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Abstract
A focusing device is one of the key elements for terahertz applications, including homeland security, medicine, industrial inspection, and other fields. Sub-wavelength tight-focusing of terahertz waves is attractive for microscopy and spectroscopy. Flat optical lenses based on metasurfaces have shown potential in diffraction-limit focusing and advantages of ultrathin thickness and lightweight for large-aperture optics. However previously reported THz metalenses suffered from either polarization-dependency or small numerical aperture (NA), which greatly limits their focusing performance. In this paper, to achieve high-NA and polarization-free operation, we proposed a polarization-independent dielectric metasurface with a sub-wavelength period of 0.4λ. A planar terahertz lens based on such metasurface was designed for a wavelength of λ = 118.8 μm with a focal length of 100λ, a radius of 300λ, and a high NA of 0.95, which was fabricated with a silicon-on-insulator wafer. The experimental results demonstrate a tight focal spot with sub-wavelength full widths at half-maxima of 0.45λ and 0.61λ in the x and y directions, respectively, on the focal plane. In the x direction, the size of 0.45λ is even smaller than the diffraction limit 0.526λ (0.5λ/NA). Such a metalens is favorable for sub-wavelength tight-focusing terahertz waves with different polarizations, due to its polarization independence. The metalens has potential applications in THz imaging, spectroscopy, information processing, and communications, among others.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Owing to the unique properties of low photon energy, fingerprint spectrum of biological and medical materials, and high penetrating ability inside a wide variety of materials, terahertz (THz) waves have promising potential in many applications, including material identification [1,2], imaging [3,4], and nondestructive sensing [5,6]. However, developing high performance THz elements, such as waveplates, lenses, and others remains a challenging issue of THz technologies. Dielectric refractive lenses and parabolic mirrors are conventional optical elements available for THz focusing and imaging. Despite the advantage of high reflective efficiency, the application of a parabolic mirror is limited by its off-axis operation, size, and weight. Like other bulky optical elements, conventional THz lenses are not only bulky and heavy, but have low efficiency due to material absorption within thick refractive lenses.
Metasurfaces, consisting of periodic sub-wavelength artificial structures, are quasi-two-dimensional artificial materials grown on ultrathin substrate. They provide an alternative way of manipulating the light wavefront, including amplitude [7,8], phase [9,10], and polarization [11,12], with sub-wavelength resolution, and therefore can realize optical elements with light weight and small thickness, compared to their conventional counterparts. In particular, optical elements realized with metasurfaces are flat and thin, which have no aberration caused by curved surfaces and have low material loss due to their short absorption length inside. Compared with the metasurfaces based on metallic resonators, in which the Ohmic loss is an intrinsic limit for high-efficiency wavefront manipulation, all-dielectric metasurfaces might reach a high-throughput efficiency of over 95% [13]. THz all-dielectric magnetic mirror metasurfaces have been proposed for the generation of vortex and Bessel beams [14]. A metalens consisting of a GaAs pillar was proposed for the THz wave focusing at frequency of 1 THz, and its focusing performance was theoretically investigated [15]. Single-layer metasurfaces were suggested for focusing THz waves for dual-wavelength operation [16]. At the frequency of 0.17 THz, a focusing lens was reported with a parallel stack of 100-μm-thick metal plates, which form an array of parallel-plate waveguides [17]. Focusing of THz waves was also experimentally demonstrated by a lens based on a C-shaped split-ring-resonator [18], quadratic phase metasurfaces [19], reflective metasurfaces [20], and dielectric metasurfaces [21].
Sub-wavelength tight-focusing is of great interest in high-resolution applications. Theoretical studies have been conducted for sub-wavelength focusing of THz waves with lenses based on annular copper slots [22], silicon hyperbolic metamaterials [23], grapheme metasurfaces [24], and two-dimensional dielectric metamaterials [25]. Sub-wavelength focusing of THz waves has been demonstrated using polarization-dependent dielectric metasurfaces with a smallest spot size of 0.87λ [26], where λ is the operating wavelength. Recently, we experimentally demonstrated a dielectric metalens for THz focusing and imaging that is based on a rotational geometric phase for circularly polarized waves [27]. However, polarization-dependent lenses only work for a particular polarization. Focusing of complex vectorial waves requires polarization-independent manipulation of wavefronts.
The previously reported THz metalenses suffered from either polarization-dependent or small NA, which greatly limits the focusing performance. To overcome these problems, in the present work, for sub-wavelength focusing of THz waves, a high-NA lens based on a polarization-independent dielectric metasurface with a period of 0.4λ was proposed, which was then fabricated using a silicon-on-insulator (SOI) wafer. Experiments were conducted to evaluate the focusing performance of the lens, which demonstrated the subwavelength focusing ability of the lens with the smallest full widths at half-maxima (FWHMs) of 0.45λ and 0.61λ in the x and y directions, respectively, under the normal incidence of a linearly polarized (in y direction) wave at a wavelength of λ = 118.8 μm.
2. Lens design
To achieve polarization-independent wavefront control, a series of metasurfaces were proposed for operation at a wavelength of λ = 118.8 μm. The unit size of the metasurfaces is the same. The basic structure of the proposed metasurfaces is illustrated in Fig. 1, where P is the unit size. On a Si substrate, there is a 2-μm-thick layer of SiO2 that serves as a stop layer in the later fabrication process. On the SiO2 layer, it is a Si cube with a height of t and side length of Lx = Ly = L, where the Lx is equal to Ly to guarantee the polarization-independent wavefront control.
To optimize the performance of metasurfaces, for given t = 65.7 μm, the dependency of phase-shift and amplitude transmittance on the unit size P and the cubic transverse size L were investigated utilizing finite-difference time-domain numerical simulation with commercial software FDTD Solutions (Lumerical Inc.) . The refractive index of Si used in the simulation is 3.4, and the 2-μm-thick SiO2 layer also was taken into account in the simulation. In the simulation, periodical boundary condition was applied. Figures 2(a) and 2(b) illustrate the theoretically predicted amplitude transmittance and phase shift as functions of both P and L. It is clearly seen that a phase-shift range of 2π can be achieved while keeping a relatively high amplitude transmittance. Trade-offs have been made among the metasurface period P, metasurface structure size L, and amplitude transmittance to guarantee a comparative high-efficiency and feasible fabrication. To achieve sub-wavelength modulation, while avoiding the coupling between neighboring elements, the period was chosen as P = 46.9 μm. This period is smaller than 0.4λ, which is critical for realization of high-NA lens.
Fig. 2 (a) Amplitude and (b) phase-shift (in rads) dependence on the parameters of metasurface period P and metasurface structure size L.
To simplify the fabrication, only eight different metasurfaces with different transverse sizes L are chosen to realize eight different phase shifts with nearly equal phase intervals (i.e., 0, 0.24π, 0.46π, 0.76π, π, 1.25π, 1.49π, and 1.75π) and with high amplitude transmittance. Table 1 lists the corresponding values of the amplitude transmittance, phase shift, and transverse size L for the eight metasurfaces. Figure 3 plots the amplitude transmittance and phase shift with respect to the eight metasurface elements. According to Table 1 and Fig. 3, the overall amplitude transmittance is greater than 80% for all eight metasurface elements. The transverse size of the metasurface structures varies from 5.6 to 31.9 μm. Simulation also reveals that a fabrication tolerance of ± 0.5 μm only leads to a maximum deviation of 0.17 rad in phase shift for all eight metasurface elements.
Table 1. Major parameters of the eight elements.
Fig. 3 Phase shift and amplitude transmittance of the eight metasurface structures, listed in Table 1, at normal incident wave.
To study the dispersion property of the metasurfaces, the phase shift and the amplitude transmittance were obtained for the eight metasurface elements in the wavelength range from 100 to 124 μm by numerical simulation. As plotted in Fig. 4(a), it is found that the phase shift varies with wavelength in different slopes, indicating deviation from the designed phase value at wavelengths other than the optimal wavelength of λ = 118.8 μm. However, the phase shift can still cover a 2π range. Within the simulated wavelength range, the amplitude transmittance remains at a relatively high value above 0.8.
Fig. 4 (a) Amplitude transmittance and phase shift of transmitted wave with respect to THz-wave wavelength for the eight different metasurface elements under study; (b) amplitude transmittance and phase shift of transmitted wave at different incident angles, i.e., 0°,22.5°, 45°, and 67.5°, for the eight different metasurface elements.
The phase shift and amplitude transmittance were also investigated at different incident angles of 0°, 22.5°, 45°, and 67.5°, where the reflection on the substrate bottom surface was ignored. It was seen that, for a small incident angle of 22.5°, the amplitude transmittance is approximately 0.8, while its value experiences an obvious decrease as the incident angle further increases. Generally, the elements with a larger size undergo a faster reduction in amplitude transmittance as the angle increases. As seen from Fig. 4(b), element no. 8 only has an amplitude transmittance of 0.3 at an incident angle of 67.5°. As for the phase shift, its relative value between the eight elements only has a small variation for all simulated incident angles.
Based on the proposed polarization-independent dielectric metasurface structures, a planar dielectric lens was designed. The lens phase distribution follows the commonly used hyperboloidal profile, i.e.
where xi,j and yi,j denote the center coordinates of the unit located in the ith column and jth row, respectively; f is the focal length; λ is the operation wavelength of 118.8 μm; and n is an integer. For each unit of the array, its ideal phase was approximated by one of eight metasurface elements with the nearest phase value. For sub-wavelength tight focusing, the lens was designed with a large NA of 0.95, radius of 300λ (35.64 mm), and focal length of 100λ (11.88 mm), whose diffraction limit is 0.526λ (0.5λ/NA).Figure 5 illustrates the focal-plane (at z = 100λ) intensity distribution of the under illumination of a plane wave with linear polarization in y- direction, which is obtained with numerical simulation based on the angular spectrum method [28]. In the simulation, the amplitude and phase distribution were assumed to be uniform within each metasurface unit. For the y-linearly polarized wave, the longitudinal components Ez generated on the focal plane will lead to broadening of focal spot transverse size in y-direction. In addition, the incident beam profile can also have effect on the spot size. In the simulation, the cross-section of the incident wave was in elliptical shape. The ratio between the semi-major axis and semi-minor axis in x- and y- directions is defined as α. Figures 5(a)-5(d) give the intensity profiles on the focal plane for different α values of 1, 2, 3 and 4. There is a clear variation in the shape of the focal spot as α changing, while the semi-major axis remains 300λ. The inset of Fig. 5(a) is the focal plane intensity distribution of the transverse electrical field only, which shows a good symmetry at the absence of longitudinal components. Figure 5(e) and 5(f) plot the intensity distribution curves along the x- and y- directions for different α value of 1, 2, 3 and 4, where Etrans and Etotal denote transverse and total electrical field components, respectively. As shown in Fig. 5(e), the FWHM of the focal spot is almost the same for all cases in the x-direction. The corresponding value of FWHM is approximately 0.46λ, which is slightly smaller than the diffraction limit of 0.526λ (0.5λ/NA). However, in the y- direction, the FWHM is 0.96λ, 0.81λ, 0.86λ and 0.96λ respectively, which is larger than the FWHM of the transverse field for α = 1. It was found that, with the increase of α, the FWHM decreases first due to suppressed longitudinal components on the focal plane, and then increases because of reduced actual aperture in y-direction. However, for all four case, the FWHM of the full field is sub-diffraction in x-direction and sub-wavelength in y-direction. According to numerical simulation, the device focusing efficiency was estimated to be approximately 46.8%. For a real device, the efficiency could be lower due to the substrate reflection. Efficiency improvement is expected with low reflective metasurfaces fabricated on the substrate bottom side. It was also found that the FWHM of the transverse component remains a value smaller than the diffraction limit, i.e. 0.482λ, when the NA of the lens reduced to 0.93, corresponding to a lens with focal length of 100λ and an effective radius of 250λ. Since the transverse component plays key role in high-NA microscope [29], it is important to achieve sub-diffraction focusing of transverse electrical field for microscopy applications.
Fig. 5 Theoretical simulation results: the optical intensity on the focal plane obtained under the illumination of linearly polarized (in y-direction) plane wave using the angular spectrum method. (a)-(d) Two-dimensional intensity profile on the focal plane for different α value of 1, 2, 3 and 4 ; The inset of (a) is the focal plane intensity distribution with transverse electrical fields only; (e) and (f) are the intensity curves on the x and y axis, respectively, for different α value of 1, 2, 3 and 4, where Etrans and Etotal denote transverse and total electrical field components, respectively
Figure 6 gives the major parameters, including peak intensity (red), FWHM (blue), and sidelobe ratio (green, the ratio of maximum sidelobe intensity to the peak intensity), of the transverse polarization along the optical axis, where the corresponding lens diffraction limit (0.5λ/NA, black-dashed) and super-oscillation criteria (0.38λ/NA, brown-dashed) [30] were also plotted for comparison. It was seen that, in addition to the major peak located at z = 100λ, there are two local peaks at z = 98λ and 102λ respectively, and the FWHM of the three spots is 0.46λ, which is smaller than the diffraction limit, as indicated by the black-dashed line. The sidelobe ratio is approximately 6%, 23% and 20% for the spots located at z = 100λ, 98λ, and 102λ respectively. It is also interesting to note that at the two local minimum located at z = 98.6λ and 101.4λ, the corresponding FWHM of the in-plane optical intensity is 0.32λ, which is smaller than the super-oscillation criteria of 0.4λ (0.38λ/NA), as indicated by the brawn-dashed line. However, at those two points, the sidelobe ratio is as large as 121%. Such subdiffraction feature with large sidelobe can be understood by optical super-oscillation phenomenon [31].
Fig. 6 The major parameters of the focused transverse electrical field: the peak intensity (red), FWHM (blue), and sidelobe ratio (green) along the optical axis, where the corresponding lens diffraction limit (blacked-dashed) and super-oscillation criteria (brawn-dashed) were also plotted for comparison.
3. Experimental results
In the fabrication, to guarantee the large-area uniformity in the height of the cubic metasurface structures, a SOI wafer with a top-layer thickness of 65.7 μm was employed. The pattern of the mask was firstly transferred onto the photoresist by ultraviolet lithography and developing steps. Inductively coupled plasma etching was applied to generate the proposed cubic metasurface arrays, and thus form the designed dielectric planar lens. During the etching process, the buried oxide layer acts as an etching stop layer in the fabrication to guarantee the height of the structures. Figures 7(a) and 7(b) are photographs of the entire lens and a magnified image of the lens central area, as indicated by the red square in Fig. 7(a). Figure 7(b) demonstrates the formation of all eight metasurface elements. It is also found that the even smallest structure is in a good cubic-shape.
Fig. 7 (a) Photograph of the metalens based on a dielectric metasurface; (b) optical microscope image of central area of metalens; (c) diagram of the focusing experimental layout.
To evaluate the focusing performance of the lens, experiments were conducted at a wavelength of 118.8 μm. The incident THz wave was linearly polarized in the y direction and produced by a FIRL100 laser (Edinburgh Instruments Ltd., UK). To observe the tight focusing, a THz focal-plane-array microbolometer (Microcam 3 THz, Thermoteknix, UK) with a high spatial resolution of 640 × 480 and pixel size of 17 × 17 μm2 was used to capture the optical intensity of the focused wave. The diameter of the THz laser beam was first expanded into approximately 420λ (approximately 50 mm) with two 90° off-axis parabolic mirrors, and then was normally impinged on the lens. The microbolometer was set directly after the lens to collect the focal-plane intensity distribution. The corresponding experimental setup is depicted in Fig. 7(c).
In the experiments, a focal spot was found near the designed focal plane. The corresponding two-dimensional optical intensity obtained in the experiment is presented in Fig. 8(a), which shows a clear bright spot surrounded by a weak ring-shaped sidelobe, which agrees well with theoretical prediction. The inset of Fig. 8(a) depicts the incident beam profile before expansion, which shows clear non-uniform intensity distribution in the beam cross-section. To evaluate the size of the focal spot, the optical intensity distributions across the central peak in the x and y directions were plotted in Figs. 8(b) and 8(c), respectively, which are extracted from the two-dimensional intensity profile illustrated in Fig. 8(a). It is noted that there are only three to five data points in the peak area in both intensity curves. To calculate the size of the focal spot, Gaussian fits were applied to the two intensity curves around the peak area, as indicated by the red curves in Figs. 8(b) and 8(c), which give values of 0.45λ and 0.61λ in the x and y directions, respectively. The FWHM in the x direction is smaller than the diffraction limit of 0.526λ (0.5λ/NA) and quite close to that predicted by the numerical simulation as in Fig. 5, while the FWHM in the y direction is slightly larger than the diffraction limit. The non-symmetry of the measured focal spot is attributed to the longditudinal polarized component on the focal plane caused by the linearly polarization (in y direction) incident wave and non-symmetrical profile of the incident THz wave in the experiment. In the focusing experiments, no obvious change was observed in the optical intensity on the focal plane when rotating the lens around its optical axis.
Fig. 8 Experimental results: the optical intensity on the focal plane obtained under the illumination of a linearly polarized wave by a THz focal-plane-array microbolometer. (a) Two-dimensional intensity profile on the focal plane, and the inset is the beam profile before expansion; (b) and (c) are the intensity curves (blue) in the x and y directions, respectively, where the red curves are the corresponding Gaussian fits of the peaks.
4. Discussions and conclusions
In conclusion, we have proposed sub-wavelength polarization-independent Si cubic metasurfaces with comparatively high theoretically predicted efficiency. The sub-wavelength size is key for fine modulation of the wavefront and is crucial for realizing high-NA lenses. Polarization-independence allow the lens to focus waves with different polartizations, especially vectorially complex polarizations. Our numerical simulation shows that the beam profile has a great effect on the spot size, especially when the longitudinal electrical field is taken into account. Based on the proposed polarization-independent metasufraces, a dielectric metalens was designed and fabricated with a high NA value of 0.95. Experimental investigations demonstrated a tight sub-wavelength focusing performance, achieving FWHMs of 0.45λ and 0.61λ in the x and y directions on the focal plane. In the x direction, the FWHM of the spot is smaller than the diffraction limit of 0.526λ (0.5λ/NA). Due to the longditudinal component, the FWHM in y direction is larger than that in x direction. Fortunately, in high-NA microscope, only transverse electrical fields contribute to the microscopy applications. The lens has potential applications in high-resolution THz imaging, spectroscopy, information processing, communications, and other fields.
Funding
China National Natural Science Foundation (61575031); National Key Basic Research and Development Program of China (Program 973) (2013CBA01700); Fundamental Research Funds for the Central Universities (106112016CDJXZ238826, 106112016CDJZR125503); National Key Research and Development Program of China (2016YFED0125200, 2016YFC0101100).
Acknowledgments
Authors also thank LetPub (www.letpub.com) for their linguistic assistance during the preparation of this manuscript.
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