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Abstract
In the super resolution imaging system, a lens and an axicon that can generate spherical wavefronts and non-diffracting Bessel beams respectively are both essential yet difficult to integrate using the traditional approach. We propose a new concept of a “Bessel-lens” to indicate unique optical elements that merge the functionalities of lenses and axicons simultaneously. The Bessel-lens is a mission that is extremely difficult if not impossible for state-of-the-art technology because of the exotic phase profile. Via the geometric phases in space-variant nanoslits, planar Bessel-lenses are designed and experimentally characterized for the first time to generate subdiffraction beams. Compared with a planar lens and axicon with the same dimensions and numerical aperture, the proposed Bessel-lens possesses a higher imaging resolution, which may find applications in microscopy, nanofabrication and dense data storage.
© 2017 Optical Society of America
1. Introduction
There have been several methods to achieve optical super resolution imaging in the past decades, including super-oscillatory [1, 2], stimulated emission depletion (STED) microscopy system [3–7], and so on. As an alternative access to super resolution imaging [8–11], Bessel beam is one kind of non-diffraction beams that commonly generated by conical shaped lens i.e., axicons [12–15], which could convert Gaussian beams into nondiffracting Bessel beams [16, 17]. Cascading lens and axicon has been utilized for the Bessel beam microscopy (BBM) [16, 17]. Although BBM has been proved fruitful to achieve super resolution imaging, the existing solutions based on cascading lens and axicon suffer from many drawbacks. For example, cascading configuration impedes the integration and increases the difficulties of practical implement.
Metasurfaces is a kind of metamaterials, which has been proved to be able to achieve full control of the state of electromagnetic waves, including the amplitude, phase and polarization [18–21]. As near ideal two-dimensional materials, metasurfaces are thin and tiny enough to construct a compact optical system. Furthermore, metasurfaces are promising to realize many fantastic functions which are nearly impossible for traditional optical elements [22–33]. In particular, there are some impressive works to achieve planar lens and planar axicon by metasurface [13, 29–31]. For example, Li et al. have shown a design of compact Bessel beam generators with catenary nanostructures recently [32].
In this paper, we report the design and experimental demonstration of the Bessel-type focus and imaging system based on the metasurface, which is composed of dense nano-apertures. As the theoretical basis of the design, geometric phases in nano-aperture array help us to encode the phase information into metasurface [19]. We design the planar axicon to generate Bessel beam, as well as a common planar focusing lens with the same diameter for comparison. More importantly, we present for the first time a design of “Bessel-lens” which superimposes the phase of focusing lens and axicon simultaneously for super resolution imaging.
2. Planar lens and planar axicon
The building blocks of metasurfaces in our design are spatially variant rectangular nano-apertures. The unit cell is a single nano-aperture and they are arranged in a hexagonal lattice to increase the symmetry. As presented in Fig. 1(a), under circularly polarized light (CPL) illumination, the anisotropic plasmonic structures could generate geometric phases. In details, the cross-polarized component of the transmitted light acquire a phase shift of φ = 2σα, where σ = ± 1 represents the left and right handed circular polarization (LCP and RCP), α is the inclination angle of the aperture. By optimizing the size of rectangular nano-aperture, we get specific P & Q combination in Fig. 1(b) to realize proper transmission efficiency. The designed operation wavelength is λ = 10.6 µm, so we chose silicon as substrate. According to the unit cell simulation of CST MICROWAVE STUDIO (CST MWS), we determine the parameters of rectangular nano-apertures: h = 200 nm, L = 5.3 µm, P = 3.8 µm and Q = 1 µm. The relationship between the orientation of the aperture and imparted phase shift is shown as green line in Fig. 1(d), which agrees well with the geometric phase shift theory. The transmission coefficient of cross polarized light was around 26% as shown in the Fig. 1(d) by red line, corresponding to energy efficiency around 7%.
Fig. 1 Schematic of the metasurface based on geometric phases. (a) Sketch of metasurface. (b) Rectangular nano-apertures are arranged in a hexagonal lattice to increase the symmetry. The angle between the main axis and the x-axis is α. The distance between two adjacent apertures is L = 5.3 µm. The size of each nano-aperture is P = 3.8 µm and Q = 1 µm (c) The substrate of metasurface is Si, and metal layer is Au which thickness is h = 200 nm. (d) The simulation result of relationship between phase shift and orientation of the aperture α in green line, as well as with amplitude transmittance efficiency in red line.
To achieve the focal light spot beyond diffraction limit we design and calculate the phase map of common planar focusing lens and planar axicon, with corresponding phase distributions illustrated in Fig. 2(a). The yellow part is the x-y plane of the designed sample of planar lens and planar axicon. The distance between the center of sample and point PS on the plane is . In order to get a single focal point PF at distance f, we design a hyperboloidal phase profile imparted onto the incident wave front [13]. The red element is a spherical surface with radius equal to the focal length of f. And the phase shift at the point PS is related to the distance of , where PL is the projection of PS onto the spherical surface. Therefore, the phase distribution of common focusing lens can be calculated by [13]
where f is the focal length of lens, and k = 2π/λ means the wave vector. The phase profile of planar lens is a hyperboloidal radial surface after modulo operation by 2π, as shown in Fig. 2(b). When f >> R, based on the theory of Taylor expansion, Eq. (1) can be simplified into:Fig. 2 Schematic of designing planar lens and planar axicon. (a) To focus a plane wave into a focal spot, a hyperboloidal phase profile should be impacted onto the incident light. The blue conical surface and red spherical surface present different wave front, on which the projections of point PF were PA and PL separately. The phase shifts of point PS on metasurface are proportional to the distance and for planar axicon and planar lens separately. The distance between point PS and point O was r. In our design, the radiuses of planar lens and planar axicon are equal, R. The focus spot of planar lens is PF, so in this figure, we present the condition of r = R. The focal length of planar lens is f, and meanwhile the focus of planar axicon is along the optical axis from point O to PF. (b) The phase profile of planar lens after modulo operation by 2π. (c) The phase profile of planar axicon after modulo operation by 2π. (d) The scanning electron microscope (SEM) image of the metasurface.
In theory, the phase distribution of axicon has a linear correlation with radius r. The blue part in Fig. 2(a) is a conical surface, and the point PA is the projection of PS onto it. Strictly speaking, because of the long depth of focus, the planar axicon doesn’t have a specific focal length. So here we use the concept of focal length in planar lens to calculate the target angle β. Assuming the radius of metasurface both was R in planar lens and planar axicon, then the angle β = tan−1(R / f). Therefore, we can get the phase map of axicon lens [13]:
where r and k in Eq. (3) have same meaning with those in Eq. (1), and when f >> R, Eq. (3) can be simplified to:Therefore, the phase profile of planar axicon is a conical surface, which is shown in Fig. 2(c) after modulo operation.
The designed radius of planar lens and planar axicon is R = 2.5 mm and the focal length is f = 100 mm. Therefore, the designed angle of planar axicon is β = 1.43°.
To prove our method and design, a metasurface was fabricated. The silicon wafer was coated with 200 nm gold layer through magnetron sputtering, and then a layer of AZ-3100 photoresist was spun onto the substrate and baked at 100°C for 10 min. Afterwards, the array of rectangular nano-apertures was patterned onto the photoresist by contact exposure method. Eventually, the rectangular nano-apertures array was obtained by ion beam etching (IBE) and the residual photoresist was removed with acetone. A scanning electron microscope (SEM) image of the metasurface is shown in Fig. 2(d).
The performance of designed planar lens and planar axicon is simulated in theory. In details, Fig. 3(a), (b) show the simulated focus spots of planar lens and planar axicon at the distance z = f separately. Intensity distributions along the black lines (y = 0) of Fig. 3(a), (b) are shown in Fig. 3(c). It’s obviously that the focus spot of planar axicon is smaller than that of planar lens. Subsequently, we measure the focus spot of planar lens and planar axicon at the distance z = f. In our experiment, the incident light was converted to circular polarization via the combination of a linear polarizer and a reflective quarter-wave plate. The light field was recorded by a charge coupled device (CCD). The experiment results of planar lens and planar axicon are shown in Figs. 3(d) and 3(e) separately. The display range of focus spot was 400µm × 400µm and the transversal lines across the beam center on every intensity map are shown in Fig. 3(f). The experiment results agree well with simulation and imply that the planar axicon could achieve focus spot beyond diffraction limitation when illuminated by parallel beam.
Fig. 3 The focus results of planar lens and planar axicon. (a) The simulated focus result of planar lens by vector diffraction method. (b) The simulated focus result of planar axicon by vector diffraction method. (c) The transversal line of y = 0 µm in (a) and (b). (d) The experimental focus result of planar lens. (e) The experimental focus result of planar axicon. (f) The transversal line of y = 0 µm in (d) and (e).
3. Super resolution imaging with Planar Bessel-lens
With the focal spot beyond diffraction limitation we got above, we could design a metasurface element to achieve super resolution imaging based on the super-resolution focusing of Bessel beam. To explain our theory, we assume that one piece of lens L can be split into two ideal thin lenses, l1 and l2, and imaging progress of them is decomposed into two steps as shown in Fig. 4(b): The first lens l1 is placed its focal length f1 from object plane to access the infinity space; then second lens refocus these series parallel light into focus spot on the imaging plane, so the distance between second lens and imaging plane is f2. For clarity of drawing, the distance between two ideal thin lenses is denoted as D. If the distance D = 0, the phase of two lenses could be added as:
Setting the focal lengths of two ideal thin lenses equal to each other, f1 = f2 = f, we know that the focal length of lens L is F = f / 2.Fig. 4 The experimental setup and method of planar lens and planar Bessel lens for imaging. (a) The phase profile of planar lens. (b) The schematic designed method of planar lens. (c) The illustration of experimental setup. The part included by green cuboid is detailed shown in (b) and (e). (d) The phase profile of planar Bessel lens. (e) The operation principle of planar Bessel lens.
So retaining the first step of imaging progress above, and replacing the second ideal thin lens by an axicon, we can realize super resolution imaging. Actually, a lens and an axicon both are essential elements for practical Bessel beam microscopy. In theory, we can combine them into one piece of optical device, and for the convenience of expression, here we introduced a new concept of “Bessel-lens” to express this new kind of optical element. However, Bessel-lens is difficult to fabricate for traditional optical machining because of its complex surface. But it is very easy for metasurface to achieve this irregular phase profile. The phase maps of planar lens and planar axicon can be calculated by theory in Fig. 2 as we discussed before. And therefore the phase distribution of planar Bessel-lens can be derived as:
where R means the maximum of r.Based on the theory above, we designed and fabricated two metasurfaces to validate our method. The parameters of planar lens and planar Bessel-lens in Eq. (5) and (6) are: R = 10 mm, f1 = f2 = f = 100 mm, λ = 10.6 µm. The phase profiles of planar lens and planar Bessel-lens are shown in Figs. 4(a) and 4(d). Also we designed a series of 5 × 5 aperture arrays as an object, as shown in Fig. 4(c); the radius of each aperture is 25 µm and the distance between two adjacent apertures is variant from 110 µm to 250 µm.
The light path in our experiment is shown in Fig. 4(c). The light source is CO2 laser whose wavelength is 10.6 µm and output polarization is linear state. After passing through the adjustable attenuator, the polarization of laser is refined by a polarizer, which main axis direction is orientated at 45°with respect to the horizontal plane. The quarter wave plate we used in experiment is reflecting mode to convert the linearly polarized light into circularly polarized light. The object consisted of aperture array is illuminated by laser and the distance between two adjacent apertures is 110 µm.
Planar lens and planar Bessel-lens are placed at same position and the imaging results of simulation and experimental are compared in Fig. 5. Besides, the simulated results of planar lens and planar Bessel-lens are shown in Figs. 5(a) and 5(b) separately. From the simulated results we find that planar lens couldn’t distinguish apertures with each other, but planar Bessel-lens obtain trenchant image meanwhile. The experiment results are presented in Figs. 5(c) and 5(d) and agree well with simulation. The transmitted intensity patterns were recorded by a CCD camera (384 × 288 pixels, UA330, Guide-Infrared Inc.). And the size of each pixel was 25 µm × 25 µm.
Fig. 5 The imaging results of planar lens and planar Bessel-lens. (a) Simulated imaging result of planar lens (b) Simulated imaging result of planar Bessel-lens. (c) Experimental imaging result of planar lens. (d) Experimental imaging result of planar Bessel-lens.
4. Discussions
From the simulation and experiment results, we find that the image of 5 × 5 aperture array formed by the planar Bessel-lens becomes a checkerboard pattern because of the interference of light field which reduces the performance of image. However, the phenomenon would not exist if we use the incoherent light source to illuminate the object. As we not possess an incoherent light source at wavelength of 10.6 µm, simulations based on the point source method is performed to further confirm the validity of Bessel lens in super resolution imaging. The simulated parameters of planar lens and planar Bessel-lens in Eq. (5) and (6) are R = 5 mm, f1 = f2 = f = 100 mm, λ = 10.6 µm which means the numerical aperture (NA) is 0.05. The simulation result is displayed in Fig. 6. The object shown in Fig. 6(a) is random distribution of apertures whose radius is 25 µm. Some representational parts are circled by blue ellipse line. Figure 6(b) shows the simulated imaging results of planar lens. A typical zone is zoomed in and presented as an inset which includes a pair of adjacent apertures. The planar Bessel-lens can distinguish them with each other well in Fig. 6(c) while planar lens cannot. Although there are water ripples around the image point in Fig. 6(c), they don’t reduce the imaging resolution.
Fig. 6 The simulated imaging results of planar lens and planar Bessel-lens illuminated by incoherent light. The typical parts are circled by blue ellipse. (a) The object is composed of random distributed apertures whose radius is 25 µm. (b) The simulated imaging result of planar lens. (c) The simulated imaging result of planar Bessel-lens. (d) The imaging result of two point sources spacing at 92μm for planar lens with NA = 0.05. (e) The imaging result of two point sources spacing at 92μm for planar Bessel lens with NA = 0.05. (f) The transversal lines of y = 0 µm in (d) and (e).
The optical resolution of traditional lens is about Rlens = 0.61λ/NA due to the diffraction limit, and the resolution of planar lens with NA = 0.05 is around Rlens = 129.3μm. Thus the planar lens can’t distinguish two point source spacing at 92μm while planar Bessel lens can as Figs. 6(d)-6(f) showing. According to the numerical calculation result by PSF function, the resolution of Bessel lens is about RBessel-lens = 0.7 Rlens.
In principle, the method of super resolution imaging by planar Bessel-lens can be used in visible light range in practical impact optical system, and the imaging performance can be further improved by using the incoherent light source.
5. Conclusions and outlook
In summary, we have proposed an approach to achieve super resolution imaging based on the new concept of Bessel-lens which merges the phase maps of a lens and an axicon. Benefited from the local geometric phase modulation, it is simple to access the targeted complex phase profile but rather difficult for traditional optical machining. We designed and fabricated some metasurfaces as planar axicon and planar Bessel-lens to confirm our method. In the meanwhile, planar lenses with the same size were also designed and fabricated for performance comparison. We found that the Bessel lens can distinguish small distance that planar lens with corresponding size and numerical aperture cannot. Considering the interference of coherent light would reduce the imaging performance, the imaging progress with incoherent light is simulated to confirm the validity of Bessel lens in super resolution imaging.
Note that, the proposed method can be expanded to visible light range to achieve super resolution imaging. In the future work, the metasurface devices which work in visible range would be designed and fabricated to prove the validity of this method.
Funding
973 Program of China (No. 2013CBA01700), National Natural Science Foundation of China (No. 61622508 & 61575201).
Acknowledgements
The authors would like to thank Dr. Wang Jiong and Dr. Zhang Zuojun for the help on the fabrication of samples.
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