Metasurfaces are widely employed in planar optics due to their ability to regulate the phase of incident light waves at sub-wavelength sizes, and are now adopted in beam generators, holographic imaging, beam shaping, and other aspects. Meanwhile, metalenses are metasurfaces that can be focused, and the produced light waves are characterized by hyperbolic phases and can provide greater diffraction efficiency than conventional lenses. Unlike traditional lenses that utilize the thickness of the material to achieve spatial focusing, metalens can adjust the phase distribution of incident light in the plane. Additionally, compared with traditional lenses, metalenses can reduce the system volume, and they are easy to integrate into other components. However, in the applications, due to the properties of the material itself, the chromatic aberration will be large, and eliminating the effect of chromatic aberration is essential for the metalens application.Metalens achieves focusing by regulating the phase of incident light, and the varying wavelengths cause changing phase, resulting in chromatic aberration. Since there is no spherical aberration in metalens, chromatic aberration is the most important source of aberration in imaging. More researchers are concentrating on the achromatic metalens design.

We design the metalens structure of all-silicon medium, with the phase modulation of the design band being 3-5 μm. The achromatic metalens design with a size of 37 μm×37 μm is realized by the transmission plate principle. The numerical aperture NA =0.24 of the designed metalens device concentrates 3-5 μm plane waves to the same focal point on the axis under the positive incidence, which can keep the focal length f=150 μm unchanged.

The designed nanopillar structure is an all-silicon medium, and silicon is a common infrared material, with high light transmittance in the 3–5 μm band. Meanwhile, the optical loss is very small, which can be ignored, and the metalens processing technology of silicon materials is relatively complete. Additionally, the nanopillar structure is periodically arranged, and the transmission phase theory is employed to change the equivalent refractive index by varying the nanopillar radius. Then, the metalens phase is regulated, and the phase compensation corresponding to different positions is provided to realize the achromatic function of the metalens (Fig. 2).

Finally, the shape of unit structure is a square substrate, and the nanostructure is a cylinder with spatial symmetry. Increasing the height of the element structure can both augment the corresponding phase change and expend the aspect ratio of the element structure (H/d), thereby increasing the processing difficulty. It is necessary to balance the relationship between height and phase to realize large enough phase change and reduce the height of element structure. Since the processing technology of the metalens is not perfect, the height of all nanopillars of metalens is selected.

The geometric parameters of the unit structure are optimized by finite-difference time-domain (FDTD) simulation software, the transmission phases of the element structure of different geometric parameters are obtained, and then the data such as phase and amplitude are utilized to establish the database required for the full-mode design.

The height H of the fixed cell structure is 6 μm and the period p is 0.8 μm, with the changed radius of the unit structure. Meanwhile, the phase distribution and transmission corresponding to different radii are obtained, and the nanopillar radius is determined to be 0.05-0.35 μm (Fig. 3). The radius of the fixed nanopillar is 0.05-0.35 μm and the cell structure period p=0.8 μm. The height of the element structure is selected as 4, 5, and 6 μm for simulation, and the height of the unit structure is determined to be 6 μm during the metalens design. Under the unchanged height H and radius r of the nanopillar, the structural periods of the selected elements are 0.8, 1.0, and 1.2 μm respectively, and the period is determined to be 0.8 μm. This ensures the phase coverage of 2π and the high transmission of the structure.

This shows the focal length curves of different wavelengths when circularly polarized light, X-linearly polarized light, and Y-linearly polarized light are normal incident (Fig. 7). The figure on the right reveals that the focal length values of the three polarized lights are very close, indicating the consistent designed metalens structure. The polarization is independent due to the high spatial symmetry of the cylindrical cell structure.

The focusing efficiency is the ratio of the light intensity of the focused circular polarized beam in an Airy spot to the light intensity of the transmitted beam. It is shown that the focusing efficiency curve changes with varying working wavelengths (Fig. 8). The lowest focusing efficiency is 44.64% under the wavelength of 3.5 μm, and the highest focusing efficiency is 65.2% under the wavelength of 5 μm, with sound focusing effect. The focusing efficiency is about 54% over the entire operating bandwidth, and this change is mainly caused by the interaction between different cell structures. The geometric parameters (p and H) of element structure are optimized to achieve high and uniform focusing efficiency.

We design a broadband achromatic metalens, which employs the transmission phase theory and the periodic arrangement of the unit structure to realize the dispersion control of the mid-infrared broadband. The designed element structure is an all-silicon medium, and the geometric parameters of the unit structure are optimized by FDTD commercial simulation software. Meanwhile, the influence of different parameters of the element structure on phase and transmission is analyzed, and a database of the geometric parameters and phase and amplitude response of the nanopillar is established. The achromatic focusing function is realized in the 3-5 μm band, and the full-wavelength focusing efficiency is about 54%. The proposed unit structure of the broadband achromatic metalens is simple and not affected by the polarization state, improving the utilization efficiency of the device. Subsequently, more types of cell structures can be introduced to achieve an achromatic focusing effect with larger bandwidths, which has certain application prospects in color display imaging systems. Although we only perform simulation verification in the visible region, the design principles and methods of the device can be generalized to other bands such as long-infrared bands.