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Abstract
Non-invasive imaging beyond the diffraction limit and free from fluorescent labels in the visible is highly desired for microscopy. It remains a challenge to obtain such super-resolution focusing along with multiwavelength achromatic performance in the far field using an integratable and easily designed system. In this work, we demonstrate a straightforward metasurface-based method to realize multiwavelength achromatic generation and focusing of cylindrically polarized vortex beams (CPVBs). Attributed to the extra degrees of freedom of CPVBs and multi-section design, we have realized multiwavelength achromatic super-resolution focusing in the air with focal size tighter than that of normally used schemes like immersion metalenses or focused radially polarized beams. It is expected that this metasurface-empowered ultra-compact design will benefit potential applications which call for high resolution, like optical microscopy, laser processing, etc.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Structured light with tailored intensity, phase and polarization has brought extra light modulation freedom and novel optical phenomena unseen in nature [1,2]. These extra degrees of freedom have benefited various applications including microscopy [3], optical communication [4], optical particle manipulation [5,6], quantum photonics [7,8], etc. Among the diversified structured light, beams depicted on the the hybrid-order Poincaré sphere [9] are the most active members of the family tree. The hybrid-order beams, in short, are combinations of two types of classical structured light, vector vortex beams (VVBs) [10] and vortex beams (VBs). They can also be named as cylindrically polarized vortex beams (CPVBs). Conventional methods to generate such beams involve Q-plate [11], spatial light modulator [12,13], digital micro mirror [14], intracavity generation [15], interference of fiber modes [16], etc. These methods consist of multiple cascaded elements, which make them intrinsically bulky and sensitive to optical path misalignment. Metasurfaces, composed of periodic subwavelength meta-atoms, have been suggested for the complete control of phase and polarization [17]. They stand out as great candidates to offer compact and integratable solutions due to their powerful light modulation and multiplexing capabilities. Diverse approaches to generate CPVBs based on metasurfaces have been discussed including utilizing two cascaded metasurfaces [18], reflective gap-surface plasmon metasurfaces [19], etc. However, their works are susceptible to optical path misalignment or work in reflection mode posing extra complications for practical applications. In addition, an ingenious design based on coherent double-nanoblock unit has demonstrated powerful ability for arbitrary control of CPVBs [20]. Nevertheless, aforementioned approaches are all limited to plane wave output lacking achromatic focusing performance. Full potentials of CPVBs under conditions with high numerical aperture (NA) are yet to be explored for further applications.
Although the achromatic metalenses have been extensively investigated [21–25], they are all limited to homogeneously polarized incident light with relatively small NA. We believe few attempts have been made to involve multiwavelength achromatic focusing of CPVBs to tighten the focus under high-NA conditions.
In this article, we have suggested a single-layered transmissive metasurface capable of multiwavelength achromatic generation of focused CPVBs based on a multi-section scheme [26]. The proposed metasurface is composed of specially selected anisotropic titanium dioxide (TiO2) nanofins to modulate phase and polarization with high freedom in the visible range. Based on the powerful metasurface-based platform, we have obtained improved super-resolution focusing in the air utilizing a special case of CPVBs, which is azimuthally polarized vortex beam (APVB) possessing opposite polarization order and topological charge $(\ell _p$=$-\ell _t$=$\pm 1)$. Although there have been researches regarding superoscillation effect for focusing beyond diffraction limit in the far field [27,28]. Whether metasurface-based or not, they all demand numerous computational resources and have to go through time-consuming optimization process. By contrast, our design needs only a straightforward unit-cell-based method to readily achieve bettered super-resolution focusing comparing to cases based on radially polarized beam [29] or immersion metalenses [30,31]. This label-free super-resolution focusing strategy without contacting the sample holds its undeniable value in practical applications, despite the outstanding performance of near-field methods [32] or fluorescent methods [33,34]. The metasurface design process is thoroughly demonstrated along with the comprehensive analysis of the super-resolution focusing effect. We believe this work exhibiting easy design and high modulation freedom will go a long way in benefiting applications in microscopy, laser micro/nano fabrication [35–37], etc.
2. Structure design scheme
2.1 Polarization and phase modulation theory
Shape birefringent optical elements like TiO$_2$ nanofins have been widely used in metasurfaces to provide geometric phase or polarization conversion. The schematic diagram of one single nanofin with hexagonal lattice is displayed in Fig. 1. When the incident light propagates through a nanofin, the optical response of such nanofin can be expressed as a 2$\times$2 complex Jones matrix $J$, which is of the form:
Fig. 1. Schematic diagram of a TiO$_2$ nanofin on SiO$_2$ substrate. H stands for the height of nanofin, P for the hexagonal lattice period, W for the nanofin width, L for the nanofin length, $\theta$ for the relative rotation angle to x axis.
When $\eta _{conversion}\approx$1, we can safely treat the corresponding nanofin as a standard half-wave-plate. However, $\eta _{conversion}\ge$0.9 can suffice for the later unit cell selection process. In this case, we can transform Eq. (2) to the following equation with a more generalized incident polarization:
According to Eq. (4), by carefully controlling the geometric parameters and rotation direction of the nanofin, we can easily achieve independent modulation of polarization and phase. In other words, if we spatially distribute the nanofins with varied $\theta$ related to the azimuthal angle $\phi$, as in $\theta (x,y)$=$q\phi +\alpha _0$, and select suitable nanofins with different geometric parameters to meet the phase requirement $(\varphi _c(x,y)$=$\ell _t \phi )$ for the VB generation, we can then easily derive arbitrary CPVBs with polarization order of $\ell _p$=$2q$ and topological charge of $\ell _t$. Even though we will not be able to change nanofins after finishing distributing, we can still vary the incident polarization to actively tune the output polarization state. The active tuning of incident polarization can be easily carried out by integrating liquid crystal retarder system with metasurfaces to guarantee the compactness of the integrated system [38,39].
2.2 Nanofin selection
To begin with, all units or metasurfaces related simulations are conducted using a commercial finite difference time domain solver (Lumerical FDTD solutions). The refractive indices ($n$) of TiO$_2$ and SiO$_2$ depicted in Fig. 2 are obtained by ellipsometry using J.A.Woollam VASE ellipsometer.
According to Eq. (1), we need to obtain $\widetilde {t_{l}}$ and $\widetilde {t_{s}}$ for further design. The normal incident light is set to illuminate the TiO$_2$ nanofin from the side of SiO$_2$ substrate. The background is air. Considering further design will concern the focusing of CPVBs, the period of the hexagonal lattice (P) must satisfy the Nyquist sampling criterion at the working wavelength ($\lambda$), $P$<$\lambda /(2NA)$. This criterion is necessary for the metasurface to render accurate phase shift per pixel. Here, we set the P as 240 nm with $\lambda$ set as 473 nm and the NA as 0.94. We then use periodic boundary condition to approximate the response of nanofins to the incident light within the metasurfaces. Based on the concept of high-contrast transmitarrays (HCTA) [40], we fix the height as 600 nm for adequate phase modulation. Then, we conduct a sweep of transmission coefficient as a function of width and length of the nanofin, both ranging from 50-190 nm with 5 nm variation step. To facilitate the data sweeping, we only need to simulate the electromagnetic response of the nanofin under XLP considering the intrinsically symmetrical response of the nanofin between XLP and vertical polarization (YLP). Figure 3(a)$\&$(b) demonstrate the simulated results of transmission and phase modulation. Utilizing the original data of transmission and phase, we can derive $\eta _{conversion}$ and additional phase $(\varphi _c)$ according to Eq. (3)$\&$(4). The calculated results versus nanofin width and length are depicted in Fig. 3(c)$\&$(d).
Fig. 3. (a)$\&$(b) Simulated nanofin transmission modulation and phase modulation normalized by 2$\pi$ as a function of its width and length. (c)$\&$(d) Calculated nanofin $\eta _{conversion}$ and rotation-insensitive additional phase $(\varphi _c/2\pi )$ as a function of its width and length. The incident $\lambda$ is 473 nm, $P$=240 nm, and $H$=600 nm.
Among all the varied nanofins, we sieve out those possessing quasi-half-wave-plate characteristic, in other words, high $\eta _{conversion}$, along with $\varphi _c$ covering 0-2$\pi$ phase range. In order to verify the design theory, we demonstrate, for example, the calculated output phases of 8 selected nanofins satisfying above requirements as a function of rotation angle $(\theta )$ in Fig. 4. The calculated data shows distinct rotation-insensitivity which is predicted in Section 2.1. As can be seen in Fig. 4, due to the influence of geometric phase, there is a fixed phase difference of $\pi$ when each nanofin rotates to a relatively perpendicular position. The abrupt phase shift also verifies the integrity of design theory and can be further explained using Eq. (4) with XLP incidence and $\theta$=$\theta +\pi /2$. The output light is as follow:
Fig. 4. Calculated output phases of 8 selected nanofins as a function of $\theta$. Despite the abrupt phase change of $\pi$, the data shows distinct rotation-insensitivity which confirms the accuracy of the design theory.
3. Multiwavelength achromatic super-resolution focusing
Radially polarized beam (RPB) has been widely utilized to generate far-field super-resolution focus due to the sharp and round longitudinal component under high-NA conditions. On the contrary, azimuthally polarized beam (APB) lacking longitudinal component will only be focused into a hollow spot, which hinders its application in direct light field imaging. With existence of $\ell _t$ providing extra degree of freedom, CPVBs can offer more possibilities in modulating focusing performance. Here, we introduce opposite $\ell _t$ and $\ell _p$ with $(\ell _t,\ell _p)$=$(-1,1)$ to APVB, in order to compensate the hollow focus of APB into a tight super-resolution focus. We conduct a calculation using vectorial angular spectrum method (VASM) [41] to compare the focusing performance of suggested APVB possessing $(\ell _t,\ell _p)$=$(-1,1)$ with RPB/APB with $(\ell _t,\ell _p)$=$(0,1)$ and left circularly polarized beam (LCPB) with $(\ell _t,\ell _p)$=$(0,0)$. These cases share the same lens parameters with a lens radius of 15.6 $\mu$m and a focal distance ($f$) of 5.5 $\mu$m, which yields NA$\approx$0.94, under the incident $\lambda$=473 nm. As can be seen in Fig. 5, due to the existence of extra $\ell _t$, we have transformed the original hollow focus of APB into a round spot tighter than that of RPB/LCPB. The calculated FWHM of focused APVB is 224 nm (0.47$\lambda$), smaller than the Abbe diffraction limit of $0.5\lambda /NA=0.53\lambda$.
Fig. 5. depicts the normalized intensity distribution calculated by VASM at focus and along the x axis at focus of the LCPB $((\ell _t,\ell _p)$=$(0,0))$, RPB/APB $((\ell _t,\ell _p)$=$(0,1))$ and APVB $((\ell _t,\ell _p)$=$(-1,1))$. The FWHMs are listed beside the illustrations of focus intensity along the x axis at focus. The incident $\lambda$ is 473 nm.
In order to generate the special APVB with $(\ell _t,\ell _p)$=$(-1,1)$ based on principle provided in Section 2, we should first select out quasi-half-wave-plate units possessing $\varphi _c$ satisfying the generation of focused vortex beam with $\ell _t$=−1. In other words, we select out units to match the phase distribution as:
where the first part on the right provides the vortex phase and the second part functions as a focusing phase. Then, we distribute the nanofins with spatially varying rotation angles as $\theta (x,y)$=$\phi /2$ which yields $\ell _p$=1. When we set the incident polarization as YLP, we will then obtain an output of focused APVB with $(\ell _t,\ell _p)$=$(-1,1)$.Since the phase modulation of individual nanofin stems from the propagation phase which is chromatic by nature. It means if we design the metasurface to provide polarization-conversion and focusing at 473 nm, the performance will not hold when incident $\lambda$ changes, especially the focusing performance. To cater to this problem, we employ spatial multiplexing as the solution to ensure multiwavelength achromatic performance at $\lambda _1$=473 nm, $\lambda _2$=532 nm, and $\lambda _3$=632.8 nm. Different from normally used interleaved spatial multiplexing method or meta-molecule design, which faces strong couplings among three $\lambda$ channels, we exploit multi-section design to divide a single metasurface into three rings, as illustrated in Fig. 6. The working wavelengths of three rings are indicated with different colors, blue for 473 nm, green for 532 nm, and red for 632.8 nm. They respectively are responsible for turning the incident light into focused APVB possessing $(\ell _t,\ell _p)$=$(-1,1)$ with the same $f$=5.5 $\mu$m at designated wavelength. The multi-section design only demands three zones to have little crosstalk among varied $\lambda$ channels, which can be easily realized in detailed unit selection as follows.
Fig. 6. Schematic diagram of multi-section metasurface and magnification of varied composing units. Ring 1 has a radius of R$_1$=15.6 $\mu$m, ring 2 of R$_2$=37.8 $\mu$m and ring 3 of R$_3$=61 $\mu$m. The intervals between rings are 1.4 $\mu$m and 2.2 $\mu$m. The $f$ of three channels are all 5.5 $\mu$m to realize achromatic performance, which yields NA of 0.94, 0.99 and 0.996, respectively. The respective ring color of blue, green and red indicates the working wavelength being 473 nm, 532 nm and 632.8 nm. The hexagonal lattice periods of three rings are set differently as 240 nm, 260 nm and 300 nm in order to satisfy the Nyquist sampling criterion.
First, we need to calculate the $\eta _{conversion}$ at three wavelengths for each type of unit in respective ring. The calculation scheme is the same as demonstrated in Section 2. Then, we define root mean square error (RMSE) of the calculated $\eta _{conversion}$ at three wavelengths to the ideal composition to determine the nanofins with the least cross talk and best required conversion performance. The RMSE can be expressed as:
Fig. 7. (a-c) represent RMSEs for units in ring 1, ring 2 and ring 3 accordingly. Due to the varied hexagonal lattice period in respective ring, the ranges of width/length depicted in the illustration are different.
According to the steps mentioned above, we spatially arrange the suitable units to match the requirements for generating APVB with $(\ell _t,\ell _p)$=$(-1,1)$. The FDTD simulation results are demonstrated in Fig. 8 (see section 1 in Supplement 1 for detailed mode analysis). Due to the limited calculation resources, we can only set the R$_1$=15.6 $\mu$m, R$_2$=37.8 $\mu$m and R$_3$=61 $\mu$m for a complete FDTD simulation of the metasurface, which yield NAs being 0.94, 0.99, and 0.996, respectively. The FWHMs of the three foci we achieve in air are 0.47$\lambda _1$ (224 nm), 0.39$\lambda _2$ (206 nm) and 0.37$\lambda _3$ (235 nm), all smaller than Abbe diffraction limit of 0.5$\lambda /NA$. As can be seen from Fig. 8(b)$\&$(c), due to the configuration of our metasurface design, the loss of low spatial frequencies in ring 2 and ring 3 render the shapes of non-diffracting optical needles with a 5.55$\lambda _2$ and 6.48$\lambda _3$ filed of view (calculated from the FWHM along z axis shown in Fig. 8), and more obvious sideband oscillations. The focusing efficiency of designed three rings can be derived using the ratio between the total light intensity after passing through a filter at focus with a radius of three times the focal FWHM and the total intensity immediately after the metasurface. The calculated focusing efficiencies are 16.55$\%$ for ring 1, 1.73$\%$ for ring 2 and 1.98$\%$ for ring 3. Compared to metallic achromatic metalenses [21] which suffer high intrinsic loss in the visible or near-infrared range, dielectric-structure-based designs can offer better focusing efficiency [42,43]. However, the focusing efficiencies of our design are relatively smaller than its transmissive dielectric counterparts due to the inevitable trade-off between the NA and the focusing efficiency. In order to achieve the super-resolution focusing using our scheme, we have to set the NA at least larger than 0.9 (the NAs of broadband achromatic metalenses mostly are smaller than 0.3). Under this high-NA condition, the rapid-changing phase profile will render obviously different geometric parameters between adjacent units, which will create adjacent couplings, and unwanted phase discrepancies will occur. Thus, the non-ideal phase profile results in the inevitable decreasing of the focusing efficiencies of our design. This limitation can possibly be addressed with inverse design [44] or adjacent optimization [45]. Besides, the loss of low spatial frequencies in the ring 2 and ring 3 further reduces their focusing efficiencies. The difference between the focusing efficiency of ring 1 and those of the two outer rings can be tempered by an interleaved multi-section scheme [46]. When ring 1 which has a complete lens profile is considered only, its performance is comparable to the recently presented research in subdiffraction focusing metalens [47], which is monochromatic and has a focusing efficiency of 17.2$\%$ with the focal FWHM being 0.429$\lambda$.
Fig. 8. (a) depicts the normalized intensity distributions of ring 1. The illustrations from left to right are xz cross section of the propagation field, xy cross section of the focus at $f$=5.5 $\mu$m and the intensity distribution along the x axis at focus. (b-c) depicts the normalized intensity distributions of ring 2, ring 3, and three foci incoherently superposed. The illustrations are set in the same order as (a). The FWHMs of foci are illustrated above the third column. The designed $f$=5.5 $\mu$m is marked with white dashed line. The dashed frame displays the peak intensity difference between the foci of three rings (each intensity is normalized to the peak intensity of the ring 1 focus).
As is shown in the dashed frame in Fig. 8, the difference of coverage areas and the loss of low spatial frequencies in the two outer rings will inevitably lead to different peak intensities. The lens center accommodating the low spatial frequencies carries most of the focusing energy. Therefore, ring 1 holding a complete lens configuration demonstrates the highest peak intensity, followed by the foci of ring 2 and ring 3. In order to realize balanced intensities, we have to offer more coverage area for the two outer rings, which will allow more light energy to be captured by the two rings, and in turn, increase the respective focal intensity. We approximate the ideal coverage area needed for balanced intensities using VASM (see section 2 in Supplement 1 for a detailed explanation), and we obtain the suitable radii are R$_1$=15.6 $\mu$m, R$_2$=65 $\mu$m and R$_3$=308 $\mu$m, which yields NAs being 0.94, 0.996 and 0.9997. The differences between the NAs of the ideal case and the FDTD case presented above are tiny, which will only create marginal difference in focal size. Thus, for better illustration of the multiwavelength achromatic performance of the metasurface, we normalize the FDTD results of three foci to their own intensity maximums. After we incoherently superpose the intensities of three foci, we get a round white focus at $f$=5.5 $\mu$m, as illustrated in Fig. 8(d). The FWHM of the white focus can be determined from the superposed intensities and here in the normalized case is 221 nm (219 nm in the ideal case), which can be further reduced by increasing the NAs of three foci. To prove the integrity of our design, we also provided another design of a three-ring metasurface capable of balanced intensities without the normalized operation. The design is based on the same principles discussed above, but with a reduced size and shorter $f$ suitable for complete rigorous FDTD simulation(see section 3 in Supplement 1).
We compare the FDTD simulation results of intensity distributions along the x axis at $f$=5.5 $\mu$m presented above with ideal results obtained from VASM. In the VASM simulation, we introduce an ideal transmission function with same lens parameters and three-ring configuration. The incidence mode is also APVB with $(\ell _t,\ell _p)$=$(-1,1)$. As is shown in Fig. 9, from the comparisons of normalized intensity distributions, we can see the lineshapes agree well, and the peak intensities given by FDTD are a bit lower. This is natural, because in FDTD simulation, besides the unwanted adjacent interference, we sacrifice certain phase accuracy for the minimized RMSEs. Thus, the imperfect metasurface transmission function will inevitably cause the decreasing of peak intensities. Nevertheless, these results along with detailed mode analysis presented in Supplement 1 prove that the light modulation provided by our metasurface design is accurate, and we have indeed generated focused APVB with $(\ell _t,\ell _p)$=(−1,1) based on the design principles discussed above.
Fig. 9. (a-c) illustrates comparisons of normalized intensity distributions along the x axis at three foci (intensities here are all normalized to the respective maximum intensity in VASM simulation result). Blue lines represent FDTD simulation results, and red lines represent VASM simulation results.
4. Conclusion
In this article, we have verified anisotropic-nanofin-based design scheme using the unit rotation and geometric parameters for independent phase and polarization manipulation. Based on the scheme, we have proposed a single-layered transmissive metasurface possessing multi-functional properties of lens, cylindrical polarization converter and vortex waveplate. By introducing a multi-section scheme and utilizing the extra degrees of freedom provided by CPVBs, we successfully demonstrated multiwavelength achromatic super-resolution focusing at wavelength of 473 nm, 532 nm, and 632.8 nm by modulating the vertically polarized beam into APVB with $(\ell _t,\ell _p)$=$(-1,1)$. The FWHMs of three foci are 0.47$\lambda _1$, 0.39$\lambda _2$, and 0.37$\lambda _3$, all fall below the Abbe diffraction limit. The capability of multi-section scheme is limited when coping with achromatic focusing involving a large number of wavelengths, and the high-NA requirement makes it difficult to utilize dispersion engineering for broadband achromatic focusing. By integrating multiwavelength achromatic metalenses with bandpass filter [48] can circumvent this problem to some extent, and provide better signal-to-noise ratio in practical applications. Still, it remains an endeavour to find broadband solutions for achromatic super-resolution focusing.
All in all, this label-free and ultra-compact method for multiwavelength achromatic super-resolution focusing in the far field is expected to benefit extensive applications and offer possibilities at building compact imaging devices like miniature microscope or endoscope, improving resolution in laser processing, etc.
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.
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