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Abstract
Metasurfaces have shown great potential in versatile areas such as vortex-beam generators, metalenses, holograms and so on. However, chromatic error hinders metasurfaces, especially metalenses, from wider applications. In this paper, we demonstrate a novel design for a transmissive mid-infrared achromatic bifocal metalens with polarization sensitivity. The compensation phase is used to eliminate the chromatic aberration. Simulation results show that, over a continuous waveband from 3.9 to 4.6µm, the focal length only changes by 2.26% with an average focusing efficiency of about 18%. This work can push the practical application of mid-infrared metasurfaces.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Metasurfaces are artificial two-dimension planar material with sub-wavelength nano-structures. With ability of engineering the amplitude, phase and polarization of electromagnetic waves [1–4], metasurfaces allow researchers to realize unconventional optical phenomena. Thanks to the mature fabrication technology, the cost of metasurfaces is greatly reduced compared with traditional optical glass, and lots of metasurface designs have been demonstrated, such as vortex-beam generators [5–8], metalenses [9–11], holograms [12–14], beam deflectors [15–17] and so on. However, these plasmonic metasurfaces suffer from ohmic loss and demonstrate a low device efficiency. Therefore, dielectric metasurfaces have gradually attracted researchers’ attention, and series of progress has been made [18–21].
Despite showing great potential in replacing traditional optical elements, it is difficult for metasurfaces to have wider applications due to the chromatic aberration, especially in imaging. Several efforts have been paid on achromatic focusing [22–30]. A pioneering work by Aieta et al. in 2015 demonstrated a multi-wavelength metalens without chromatic errors in the near-infrared by using dispersive phase compensation [27]. In 2018, Wei Ting Chen et al. showed the achromatic imaging from 470 to 670nm by simultaneously controlling the phase, group delay and group delay dispersion of the light [28]. In the same year, Shane Colburn et al. proposed to combine computational imaging and metasurface optics, and realized broadband achromatic imaging in visible light [29]. Compared with dispersion engineering works, this method can help realize achromatic imaging with larger numerical aperture (NA). However, the real-time imaging cannot be achieved for the required image filtering. In 2020, Abdoulaye Ndao et al. enforced the slopes of the phase-shift from the center of the metalens to its edge and realized achromatic focusing from 640 nm to 1200 nm [30].
Mid-infrared (MIR) wavelength covers the rotational–vibrational spectroscopy of most molecules and an atmospheric window. Therefore, MIR metasurfaces can play an important role in photoelectric countermeasure [31], night vision imaging system and optical free-space communications [32]. In particularly, the achromatic bifocal metalens (ABM) with polarization sensitivity in MIR can obtain polarization information while keeping imaging systems compact, and the polarization information is benefit for background suppression and camouflage recognition. Experimentally, the MIR broadband ABM with linearly polarization sensitivity has been demonstrated through simultaneously and elaborately manipulating the phase dispersion and the polarization based on all silicon metasurface in the Ref. [33].
Benefiting from the spatial multiplexing arrangement and multitasking design with elaborate arrangement of the unit cells, we realize a transmissive ABM with circular polarization sensitivity, shown in Fig. 1. The proposed ABM is based on amorphous silicon (α-Si) nanopost and calcium fluoride (CaF2) substrate with NA of 0.6. By changing the chirality of the incident light, different focal points will appear with the same longitudinal distance but different horizontal positions. Simulation results show that, over a continuous waveband from 3.9 to 4.6µm, the designed ABM demonstrates a variation of 1.05µm in focal length and sub-wavelength imaging characteristic.
2. Principle of achromatic bifocal metalenses
The phase profile of a normal transverse bifocal metalens follows
where, x and y are the horizontal and vertical coordinates in Cartesian coordinate system on the metasurface plane, f is frequency of the incident electromagnetic wave, c is the speed of the electromagnetic wave in vacuum, F is the focal length and 2d is the distance between two focal points along the x-axis. Similar to the standard metalens, the bifocal metalens can maintain the capacity of electromagnetic wave converging near the designed frequency, but the focal length changes. As the frequency decreases, the focal length gradually becomes shorter. This dispersion characteristic is almost the same to the diffractive lens, but opposite to the refractive lens. For a refraction lens with normal dispersion, as the frequency decreases, its refractive index decreases, thus showing a longer focal length. However, the diffractive lenses and metalenses follow Fresnel-Kirchhoff integral. For the constant phase profile, interference makes electromagnetic waves with lower frequency more concentrated and demonstrate shorter focal length. However, an achromatic lens can converge the electromagnetic wave with different frequencies to the same focal points, and this kind of lens is more desired in continuous waveband imaging.In order to achieve an ABM, the phase profile should be modified by a compensation phase expressed by [28,34–37]
The Eq. (2) can be seen as two parts. The former part is the hyperbolic phase profiles at the reference frequency, set to be fref=76.9Thz (λref=3.9µm) in this paper. The latter part is the function of the working frequency. Both the two parts have different values at different positions.
To obtain the former part, we use the Pancharatnam-Berry (PB) phase [38–41]. When the right-handed circularly polarized (RCP) light passes through the unit cell, the transmitted light can be expressed by the Jones vector [42]:
For the latter compensation phase, we use the propagation phase, which originates from the waveguide effect. This phase is given by ${\varphi _{pro}} = 2\pi {n_{eff}}{H_m}/{\lambda _0}$, where ${n_{eff}}$ is the effective index, Hm is the height of the nanopost, and λ0 is the wavelength in vacuum [43]. The propagation phase is the function of frequency, and the difference between its value at the maximum frequency (fmax) and the minimum frequency (fmin) is phase compensation value. As the size of the nanopost changes, the dispersion characteristics of its propagation phase will also change. This makes it possible to realize different compensation phase at different positions. It should be noted that, while introducing different compensation phase, the propagation phase at reference frequency will also change. Therefore, Eq. (5) should be modified to
By combining PB phase and propagation phase, the Eq. (2) is satisfied.
3. Characteristic of unit cells
The structure of the unit cell is the α-Si nanopost with an ellipse cross-section sitting on the CaF2 substrate, showing in Fig. 2. In the MIR waveband, α-Si and CaF2 have high (n=3.43) and low (n=1.35) refractive indexes respectively, and both show low optical loss. Firstly, the high contrast in refractive indexes between nanoposts and surrounding material (air) helps confine the electromagnetic wave within nanoposts, and the coupling between adjacent nanoposts will be suppressed. Secondly, the high refractive index of nanoposts can extend the optical path along the z-axis and support more modes in the Fabry-Perot resonator, which is benefit for expanding the range of phase compensation value [44]. In addition, the low refractive indexes of the substrate can decrease the reflection of incident electromagnetic waves, and therefore enhance the efficiency of metasurfaces. Considering the fabrication difficulty, the height of nanoposts (Hm) and substrate (Hs) should be fixed values. The Hm should be tall enough to provide enough range of phase compensation value, but the actual fabrication capacity will limit the aspect ratio not to be too large. The Hm and Hs are set to be 4µm and 5µm respectively, and the period (P) is set to be 1.8µm.
Fig. 2. (a) Diagram of the ABM with polarization sensitivity. Unit cells of two colors converge RCP and LCP electromagnetic wave respectively and are arranged alternately. The (b) perspective view, (c) front view and (d) top view of a unit cell.
To realize different compensation phase at different positions, we choose 15 unit cells by sweeping the major axis (Dx) and the minor axis (Dy) of the ellipse simultaneously, shown in Table 1. Characteristics of the unit cells are simulated by finite difference time-domain (FDTD) method [45–48], which is a numerical analysis technique used for modeling computational electrodynamics. The plane wave is perpendicularly incident from the substrate side along the z-axis. Periodic boundary condition is used along the x-axis and the y-axis, while the perfectly matched layers (PML) are used along the z-axis. Figure 3(a) shows the phase modulation capacity of three unit cells at different frequencies.
Fig. 3. (a) Phase modulation capacity of three unit cells with different parameters (Dx Dy). The shaded region marks the design waveband 65.2–76.9THz (3.9–4.6µm). (b) The required range of CPWW is a function of F and (r0+d), when the working waveband is determined to be 3.9–4.6µm. The phase profile of an ideal ABM at three different frequencies. The required group delay in (c) is positive, and the required group delay in (d) is negative. r0 is the radius of the metalens. The dotted lines indicate the position of the optical center.
Table 1. Characteristics of the selected unit cells
It can be seen that, among the design waveband 65.2–76.9THz (3.9–4.6µm), the phase modulation capacity increases almost linearly as the frequency increases. Therefore, the phase profiles of an ideal ABM at different frequencies should conform the form in Fig. 3(c), whose required group delay is positive. The phase compensation value difference between two adjacent unit cells is 15°, and the range of phase compensation value reaches 210°. It should be noted that, after the provided phase compensation value range is determined, the required range of compensation phase among whole waveband (CPWW) cannot exceed this value. Figure 3(b) shows the relationship between the required range of CPWW and F, (r0+d), when the working waveband is determined to be 3.9–4.6µm. The left part of the contour line with a value of 210° is the parameter of an achievable ABM with the unit cells in Tab. 1.
4. Result and discussion
Based on the aforementioned method and unit cells, two metalenses for the LCP and RCP light with different focal points are designed respectively. Shown in Fig. 2(a), the ABM with polarization sensitivity is designed by alternately placing unit cells of these two metalenses. So that the two metalenses can maintain the similar structure and demonstrate the similar imaging characteristics. The designed ABM, working at the waveband of 3.9–4.6µm, has a diameter of 68.4µm, an achromatic focal length of 46µm, and the distance between two focal points is set to be 9µm. For each metalens, the required range of CPWW is 203°, and the chosen unit cells can satisfy. The nanopost with a phase compensation value of 440° is arranged in the optical center of the metalens, where the max CPWW is need. The farther away from the optical center, the smaller the required CPWW.
The characteristic of the designed ABM is also simulated by FDTD method. The XLP plane wave is normally incident along the z-axis, and the PML are used along all three axes. Figure 4(a) shows the normalized electric field intensity distribution on the y=0 plane of the ABM among 3.9–4.6µm, and the green dotted line indicates the focal plane position. As comparison, the simulation results of a transmissive ordinary bifocal metalens (OBM) with the same parameters are shown in Fig. 4(b). Figure 4 intuitively illustrates that, the focal length of an OBM decreases dramatically, as the incident wavelength increases. In contrast, the focal length of the proposed ABM remains almost unchanged. In addition, the distance between two focal points of these two bifocal metalenses is closed to the design value of 9µm, and the maximum error does not exceed 0.2µm.
Fig. 4. The normalized electric field intensity distribution on the y=0 plane of (a) the ABM and (b) the OBM among 3.9–4.6µm when the XLP light is incident.
Figure 5(a) quantitatively descripts the change in focal length. Within the whole working waveband from 3.9µm to 4.6µm, the focal length of the OBM has a variation of 9.91µm (19.67% to the average focal length), while the focal length of the ABM demonstrates a variation of 1.05µm (2.26% to the average focal length). The focusing efficiency and the full width at half maximum (FWHM) of the ABM is shown in Fig. 5(b). The focusing efficiency is defined as the ratio of the energy in three times the FWHM range with the focal point as the center to the total incident energy. Compared with the increased focusing efficiency with increasing wavelength in Ref. [49], the focusing efficiency in this work demonstrates a better stability. This difference stems from that, the work in Ref. [49] takes the reference wavelength at the maximum wavelength, while our work takes the reference wavelength at the minimum wavelength. Moreover, the average focusing efficiency reaches about 18%, and all the calculated FWHM are in the range of 0.8λ–0.9λ, demonstrating the sub-wavelength imaging characteristic.
Fig. 5. (a) The focal length of the designed ABM and the OBM as a function of the wavelength of 3.9–4.6µm. The gray dotted line indicates the designed focal length F=46µm. (b) The focusing efficiency and the FWHM of the designed ABM as a function of the wavelength among 3.9–4.6µm.
The imaging characteristic of the designed ABM when the RCP light is incident is shown in Fig. 6 and Fig. 7. In this situation, only one focal point appears at (4.5, 0, Fλ)µm, and the variation of the focal length is 1.24µm, which is slightly larger than that value when the XLP light is incident. This difference stems from the simulation error. It can be seen from Fig. 7(a) that, the point spread function (PSF) maintains a good rotational symmetry, but the background noise increases as the wavelength increases. This phenomenon stems from the decreased polarization conversion efficiency with increasing wavelength. Even so, the simulated intensity profile is close to the corresponding diffraction limit, and the FWHM is in the range of 0.77λ–0.85λ. In addition, the focusing efficiency demonstrates the similar tend to that in Fig. 5(b) with a slightly larger value. The improvement in imaging quality is due to the reduction of crosstalk from the LCP light. When the LCP light is incident, the variation of the focal length, PSF, focusing efficiency and FWHM are almost the same, except that the focal points falls at (−4.5, 0, Fλ)µm.
Fig. 6. The normalized electric field intensity distribution on the y=0 plane when the RCP light is incident among 3.9–4.6µm. The green dotted line indicates the focal plane position.
Fig. 7. (a) The PSF and intensity profile on the average focal plane. The solid lines represent the simulated intensity profile, while the dotted lines represent the corresponding diffraction limited intensity profile. (b) The focusing efficiency and FWHM of the ABM, when the RCP light is incident.
It can be seen that, regardless of the polarization state of the incident light, the focusing efficiency of the designed ABM is not high. The energy loss mainly comes from two aspects. On the one hand, the alternating arrangement causes the half of the energy loss. On the other hand, the unit cell is not a good half-wave plate. It means that both the anomalous mode with PB phase and the normal mode without PB phase exist simultaneously, and the latter will not be converged. In addition, the unfocused light becomes background noise and affects the image quality, which seriously limits the working waveband of the ABM. These two issues can be improved by optimizing the unit cells to obtain high polarization conversion efficiency.
5. Conclusions
In summary, we propose and design a transmissive polarization-sensitive ABM with NA=0.6 working at the MIR waveband based on dielectric metasurface. By simultaneously tuning the length of the long axis and the short axis of the nanoposts, desired compensation phase is realized at the different positions. When changing the polarization state of the incident light, the focal points appear at the different horizontal positions with almost the same focal length. In theory, this design method can be extended to other waveband by adjusting the parameters of unit cells, and the result shown in this work is a proof of concept. We believe that, the realization of achromatic bifocal imaging will play an important role in the specific fields such as vision correction, optical free-space communications and imaging systems.
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.
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