally polarized light using absorbance modulation,” Appl. Phys. Lett. 104, 061103 (2014).
[Crossref]

Feurer, T.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
[Crossref]

Flossmann, F.

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

Gan, F.

Y. Zha, J. Wei, H. Wang, and F. Gan, “Creation of an ultra-long depth of focus super-resolution longitudinally polarized beam with a ternary optical element,” J. Opt. 15, 075703 (2013).
[Crossref]

Garcia de Abajo, F. J.

F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

Gecevicius, M.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[Crossref]

Gertus, T.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[Crossref]

Gong, Q. H.

W. Liu, D. S. Dong, H. Yang, Q. H. Gong, and K. B. Shi, “Robust and high-speed rotation control in optical tweezers by using polarization synthesis based on heterodyne interference,” Opto-Electron. Adv. 3, 200022 (2020).
[Crossref]

Götte, J. B.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Grosjean, T.

C. Ecoffey and T. Grosjean, “Far-field mapping of the longitudinal magnetic and electric optical fields,” Opt. Lett. 38, 4974–4977 (2013).
[Crossref]

T. Grosjean and D. Courjon, “Polarization filtering induced by imaging systems: effect on image structure,” Phys. Rev. E 67, 046611 (2003).
[Crossref]

Gu, M.

Gupta, D. N.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368, 402–407 (2007).
[Crossref]

Hao, C.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Hao, X.

T. Wang, X. Wang, C. Kuang, X. Hao, and X. Liu, “Experimental verification of the far-field subwavelength focusing with multiple concentric nanorings,” Appl. Phys. Lett. 97, 231105 (2010).
[Crossref]

Hayazawa, N.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6241 (2004).
[Crossref]

He, P.

Y. Liu, D. Cline, and P. He, “Vacuum laser acceleration using a radially polarized CO2 laser beam,” Nucl. Instrum. Methods Phys. Res. 424, 296–303 (1999).
[Crossref]

He, Y.

G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017).
[Crossref]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

Hong, M.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref]

Hong, M. H.

Y. Zhou and M. H. Hong, “Formation of polarization-dependent optical vortex beams via an engineered microsphere,” Opt. Express 29, 11121–11131 (2021).
[Crossref]

F. Qin, K. Huang, J. F. Wu, J. H. Teng, C. W. Qiu, and M. H. Hong, “A supercritical lens optical label-free microscopy: sub-diffraction resolution and ultra-long working distance,” Adv. Mater. 29, 1602721 (2017).
[Crossref]

Huang, F. M.

F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

Huang, K.

F. Qin, K. Huang, J. F. Wu, J. H. Teng, C. W. Qiu, and M. H. Hong, “A supercritical lens optical label-free microscopy: sub-diffraction resolution and ultra-long working distance,” Adv. Mater. 29, 1602721 (2017).
[Crossref]

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref]

Jaroszewicz, Z.

V. Ravi, P. Suresh, K. B. Rajesh, Z. Jaroszewicz, P. M. Anbarasan, and T. V. S. Pillai, “Generation of sub-wavelength longitudinal magnetic probe using high numerical aperture lens axicon and binary phase plate,”J. Opt. 14, 055704 (2012).
[Crossref]

K. B. Rajesh, Z. Jaroszewicz, and P. M. Anbarasan, “Improvement of lens axicon’s performance for longitudinally polarized beam generation by adding a dedicated phase transmittance,” Opt. Express 18, 26799 (2010).
[Crossref]

Jia, B.

Jiang, S.

G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017).
[Crossref]

A. Yu, G. Chen, Z. Zhang, Z. Wen, L. Dai, K. Zhang, S. Jiang, Z. Wu, Y. Li, C. Wang, and X. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6, 38859 (2016).
[Crossref]

Jiang, X.

Z. Wu, Q. Zhang, X. Jiang, Z. Wen, G. Liang, Z. Zhang, Z. Shang, and G. Chen, “Broadband integrated metalens for creating superoscillation 3D hollow spot by independent control of azimuthally and radially polarized waves,” J. Phys. D 52, 415103 (2019).
[Crossref]

Jiang, Z.

Jiao, J.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref]

Jin, Q.

Kang, H.

Kant, N.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368, 402–407 (2007).
[Crossref]

Katzav, E.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Kawata, S.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6241 (2004).
[Crossref]

Kazansky, P. G.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[Crossref]

Kempf, A.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Kim, D. E.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368, 402–407 (2007).
[Crossref]

Kozawa, Y.

Kuang, C.

T. Wang, X. Wang, C. Kuang, X. Hao, and X. Liu, “Experimental verification of the far-field subwavelength focusing with multiple concentric nanorings,” Appl. Phys. Lett. 97, 231105 (2010).
[Crossref]

Li, D.

Li, Q.

Q. Li, X. Zhao, B. Zhang, Y. Zheng, L. Zhou, L. Wang, Y. Wu, and Z. Fang, “Nanofocusing of longitudinally polarized light using absorbance modulation,” Appl. Phys. Lett. 104, 061103 (2014).
[Crossref]

Li, Y.

S. Zhang, H. Chen, Z. Wu, K. Zhang, Y. Li, G. Chen, Z. Zhang, Z. Wen, L. Dai, and L. Wang, “Synthesis of sub-diffraction quasi-non-diffracting beams by angular spectrum compression,” Opt. Express 25, 27104–27118 (2017).
[Crossref]

A. Yu, G. Chen, Z. Zhang, Z. Wen, L. Dai, K. Zhang, S. Jiang, Z. Wu, Y. Li, C. Wang, and X. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6, 38859 (2016).
[Crossref]

Liang, F.

Z. Wu, Q. Jin, K. Zhang, Z. Zhang, F. Liang, Z. Wen, A. Yu, and G. Chen, “Binary-amplitude modulation based super-oscillatory focusing planar lens for azimuthally polarized wave,” Opto-Electron. Eng. 45, 170660 (2018).
[Crossref]

Liang, G.

Z. Wu, F. Dong, S. Zhang, S. Yan, G. Liang, Z. Zhang, Z. Wen, G. Chen, L. Dai, and W. Chu, “Broadband dielectric metalens for polarization manipulating and superoscillation focusing of visible light,” ACS Photon. 7, 180–189 (2020).
[Crossref]

Z. Wu, Q. Zhang, X. Jiang, Z. Wen, G. Liang, Z. Zhang, Z. Shang, and G. Chen, “Broadband integrated metalens for creating superoscillation 3D hollow spot by independent control of azimuthally and radially polarized waves,” J. Phys. D 52, 415103 (2019).
[Crossref]

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Z. Wu, Q. Jin, S. Zhang, K. Zhang, L. Wang, L. Dai, Z. Wen, Z. Zhang, G. Liang, Y. Liu, and G. Chen, “Generating a three-dimensional hollow spot with sub-diffraction transverse size by a focused cylindrical vector wave,” Opt. Express 26, 7866–7875 (2018).
[Crossref]

Z. Wu, K. Zhang, S. Zhang, Q. Jin, Z. Wen, L. Wang, L. Dai, Z. Zhang, H. Chen, G. Liang, Y. Liu, and G. Chen, “Optimization-free approach for generating sub-diffraction quasi-non-diffracting beams,” Opt. Express 26, 16585–16599 (2018).
[Crossref]

Lindberg, J.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012).
[Crossref]

Liu, T.

Liu, W.

W. Liu, D. S. Dong, H. Yang, Q. H. Gong, and K. B. Shi, “Robust and high-speed rotation control in optical tweezers by using polarization synthesis based on heterodyne interference,” Opto-Electron. Adv. 3, 200022 (2020).
[Crossref]

R. Zuo, W. Liu, H. Cheng, S. Chen, and J. Tian, “Breaking the diffraction limit with radially polarized light based on dielectric metalenses,” Adv. Opt. Mater. 6, 1800795 (2018).
[Crossref]

Liu, X.

T. Wang, X. Wang, C. Kuang, X. Hao, and X. Liu, “Experimental verification of the far-field subwavelength focusing with multiple concentric nanorings,” Appl. Phys. Lett. 97, 231105 (2010).
[Crossref]

Liu, Y.

Lu, Y.

S. Wang, J. Xu, Y. Zhong, R. Ren, Y. Lu, H. Wan, J. Wang, and J. Ding, “Focus modulation of cylindrical vector beams through negative-index grating lenses,” Opt. Commun. 372, 245–249 (2016).
[Crossref]

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Luo, X.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017).
[Crossref]

A. Yu, G. Chen, Z. Zhang, Z. Wen, L. Dai, K. Zhang, S. Jiang, Z. Wu, Y. Li, C. Wang, and X. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6, 38859 (2016).
[Crossref]

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref]

Matsunaga, D.

Meier, M.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
[Crossref]

Meng, P. W.

Nesterov, A. V.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[Crossref]

Nie, Z.

Z. Nie, W. Ding, D. Li, X. Zhang, Y. Wang, and Y. Song, “Spherical and sub-wavelength longitudinal magnetization generated by 4π tightly focusing radially polarized vortex beams,” Opt. Express 23, 690–701 (2015).
[Crossref]

Z. Nie, G. Shi, X. Zhang, Y. Wang, and Y. Song, “Generation of super-resolution longitudinally polarized beam with ultra-long depth of focus using radially polarized hollow Gaussian beam,” Opt. Commun. 331, 87–93 (2014).
[Crossref]

Niziev, V. G.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[Crossref]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

O’Holleran, K.

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

Padgett, M. J.

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

Pereira, S.

Piché, M.

Pillai, T. V. S.

V. Ravi, P. Suresh, K. B. Rajesh, Z. Jaroszewicz, P. M. Anbarasan, and T. V. S. Pillai, “Generation of sub-wavelength longitudinal magnetic probe using high numerical aperture lens axicon and binary phase plate,”J. Opt. 14, 055704 (2012).
[Crossref]

Qin, F.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

F. Qin, K. Huang, J. F. Wu, J. H. Teng, C. W. Qiu, and M. H. Hong, “A supercritical lens optical label-free microscopy: sub-diffraction resolution and ultra-long working distance,” Adv. Mater. 29, 1602721 (2017).
[Crossref]

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref]

Qiu, C.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref]

Qiu, C. W.

G. Chen, Z. Q. Wen, and C. W. Qiu, “Superoscillation: from physics to optical applications,” Light: Sci. Appl. 8, 56 (2019).
[Crossref]

F. Qin, K. Huang, J. F. Wu, J. H. Teng, C. W. Qiu, and M. H. Hong, “A supercritical lens optical label-free microscopy: sub-diffraction resolution and ultra-long working distance,” Adv. Mater. 29, 1602721 (2017).
[Crossref]

Qiu, C.-W.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Rajesh, K. B.

V. Ravi, P. Suresh, K. B. Rajesh, Z. Jaroszewicz, P. M. Anbarasan, and T. V. S. Pillai, “Generation of sub-wavelength longitudinal magnetic probe using high numerical aperture lens axicon and binary phase plate,”J. Opt. 14, 055704 (2012).
[Crossref]

K. B. Rajesh, Z. Jaroszewicz, and P. M. Anbarasan, “Improvement of lens axicon’s performance for longitudinally polarized beam generation by adding a dedicated phase transmittance,” Opt. Express 18, 26799 (2010).
[Crossref]

Ravi, V.

V. Ravi, P. Suresh, K. B. Rajesh, Z. Jaroszewicz, P. M. Anbarasan, and T. V. S. Pillai, “Generation of sub-wavelength longitudinal magnetic probe using high numerical aperture lens axicon and binary phase plate,”J. Opt. 14, 055704 (2012).
[Crossref]

Remez, R.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Ren, R.

S. Wang, J. Xu, Y. Zhong, R. Ren, Y. Lu, H. Wan, J. Wang, and J. Ding, “Focus modulation of cylindrical vector beams through negative-index grating lenses,” Opt. Commun. 372, 245–249 (2016).
[Crossref]

Rogers, E. T. F.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

G. H. Yuan, K. S. Rogers, E. T. F. Rogers, and N. I. Zheludev, “Far-field superoscillatory metamaterial superlens,” Phys. Rev. Appl. 11, 064016 (2019).
[Crossref]

G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2015).
[Crossref]

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012).
[Crossref]

Rogers, K. S.

G. H. Yuan, K. S. Rogers, E. T. F. Rogers, and N. I. Zheludev, “Far-field superoscillatory metamaterial superlens,” Phys. Rev. Appl. 11, 064016 (2019).
[Crossref]

Romano, V.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
[Crossref]

Roy, T.

G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2015).
[Crossref]

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012).
[Crossref]

Sabadini, I.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Saito, Y.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6241 (2004).
[Crossref]

Sakashita, R.

Sato, S.

Savo, S.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012).
[Crossref]

Schwartz, M.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Shang, Z.

Z. Wu, Q. Zhang, X. Jiang, Z. Wen, G. Liang, Z. Zhang, Z. Shang, and G. Chen, “Broadband integrated metalens for creating superoscillation 3D hollow spot by independent control of azimuthally and radially polarized waves,” J. Phys. D 52, 415103 (2019).
[Crossref]

Shen, Z.

G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2015).
[Crossref]

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Sheppard, J. R.

Shi, G.

Z. Nie, G. Shi, X. Zhang, Y. Wang, and Y. Song, “Generation of super-resolution longitudinally polarized beam with ultra-long depth of focus using radially polarized hollow Gaussian beam,” Opt. Commun. 331, 87–93 (2014).
[Crossref]

Shi, K. B.

W. Liu, D. S. Dong, H. Yang, Q. H. Gong, and K. B. Shi, “Robust and high-speed rotation control in optical tweezers by using polarization synthesis based on heterodyne interference,” Opto-Electron. Adv. 3, 200022 (2020).
[Crossref]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Song, Y.

Z. Nie, W. Ding, D. Li, X. Zhang, Y. Wang, and Y. Song, “Spherical and sub-wavelength longitudinal magnetization generated by 4π tightly focusing radially polarized vortex beams,” Opt. Express 23, 690–701 (2015).
[Crossref]

Z. Nie, G. Shi, X. Zhang, Y. Wang, and Y. Song, “Generation of super-resolution longitudinally polarized beam with ultra-long depth of focus using radially polarized hollow Gaussian beam,” Opt. Commun. 331, 87–93 (2014).
[Crossref]

Stief, F.

Struppa, D. C.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Suk, H.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368, 402–407 (2007).
[Crossref]

Suresh, P.

V. Ravi, P. Suresh, K. B. Rajesh, Z. Jaroszewicz, P. M. Anbarasan, and T. V. S. Pillai, “Generation of sub-wavelength longitudinal magnetic probe using high numerical aperture lens axicon and binary phase plate,”J. Opt. 14, 055704 (2012).
[Crossref]

Tang, W. T.

Teng, J. H.

F. Qin, K. Huang, J. F. Wu, J. H. Teng, C. W. Qiu, and M. H. Hong, “A supercritical lens optical label-free microscopy: sub-diffraction resolution and ultra-long working distance,” Adv. Mater. 29, 1602721 (2017).
[Crossref]

Tian, J.

R. Zuo, W. Liu, H. Cheng, S. Chen, and J. Tian, “Breaking the diffraction limit with radially polarized light based on dielectric metalenses,” Adv. Opt. Mater. 6, 1800795 (2018).
[Crossref]

Tollaksen, J.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Uesugi, Y.

Urbach, P.

Wan, H.

S. Wang, J. Xu, Y. Zhong, R. Ren, Y. Lu, H. Wan, J. Wang, and J. Ding, “Focus modulation of cylindrical vector beams through negative-index grating lenses,” Opt. Commun. 372, 245–249 (2016).
[Crossref]

Wang, C.

G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017).
[Crossref]

A. Yu, G. Chen, Z. Zhang, Z. Wen, L. Dai, K. Zhang, S. Jiang, Z. Wu, Y. Li, C. Wang, and X. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6, 38859 (2016).
[Crossref]

Wang, H.

Y. Zha, J. Wei, H. Wang, and F. Gan, “Creation of an ultra-long depth of focus super-resolution longitudinally polarized beam with a ternary optical element,” J. Opt. 15, 075703 (2013).
[Crossref]

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Wang, J.

S. Wang, J. Xu, Y. Zhong, R. Ren, Y. Lu, H. Wan, J. Wang, and J. Ding, “Focus modulation of cylindrical vector beams through negative-index grating lenses,” Opt. Commun. 372, 245–249 (2016).
[Crossref]

J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express 18, 21965–21972 (2010).
[Crossref]

Wang, L.

Wang, S.

S. Wang, J. Xu, Y. Zhong, R. Ren, Y. Lu, H. Wan, J. Wang, and J. Ding, “Focus modulation of cylindrical vector beams through negative-index grating lenses,” Opt. Commun. 372, 245–249 (2016).
[Crossref]

Wang, T.

T. Wang, X. Wang, C. Kuang, X. Hao, and X. Liu, “Experimental verification of the far-field subwavelength focusing with multiple concentric nanorings,” Appl. Phys. Lett. 97, 231105 (2010).
[Crossref]

Wang, X.

T. Wang, X. Wang, C. Kuang, X. Hao, and X. Liu, “Experimental verification of the far-field subwavelength focusing with multiple concentric nanorings,” Appl. Phys. Lett. 97, 231105 (2010).
[Crossref]

Wang, Y.

Z. Nie, W. Ding, D. Li, X. Zhang, Y. Wang, and Y. Song, “Spherical and sub-wavelength longitudinal magnetization generated by 4π tightly focusing radially polarized vortex beams,” Opt. Express 23, 690–701 (2015).
[Crossref]

Z. Nie, G. Shi, X. Zhang, Y. Wang, and Y. Song, “Generation of super-resolution longitudinally polarized beam with ultra-long depth of focus using radially polarized hollow Gaussian beam,” Opt. Commun. 331, 87–93 (2014).
[Crossref]

Wei, J.

Y. Zha, J. Wei, H. Wang, and F. Gan, “Creation of an ultra-long depth of focus super-resolution longitudinally polarized beam with a ternary optical element,” J. Opt. 15, 075703 (2013).
[Crossref]

Wen, Z.

Z. Wu, F. Dong, S. Zhang, S. Yan, G. Liang, Z. Zhang, Z. Wen, G. Chen, L. Dai, and W. Chu, “Broadband dielectric metalens for polarization manipulating and superoscillation focusing of visible light,” ACS Photon. 7, 180–189 (2020).
[Crossref]

Z. Wu, Q. Zhang, X. Jiang, Z. Wen, G. Liang, Z. Zhang, Z. Shang, and G. Chen, “Broadband integrated metalens for creating superoscillation 3D hollow spot by independent control of azimuthally and radially polarized waves,” J. Phys. D 52, 415103 (2019).
[Crossref]

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Z. Wu, Q. Jin, S. Zhang, K. Zhang, L. Wang, L. Dai, Z. Wen, Z. Zhang, G. Liang, Y. Liu, and G. Chen, “Generating a three-dimensional hollow spot with sub-diffraction transverse size by a focused cylindrical vector wave,” Opt. Express 26, 7866–7875 (2018).
[Crossref]

Z. Wu, K. Zhang, S. Zhang, Q. Jin, Z. Wen, L. Wang, L. Dai, Z. Zhang, H. Chen, G. Liang, Y. Liu, and G. Chen, “Optimization-free approach for generating sub-diffraction quasi-non-diffracting beams,” Opt. Express 26, 16585–16599 (2018).
[Crossref]

Z. Wu, Q. Jin, K. Zhang, Z. Zhang, F. Liang, Z. Wen, A. Yu, and G. Chen, “Binary-amplitude modulation based super-oscillatory focusing planar lens for azimuthally polarized wave,” Opto-Electron. Eng. 45, 170660 (2018).
[Crossref]

S. Zhang, H. Chen, Z. Wu, K. Zhang, Y. Li, G. Chen, Z. Zhang, Z. Wen, L. Dai, and L. Wang, “Synthesis of sub-diffraction quasi-non-diffracting beams by angular spectrum compression,” Opt. Express 25, 27104–27118 (2017).
[Crossref]

G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017).
[Crossref]

A. Yu, G. Chen, Z. Zhang, Z. Wen, L. Dai, K. Zhang, S. Jiang, Z. Wu, Y. Li, C. Wang, and X. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6, 38859 (2016).
[Crossref]

Wen, Z. Q.

G. Chen, Z. Q. Wen, and C. W. Qiu, “Superoscillation: from physics to optical applications,” Light: Sci. Appl. 8, 56 (2019).
[Crossref]

Wong, A. M. H.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

Wu, J.

G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017).
[Crossref]

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref]

Wu, J. F.

F. Qin, K. Huang, J. F. Wu, J. H. Teng, C. W. Qiu, and M. H. Hong, “A supercritical lens optical label-free microscopy: sub-diffraction resolution and ultra-long working distance,” Adv. Mater. 29, 1602721 (2017).
[Crossref]

Wu, Y.

Q. Li, X. Zhao, B. Zhang, Y. Zheng, L. Zhou, L. Wang, Y. Wu, and Z. Fang, “Nanofocusing of longitudinally polarized light using absorbance modulation,” Appl. Phys. Lett. 104, 061103 (2014).
[Crossref]

Wu, Z.

Z. Wu, F. Dong, S. Zhang, S. Yan, G. Liang, Z. Zhang, Z. Wen, G. Chen, L. Dai, and W. Chu, “Broadband dielectric metalens for polarization manipulating and superoscillation focusing of visible light,” ACS Photon. 7, 180–189 (2020).
[Crossref]

Z. Wu, Q. Zhang, X. Jiang, Z. Wen, G. Liang, Z. Zhang, Z. Shang, and G. Chen, “Broadband integrated metalens for creating superoscillation 3D hollow spot by independent control of azimuthally and radially polarized waves,” J. Phys. D 52, 415103 (2019).
[Crossref]

Z. Wu, K. Zhang, S. Zhang, Q. Jin, Z. Wen, L. Wang, L. Dai, Z. Zhang, H. Chen, G. Liang, Y. Liu, and G. Chen, “Optimization-free approach for generating sub-diffraction quasi-non-diffracting beams,” Opt. Express 26, 16585–16599 (2018).
[Crossref]

Z. Wu, Q. Jin, K. Zhang, Z. Zhang, F. Liang, Z. Wen, A. Yu, and G. Chen, “Binary-amplitude modulation based super-oscillatory focusing planar lens for azimuthally polarized wave,” Opto-Electron. Eng. 45, 170660 (2018).
[Crossref]

Z. Wu, Q. Jin, S. Zhang, K. Zhang, L. Wang, L. Dai, Z. Wen, Z. Zhang, G. Liang, Y. Liu, and G. Chen, “Generating a three-dimensional hollow spot with sub-diffraction transverse size by a focused cylindrical vector wave,” Opt. Express 26, 7866–7875 (2018).
[Crossref]

S. Zhang, H. Chen, Z. Wu, K. Zhang, Y. Li, G. Chen, Z. Zhang, Z. Wen, L. Dai, and L. Wang, “Synthesis of sub-diffraction quasi-non-diffracting beams by angular spectrum compression,” Opt. Express 25, 27104–27118 (2017).
[Crossref]

G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017).
[Crossref]

A. Yu, G. Chen, Z. Zhang, Z. Wen, L. Dai, K. Zhang, S. Jiang, Z. Wu, Y. Li, C. Wang, and X. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6, 38859 (2016).
[Crossref]

Xu, H.

Xu, J.

S. Wang, J. Xu, Y. Zhong, R. Ren, Y. Lu, H. Wan, J. Wang, and J. Ding, “Focus modulation of cylindrical vector beams through negative-index grating lenses,” Opt. Commun. 372, 245–249 (2016).
[Crossref]

Yan, S.

Z. Wu, F. Dong, S. Zhang, S. Yan, G. Liang, Z. Zhang, Z. Wen, G. Chen, L. Dai, and W. Chu, “Broadband dielectric metalens for polarization manipulating and superoscillation focusing of visible light,” ACS Photon. 7, 180–189 (2020).
[Crossref]

Yang, H.

W. Liu, D. S. Dong, H. Yang, Q. H. Gong, and K. B. Shi, “Robust and high-speed rotation control in optical tweezers by using polarization synthesis based on heterodyne interference,” Opto-Electron. Adv. 3, 200022 (2020).
[Crossref]

Yang, S.

Yew, E. Y. S.

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

Yu, A.

Z. Wu, Q. Jin, K. Zhang, Z. Zhang, F. Liang, Z. Wen, A. Yu, and G. Chen, “Binary-amplitude modulation based super-oscillatory focusing planar lens for azimuthally polarized wave,” Opto-Electron. Eng. 45, 170660 (2018).
[Crossref]

G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017).
[Crossref]

A. Yu, G. Chen, Z. Zhang, Z. Wen, L. Dai, K. Zhang, S. Jiang, Z. Wu, Y. Li, C. Wang, and X. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6, 38859 (2016).
[Crossref]

Yu, M.

Yu, Y.

Yuan, G.

G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2015).
[Crossref]

Yuan, G. H.

G. H. Yuan, K. S. Rogers, E. T. F. Rogers, and N. I. Zheludev, “Far-field superoscillatory metamaterial superlens,” Phys. Rev. Appl. 11, 064016 (2019).
[Crossref]

Zha, Y.

Y. Zha, J. Wei, H. Wang, and F. Gan, “Creation of an ultra-long depth of focus super-resolution longitudinally polarized beam with a ternary optical element,” J. Opt. 15, 075703 (2013).
[Crossref]

Zhan, Q.

Zhang, B.

Q. Li, X. Zhao, B. Zhang, Y. Zheng, L. Zhou, L. Wang, Y. Wu, and Z. Fang, “Nanofocusing of longitudinally polarized light using absorbance modulation,” Appl. Phys. Lett. 104, 061103 (2014).
[Crossref]

Zhang, K.

Z. Wu, Q. Jin, K. Zhang, Z. Zhang, F. Liang, Z. Wen, A. Yu, and G. Chen, “Binary-amplitude modulation based super-oscillatory focusing planar lens for azimuthally polarized wave,” Opto-Electron. Eng. 45, 170660 (2018).
[Crossref]

Z. Wu, K. Zhang, S. Zhang, Q. Jin, Z. Wen, L. Wang, L. Dai, Z. Zhang, H. Chen, G. Liang, Y. Liu, and G. Chen, “Optimization-free approach for generating sub-diffraction quasi-non-diffracting beams,” Opt. Express 26, 16585–16599 (2018).
[Crossref]

Z. Wu, Q. Jin, S. Zhang, K. Zhang, L. Wang, L. Dai, Z. Wen, Z. Zhang, G. Liang, Y. Liu, and G. Chen, “Generating a three-dimensional hollow spot with sub-diffraction transverse size by a focused cylindrical vector wave,” Opt. Express 26, 7866–7875 (2018).
[Crossref]

G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017).
[Crossref]

S. Zhang, H. Chen, Z. Wu, K. Zhang, Y. Li, G. Chen, Z. Zhang, Z. Wen, L. Dai, and L. Wang, “Synthesis of sub-diffraction quasi-non-diffracting beams by angular spectrum compression,” Opt. Express 25, 27104–27118 (2017).
[Crossref]

A. Yu, G. Chen, Z. Zhang, Z. Wen, L. Dai, K. Zhang, S. Jiang, Z. Wu, Y. Li, C. Wang, and X. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6, 38859 (2016).
[Crossref]

Zhang, Q.

Z. Wu, Q. Zhang, X. Jiang, Z. Wen, G. Liang, Z. Zhang, Z. Shang, and G. Chen, “Broadband integrated metalens for creating superoscillation 3D hollow spot by independent control of azimuthally and radially polarized waves,” J. Phys. D 52, 415103 (2019).
[Crossref]

Zhang, S.

Zhang, X.

Z. Nie, W. Ding, D. Li, X. Zhang, Y. Wang, and Y. Song, “Spherical and sub-wavelength longitudinal magnetization generated by 4π tightly focusing radially polarized vortex beams,” Opt. Express 23, 690–701 (2015).
[Crossref]

Z. Nie, G. Shi, X. Zhang, Y. Wang, and Y. Song, “Generation of super-resolution longitudinally polarized beam with ultra-long depth of focus using radially polarized hollow Gaussian beam,” Opt. Commun. 331, 87–93 (2014).
[Crossref]

Zhang, Z.

Z. Wu, F. Dong, S. Zhang, S. Yan, G. Liang, Z. Zhang, Z. Wen, G. Chen, L. Dai, and W. Chu, “Broadband dielectric metalens for polarization manipulating and superoscillation focusing of visible light,” ACS Photon. 7, 180–189 (2020).
[Crossref]

Z. Wu, Q. Zhang, X. Jiang, Z. Wen, G. Liang, Z. Zhang, Z. Shang, and G. Chen, “Broadband integrated metalens for creating superoscillation 3D hollow spot by independent control of azimuthally and radially polarized waves,” J. Phys. D 52, 415103 (2019).
[Crossref]

Z. Wu, Q. Jin, S. Zhang, K. Zhang, L. Wang, L. Dai, Z. Wen, Z. Zhang, G. Liang, Y. Liu, and G. Chen, “Generating a three-dimensional hollow spot with sub-diffraction transverse size by a focused cylindrical vector wave,” Opt. Express 26, 7866–7875 (2018).
[Crossref]

Z. Wu, K. Zhang, S. Zhang, Q. Jin, Z. Wen, L. Wang, L. Dai, Z. Zhang, H. Chen, G. Liang, Y. Liu, and G. Chen, “Optimization-free approach for generating sub-diffraction quasi-non-diffracting beams,” Opt. Express 26, 16585–16599 (2018).
[Crossref]

Z. Wu, Q. Jin, K. Zhang, Z. Zhang, F. Liang, Z. Wen, A. Yu, and G. Chen, “Binary-amplitude modulation based super-oscillatory focusing planar lens for azimuthally polarized wave,” Opto-Electron. Eng. 45, 170660 (2018).
[Crossref]

G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017).
[Crossref]

S. Zhang, H. Chen, Z. Wu, K. Zhang, Y. Li, G. Chen, Z. Zhang, Z. Wen, L. Dai, and L. Wang, “Synthesis of sub-diffraction quasi-non-diffracting beams by angular spectrum compression,” Opt. Express 25, 27104–27118 (2017).
[Crossref]

A. Yu, G. Chen, Z. Zhang, Z. Wen, L. Dai, K. Zhang, S. Jiang, Z. Wu, Y. Li, C. Wang, and X. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6, 38859 (2016).
[Crossref]

Zhao, X.

Q. Li, X. Zhao, B. Zhang, Y. Zheng, L. Zhou, L. Wang, Y. Wu, and Z. Fang, “Nanofocusing of longitudinally polarized light using absorbance modulation,” Appl. Phys. Lett. 104, 061103 (2014).
[Crossref]

Zheludev, N.

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

Zheludev, N. I.

G. H. Yuan, K. S. Rogers, E. T. F. Rogers, and N. I. Zheludev, “Far-field superoscillatory metamaterial superlens,” Phys. Rev. Appl. 11, 064016 (2019).
[Crossref]

G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2015).
[Crossref]

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012).
[Crossref]

Zheng, Y.

Q. Li, X. Zhao, B. Zhang, Y. Zheng, L. Zhou, L. Wang, Y. Wu, and Z. Fang, “Nanofocusing of longitudinally polarized light using absorbance modulation,” Appl. Phys. Lett. 104, 061103 (2014).
[Crossref]

Zhitenev, N.

Zhong, Y.

S. Wang, J. Xu, Y. Zhong, R. Ren, Y. Lu, H. Wan, J. Wang, and J. Ding, “Focus modulation of cylindrical vector beams through negative-index grating lenses,” Opt. Commun. 372, 245–249 (2016).
[Crossref]

Zhou, L.

Q. Li, X. Zhao, B. Zhang, Y. Zheng, L. Zhou, L. Wang, Y. Wu, and Z. Fang, “Nanofocusing of longitudinally polarized light using absorbance modulation,” Appl. Phys. Lett. 104, 061103 (2014).
[Crossref]

Zhou, Y.

Zuo, R.

R. Zuo, W. Liu, H. Cheng, S. Chen, and J. Tian, “Breaking the diffraction limit with radially polarized light based on dielectric metalenses,” Adv. Opt. Mater. 6, 1800795 (2018).
[Crossref]

ACS Photon. (1)

Z. Wu, F. Dong, S. Zhang, S. Yan, G. Liang, Z. Zhang, Z. Wen, G. Chen, L. Dai, and W. Chu, “Broadband dielectric metalens for polarization manipulating and superoscillation focusing of visible light,” ACS Photon. 7, 180–189 (2020).
[Crossref]

Adv. Mater. (1)

F. Qin, K. Huang, J. F. Wu, J. H. Teng, C. W. Qiu, and M. H. Hong, “A supercritical lens optical label-free microscopy: sub-diffraction resolution and ultra-long working distance,” Adv. Mater. 29, 1602721 (2017).
[Crossref]

Adv. Opt. Mater. (1)

R. Zuo, W. Liu, H. Cheng, S. Chen, and J. Tian, “Breaking the diffraction limit with radially polarized light based on dielectric metalenses,” Adv. Opt. Mater. 6, 1800795 (2018).
[Crossref]

Appl. Phys. A (1)

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
[Crossref]

Appl. Phys. Lett. (5)

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6241 (2004).
[Crossref]

Q. Li, X. Zhao, B. Zhang, Y. Zheng, L. Zhou, L. Wang, Y. Wu, and Z. Fang, “Nanofocusing of longitudinally polarized light using absorbance modulation,” Appl. Phys. Lett. 104, 061103 (2014).
[Crossref]

T. Wang, X. Wang, C. Kuang, X. Hao, and X. Liu, “Experimental verification of the far-field subwavelength focusing with multiple concentric nanorings,” Appl. Phys. Lett. 97, 231105 (2010).
[Crossref]

F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[Crossref]

J. Opt. (3)

M. Berry, N. Zheludev, Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, J. Tollaksen, E. T. F. Rogers, F. Qin, M. Hong, X. Luo, R. Remez, A. Arie, J. B. Götte, M. R. Dennis, A. M. H. Wong, G. V. Eleftheriades, Y. Eliezer, A. Bahabad, G. Chen, Z. Wen, G. Liang, C. Hao, C.-W. Qiu, A. Kempf, E. Katzav, and M. Schwartz, “Roadmap on superoscillations,”J. Opt. 21, 053002 (2019).
[Crossref]

V. Ravi, P. Suresh, K. B. Rajesh, Z. Jaroszewicz, P. M. Anbarasan, and T. V. S. Pillai, “Generation of sub-wavelength longitudinal magnetic probe using high numerical aperture lens axicon and binary phase plate,”J. Opt. 14, 055704 (2012).
[Crossref]

Y. Zha, J. Wei, H. Wang, and F. Gan, “Creation of an ultra-long depth of focus super-resolution longitudinally polarized beam with a ternary optical element,” J. Opt. 15, 075703 (2013).
[Crossref]

J. Phys. D (2)

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[Crossref]

Z. Wu, Q. Zhang, X. Jiang, Z. Wen, G. Liang, Z. Zhang, Z. Shang, and G. Chen, “Broadband integrated metalens for creating superoscillation 3D hollow spot by independent control of azimuthally and radially polarized waves,” J. Phys. D 52, 415103 (2019).
[Crossref]

Light: Sci. Appl. (1)

G. Chen, Z. Q. Wen, and C. W. Qiu, “Superoscillation: from physics to optical applications,” Light: Sci. Appl. 8, 56 (2019).
[Crossref]

Nat. Mater. (1)

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11, 432–435 (2012).
[Crossref]

Nat. Photonics (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Nucl. Instrum. Methods Phys. Res. (1)

Y. Liu, D. Cline, and P. He, “Vacuum laser acceleration using a radially polarized CO2 laser beam,” Nucl. Instrum. Methods Phys. Res. 424, 296–303 (1999).
[Crossref]

Opt. Commun. (2)

Z. Nie, G. Shi, X. Zhang, Y. Wang, and Y. Song, “Generation of super-resolution longitudinally polarized beam with ultra-long depth of focus using radially polarized hollow Gaussian beam,” Opt. Commun. 331, 87–93 (2014).
[Crossref]

S. Wang, J. Xu, Y. Zhong, R. Ren, Y. Lu, H. Wan, J. Wang, and J. Ding, “Focus modulation of cylindrical vector beams through negative-index grating lenses,” Opt. Commun. 372, 245–249 (2016).
[Crossref]

Opt. Express (15)

Z. Nie, W. Ding, D. Li, X. Zhang, Y. Wang, and Y. Song, “Spherical and sub-wavelength longitudinal magnetization generated by 4π tightly focusing radially polarized vortex beams,” Opt. Express 23, 690–701 (2015).
[Crossref]

K. B. Rajesh, Z. Jaroszewicz, and P. M. Anbarasan, “Improvement of lens axicon’s performance for longitudinally polarized beam generation by adding a dedicated phase transmittance,” Opt. Express 18, 26799 (2010).
[Crossref]

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
[Crossref]

H. Dehez, A. April, and M. Piché, “Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent,” Opt. Express 20, 14891–14905 (2012).
[Crossref]

H. Kang, B. Jia, and M. Gu, “Polarization characterization in the focal volume of high numerical aperture objectives,” Opt. Express 18, 10813–10821 (2010).
[Crossref]

Y. Zhou and M. H. Hong, “Formation of polarization-dependent optical vortex beams via an engineered microsphere,” Opt. Express 29, 11121–11131 (2021).
[Crossref]

Z. Wu, Q. Jin, S. Zhang, K. Zhang, L. Wang, L. Dai, Z. Wen, Z. Zhang, G. Liang, Y. Liu, and G. Chen, “Generating a three-dimensional hollow spot with sub-diffraction transverse size by a focused cylindrical vector wave,” Opt. Express 26, 7866–7875 (2018).
[Crossref]

Z. Wu, K. Zhang, S. Zhang, Q. Jin, Z. Wen, L. Wang, L. Dai, Z. Zhang, H. Chen, G. Liang, Y. Liu, and G. Chen, “Optimization-free approach for generating sub-diffraction quasi-non-diffracting beams,” Opt. Express 26, 16585–16599 (2018).
[Crossref]

P. W. Meng, S. Pereira, and P. Urbach, “Confocal microscopy with a radially polarized focused beam,” Opt. Express 26, 29600–29613 (2018).
[Crossref]

Y. Kozawa, R. Sakashita, Y. Uesugi, and S. Sato, “Imaging with a longitudinal electric field in confocal laser scanning microscopy to enhance spatial resolution,” Opt. Express 28, 18418 (2020).
[Crossref]

Y. Liu, H. Xu, F. Stief, N. Zhitenev, and M. Yu, “Far-field superfocusing with an optical fiber based surface plasmonic lens made of nanoscale concentric annular slits,” Opt. Express 19, 20233–20243 (2011).
[Crossref]

T. Liu, S. Yang, and Z. Jiang, “Electromagnetic exploration of far-field superfocusing nanostructured metasurfaces,” Opt. Express 24, 16297–16308 (2016).
[Crossref]

J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express 18, 21965–21972 (2010).
[Crossref]

Y. Yu and Q. Zhan, “Optimization-free optical focal field engineering through reversing the radiation pattern from a uniform line source,” Opt. Express 23, 7527–7534 (2015).
[Crossref]

S. Zhang, H. Chen, Z. Wu, K. Zhang, Y. Li, G. Chen, Z. Zhang, Z. Wen, L. Dai, and L. Wang, “Synthesis of sub-diffraction quasi-non-diffracting beams by angular spectrum compression,” Opt. Express 25, 27104–27118 (2017).
[Crossref]

Opt. Lett. (2)

Optica (1)

Opto-Electron. Adv. (1)

W. Liu, D. S. Dong, H. Yang, Q. H. Gong, and K. B. Shi, “Robust and high-speed rotation control in optical tweezers by using polarization synthesis based on heterodyne interference,” Opto-Electron. Adv. 3, 200022 (2020).
[Crossref]

Opto-Electron. Eng. (1)

Z. Wu, Q. Jin, K. Zhang, Z. Zhang, F. Liang, Z. Wen, A. Yu, and G. Chen, “Binary-amplitude modulation based super-oscillatory focusing planar lens for azimuthally polarized wave,” Opto-Electron. Eng. 45, 170660 (2018).
[Crossref]

Phys. Lett. A (1)

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A 368, 402–407 (2007).
[Crossref]

Phys. Rev. Appl. (1)

G. H. Yuan, K. S. Rogers, E. T. F. Rogers, and N. I. Zheludev, “Far-field superoscillatory metamaterial superlens,” Phys. Rev. Appl. 11, 064016 (2019).
[Crossref]

Phys. Rev. E (1)

T. Grosjean and D. Courjon, “Polarization filtering induced by imaging systems: effect on image structure,” Phys. Rev. E 67, 046611 (2003).
[Crossref]

Phys. Rev. Lett. (2)

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

Sci. Rep. (4)

A. Yu, G. Chen, Z. Zhang, Z. Wen, L. Dai, K. Zhang, S. Jiang, Z. Wu, Y. Li, C. Wang, and X. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6, 38859 (2016).
[Crossref]

G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2015).
[Crossref]

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015).
[Crossref]

G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for superoscillatory optical hollow needles,” Sci. Rep. 7, 4697 (2017).
[Crossref]

Other (1)

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

Supplementary Material (1)

NameDescription
» Supplement 1       Supplemental Document

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1. Focusing of radially polarized optical field by a lens with a focal length of $f$, which is represented by a reference sphere $S$ (dashed curve).
Fig. 2.
Fig. 2. Radial-to-linear polarization conversion microscopy for characterizing a longitudinally polarized optical field. The objective lens and tube lens are represented by two reference spheres, i.e., ${S_1}$ and ${S_2}$, respectively. The ${S}$-wave plate (SWP) and linear polarizer (LP) are set in the light path between the objective lens and the tube lens. The dipole is an analog of the longitudinally polarized optical field.
Fig. 3.
Fig. 3. Polarization-conversion-based optical microscope for independently measuring both transversely and longitudinally polarized fields. Vertical arm and objective lens form RLPCM, and horizontal arm and objective lens form COM. COM and RLPCM are for observing transverse polarizations, i.e., ${E_x}$ and ${E_y}$, and longitudinal polarization, i.e., ${E_z}$, respectively. The lens is used to generate the focused optical field for the polarization imaging experiment.
Fig. 4.
Fig. 4. Intensity distribution of optical fields created by focusing linearly polarized light. Theoretical results for (a) $|{E_x}{|^2}$, (b) $|{E_y}{|^2}$, (c) $|{E_z}{|^2}$, and (d) $|E{|^2}$. Experimental results for (e) $|{E_x}{|^2}$, (f) $|{E_y}{|^2}$, (g) $|{E_z}{|^2}$, and (h) $|E{|^2}$. Comparison of intensity distributions between theoretical and experimental results for (i) $|{E_x}{|^2}$ and (j) $|{E_y}{|^2}$ in the $x$ direction, and (k) $|{E_z}{|^2}$ and (l) $|E{|^2}$ in the incident polarization direction, as indicated by the white lines in (c) and (g), where red curves are VASM results, and blue curves are experimental results. The scalar bar represents a length of ${1}\lambda$.
Fig. 5.
Fig. 5. Intensity distribution of optical fields created by focusing circularly polarized light. Theoretical results for (a) $|{E_x}{|^2}$, (b) $|{E_y}{|^2}$, (c) $|{E_z}{|^2}$, and (d) $|E{|^2}$. Experimental results for (e) $|{E_x}{|^2}$, (f) $|{E_y}{|^2}$, (g) $|{E_z}{|^2}$, and (h) $|E{|^2}$. Comparison of intensity distributions between theoretical and experimental results for (i) $|{E_x}{|^2}$, (j) $|{E_y}{|^2}$, (k) $|{E_z}{|^2}$, and (l) $|E{|^2}$ on the $x$ axis, where red curves are VASM results, and blue curves are experimental results. The scalar bar represents a length of ${1}\lambda$.
Fig. 6.
Fig. 6. Intensity distribution of optical fields created by focusing radially polarized light. Theoretical results for (a) $|{E_x}{|^2}$, (b) $|{E_y}{|^2}$, (c) $|{E_z}{|^2}$, and (d) $|E{|^2}$. FEM simulation results for (e) $|{E_x}{|^2}$, (f) $|{E_y}{|^2}$, (g) $|{E_z}{|^2}$, and (h) $|E{|^2}$. Experimental results for (i) $|{E_x}{|^2}$, (j) $|{E_y}{|^2}$, (k) $|{E_z}{|^2}$, and (l) $|E{|^2}$. Comparison of intensity distributions between theoretical and experimental results for (m) $|{E_x}{|^2}$, (n) $|{E_y}{|^2}$, (o) $|{E_z}{|^2}$, and (p) $|E{|^2}$, where red curves are VASM results, green curves are FEM results, and blue curves are experimental results. The scalar bar represents a length of ${1}\lambda$.
Fig. 7.
Fig. 7. Intensity distribution of optical fields on the propagation plane created by focusing radially polarized light. VASM results for (a) $|{E_x}{|^2}$, (b) $|{E_y}{|^2}$, (c) $|{E_z}{|^2}$, and (d) $|E{|^2}$. FEM results for (e) $|{E_x}{|^2}$, (f) $|{E_y}{|^2}$, (g) $|{E_z}{|^2}$, and (h) $|E{|^2}$. Experimental results for (i) $|{E_x}{|^2}$, (j) $|{E_y}{|^2}$, (k) $|{E_z}{|^2}$, and (l) $|E{|^2}$. Comparison of the major parameters (intensity, FWHM, and SR) between theoretical and experimental results for (m) $|{E_x}{|^2}$, (n) $|{E_y}{|^2}$, (o) $|{E_z}{|^2}$, and (p) $|E{|^2}$ along the optical axis, where red curves are VASM results, green curves are FEM results, blue curves are experimental results, magenta dashed lines are ADL, and black dashed lines are super-oscillation criterion.

Tables (3)

Tables Icon

Table 1. Major Parameters of Optical Intensity Created by Focusing Linearly Polarized Light

Tables Icon

Table 2. Major Parameters of Optical Intensity Created by Focusing Circularly Polarized Light

Tables Icon

Table 3. Major Parameters of Optical Intensity Created by Focusing Radially Polarized Light

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

E z ( r ) = { i k f exp ( i k f ) 0 θ max P ( θ ) ( cos θ ) 1 / 2 ( sin θ ) 2 J 0 ( k r sin θ ) d θ } × E i n c n z .
E 1 = μ z ( ω 2 ε 0 c 2 ) exp ( i k f ) 4 π f [ cos ϕ sin θ cos θ sin ϕ sin θ cos θ sin 2 θ ] ,
E 2 = ( cos θ ) 1 / 2 [ ( E 1 n θ ) n ρ + ( E 1 n ϕ ) n θ ] = { μ z ( ω 2 ε 0 c 2 ) exp ( i k f ) 4 π f ( cos θ ) 1 / 2 sin θ } n ρ ,
E 3 = ( E 2 n ρ ) n y = { μ z ( ω 2 ε 0 c 2 ) exp ( i k f ) 4 π f ( cos θ ) 1 / 2 sin θ } n y ,
E 4 = [ ( E 3 n ρ ) n θ + ( E 3 n ϕ ) n ϕ ] ( cos θ ) 1 / 2 = { μ z ( ω 2 ε 0 c 2 ) exp ( i k f ) 4 π f ( cos θ ) 1 / 2 sin θ } n y .
E i m a g e ( ρ ) = { i C 0 θ max ( cos θ ) 1 / 2 ( sin θ ) 2 J 0 ( k sin θ ρ / M ) d θ } μ z E y ,
E i m a g e ( r ) = { i C 0 θ max ( cos θ ) 1 / 2 ( sin θ ) 2 J 0 ( k sin θ ) d θ } n y ,
E x ( r ) = { i k f exp ( i k f ) 0 θ max P ( θ ) ( cos θ ) 1 / 2 × 1 2 ( 1 + cos θ ) sin θ J 0 ( k r sin θ ) d θ } E i n c n x .
E i m a g e ( r , φ ) = { i C 0 θ max ( cos θ ) 1 / 2 1 2 ( 1 + cos θ ) sin θ J 0 ( k sin θ r ) d θ } n x .
{ E n ^ ϕ ( r , z ) = 0 A 0 ( f xy ) exp ( i 2 π f z z ) J 0 ( 2 π f xy r ) 2 π f xy d f xy , E z ( r , ϕ , z ) = i cos ( ϕ ϕ ) 0 A 0 ( f xy ) ( f xy / f z ) exp ( i 2 π f z z ) J 1 ( 2 π f xy r ) 2 π f xy d f xy , A 0 ( f xy ) = 0 E n ^ ϕ ( r , z = 0 ) T ( r ) J 0 ( 2 π f xy r ) 2 π r d r ,
{ E ρ ( r , z ) = 0 A 1 ( f xy ) exp ( i 2 π f z z ) J 1 ( 2 π f xy r ) 2 π f xy d f xy , E z ( r , z ) = i 0 A 1 ( f xy ) ( f xy / f z ) exp ( i 2 π f z z ) J 0 ( 2 π f xy r ) 2 π f xy d f xy , A 1 ( f xy ) = 0 E ρ ( r , z = 0 ) T ( r ) J 1 ( 2 π f xy r ) 2 π r d r ,

Metrics