Broadband Achromatic Metalens in the Long-Wave Infrared Regime

Compared with conventional lenses, metalenses can offer smaller form factor, light weight and new functionality, and have become the focus of the recent investigations of metasurface. To date, quite a lot of studies have been reported for chromatic aberration correction of metalenses. Noting most work was carried out to satisfy the required group delay or dispersion by introducing sophisticated and integrated nanopillars, which are obtained using computationally intensive global search algorithms. Generally, the design is time-consuming. Here, we propose an alternative scheme, where metalens that corrects the chromatic aberration can be designed by optimizing the spectral degree of freedom C(ω), rather than relying on time-consuming algorithms. The proposed metalens is compatible with the complementary metal oxide semiconductor (CMOS) fabrication process. The numerically demonstrated metalens with a diameter of 200 μm and a numerical aperture (NA) of 0.24 has a small focal length fractional change (FLFC) of 4.1% over the wavelength of 9∼12 μm with an average efficiency of 51.7% in transmission mode. Moreover, the effect of fabrication deviations on the performance of the metalens is analyzed, showing critical dimension (CD) bias, sidewall angle and pillar height all have different levels of influence on the performance of the metalens.

Here, we propose a scheme, where the proposed achromatic metalens in the long-wave infrared regime of 9∼12 μm is designed by optimizing the spectral degree of freedom C(ω). The combination of the adjacent pillars, which can be considered as a supercell will redistribute when C(ω) is changed. C(ω) provides one more dimension to optimize the group delay of the metalens to suppress the achromatic aberration, which hasn't been discussed before. The achromatic metalens with a diameter of 200 μm and NA of 0.24 is proposed to fabricate on all-silicon wafer using CMOS-compatible process. To evaluate the performance of the achromatic metalenses, the focal length fractional change (FLFC) and chromatic focal shift are discussed. Moreover, we also analyze the effect of fabrication process deviation on the performance of the metalens. The proposed scheme offers perspectives to extend the operating bandwidth of achromatic metalenses to LWIR and we can envision applications in miniaturized and integrated LWIR imaging.

II. DESIGN PRINCIPLE AND RESULT
Traditional lenses composed of glass or other transparent materials rely on the accumulation of optical path in the medium to achieve phase control. Similar with a refractive lens and other diffractive optical lens, the relative phase profile for metalenses to achieve focusing can be expressed as follows: where ω, C(ω), c, r, and ƒ are angular frequency, spectral degree of freedom, speed of light, radial coordinate and focal length, respectively. C(ω) is related with central design wavelength, which can be set in a range of 0 to 2π. The relative phase profile doesn't change as C(ω) takes on a different value for a given design frequency. The required phase of the metalens is controlled by the shape and geometry of the meta-atom unit. One of the important processes in designing metalenses is to design a libary of the structures, such as diameter, to achieve a phase coverage of 0∼2π. The meta-atom unit of our metalens shown in Fig. 1(b) is a circular silicon pillar on a silicon substrate. These pillars are regarded as truncated waveguides. Different phase responses can be obtained by engineering the diameter of the nanopillars. Due to the dispersive properties of materials, effective refractive indices vary with different wavelengths for the same structure, thus causing phase differences. In order to achieve a broadband achromatic metalens, the phase profiles with a fixed focal length over a wide range of wavelengths are required.
In recent years, Chen et al. [28], [29], [30] explained dispersion by introducing group delay and group delay dispersion into the phase profile as shown in (1). Equation (1) can be further expanded as a Taylor series near a design frequency ω d as follows: The first term indicates that the wavefront of the incident light is a spherical wave. The second term depicts the time delay for different positions of the lens to reach the focus. The third term and higher ensure that the outgoing wavepackets are identical. The group delay describes how quickly the phase changes with frequency, which helps suppress chromatic aberrations over the entire wavelength range. The group delay can be derived from (1) for an achromatic metalens as: The group delay is related to the spatial position when the focal length is irrelevant to ω. Meeting a certain distribution of phase and group delay is important to suppress chromatic aberration. It is worth noting that the group delay is determined by the geometry, adjacent pillars as well as pillar material properties. With C(ω) changing, ϕ(r, ω) will cause pillar redistribution in metalenses, which further affects the group delay profile of the metalens. C(ω) may provide an alternative chance to optimize the group delay and eliminate the chromatic aberration. Here, C(ω) is determined after obtaining the FLFC through a parametric sweep within 0∼2π.
To demonstrate this concept, metalenses with different C(ω) are studied. The building block of the metalenses are circular silicon pillars with height and lattice period of 7.9 μm and 3.05 μm, respectively as illustrated in Fig. 1(a). Fig. 2(a) shows the relationship of phase shift and transmission with the pillar radius ranging from 300 nm to 1300 nm at a given design wavelength of 10 μm. The pillars cover 2π phase shift, and meanwhile maintain high transmittance, which is very beneficial to improve focusing efficiency. Fig. 2(c) shows the relative phase and group delay for the pillar with diameter D1 = 800 nm surrounded by different pillars shown in Fig. 2(b). The relative phase and group delay were obtained by changing adjacent pillars distribution of the pillar. One can see that the group delay of the pillars varies depending on the combination of the adjacent pillars while the relevant phase has no significant changes. Thus C(ω) can be considered as one dimension to optimize the group delay of the metalens to suppress the achromatic aberration, which hasn't been discussed before. The relative phase profile of the studied metalenses is achieved according to (1), where the aperture is 200 μm and the design focal length ƒ is 480 μm. To quantitatively analyze the performance of chromatic aberration correction, the where ƒ max , ƒ min , mean(ƒ) represent the maximum, minimum and average focal length over the interested bandwidth. The FLFC presents the degree of focus shift deviation over the interested wavelength range. The smaller FLFC means the better achromatic performance of the lenses. It can be obtained using (4) to calculate after the metalens with different C(ω) are simulated. Fig. 2(d) shows the FLFC is a function of C(ω) in the design metalens, where M(C(ω) = 0.15 × 2π) and N(C(ω) = 0.55 × 2π) are the two points with the smallest FLFC of 4.1% and largest FLFC of 59.8%. Fig. 2(e), (f) show the comparison between actual group delay distribution of the metalens obtained by linearly fit the phase spectrum of the pillars in metalens and target group delay under the corresponding C(ω) value of M and N points in Fig. 2(d). The actual group delay of the metalens with C(ω) in M can approach to the target group delay, but significantly deviate with value in N. It can be seen the FLFC can be minimized to correct chromatic aberration by optimizing C(ω). It should be noticed as the larger diameters or NA of metalenses are required, the proposed method may meet some limitations since the larger group delay is required to compensate. A possible solution may need to introduce combined complex nano-structures to expand the group delay library. Fig. 3(a) and (b) shows the electric field intensity distribution along the propagation direction of two metalenses with C(ω) corresponding to M and N in Fig. 2(d). Here we name the two metalenses as M metalens and N metalens, respectively. The dashed lines depict the average focal length of the metalenses working at wavelengths of 9 to 12 μm, and are defined as the effective focal plane of the metalenses. It can be seen that the suppression of chromatic aberration in M metalens is more superior that N metalens. Visible parasitic focal spots along the propagation direction are found mainly due to the imperfect match between the desired phase and the actual phase. Fig. 3(c) illustrates the normalized intensity profiles cut from the dashed line of M metalens. The full widths at half maximum (FWHM) on the focal plane are proportional to the operating wavelengths, increasing from 16.25 μm to 24.5 μm. The focal lengths and the chromatic focal shift sampled in the interested wavelengths are plotted in Fig. 3(d). The grey dashed line denotes the average focal length of 400.8 μm, yielding a simulation NA of 0.24. The deviation of the simulated average focal length from the design focal length (ƒ = 480 μm) at 10 μm is possibly due to proximity effects between adjacent meta-atoms, and phase errors caused by the mismatch between the desired and actual phase responses of meta-atoms in a metalens. Chromatic focal shift shows the variation of focal lengths with wavelengths, referenced to the designed wavelength of 10 μm. The absolute value of the maximum chromatic focal shift is 15 μm. Fig. 3(e) shows that M metalens has focusing efficiency in a range of 35%∼61% at the operation wavelengths of 9∼12 μm. Correspondingly, the average focusing efficiency of 51.7% over the entire bandwidth is achieved. Focusing efficiency here is defined as the power fraction of transmitted light through a square aperture with a width of 72 μm positioned in the focal plane of the metalens and normal incident light (we assume the incident surface (non-metasurface layer) of the metalens is coated with anti-reflective layer, and has high transmittance of 100%). 72 μm is selected as ∼3 times of the max FWHM (24.5 μm) at 12 μm wavelength. Considering the practical applications, the pixel size of LWIR sensors is usually square, and thus the square integral area is proposed to use.
Precise phase modulation of light by metalenses comes from tailored design of subwavelength meta-atoms. While defects or errors during the fabrication process can have an effect on the geometry of the meta-atoms and thus influence the performance of a metalens. Deviations in critical dimension (CD) are common during CMOS fabrication process. Due to the non-uniformity of the etching process, the height of the pillars at the center and the edge of the wafer will deviate. It is difficult to maintain vertical sidewalls for pillars with relatively large aspect ratio during etching process. Fig. 4(a)-(d) display cross-sectional schematic of ideal nanopillars, nanopillars with a changed diameter (ΔD), height (ΔH) and sidewall angle (SA), respectively. The influence caused by CD bias, pillar height variation and SA on the performance of the metalens can be analyzed by three metrics: the FLFC, average focal length and average focusing efficiency over the entire interested bandwidth. Additionally, this study applies uniform CD bias, height variation and SA on the whole nanopillars regardless of the individual dimensions.
One can see from Fig. 4(e) that CD bias has different effects on the FLFC over the interested wavelength range. The plus CD bias leads to a dramatic increase in FLFC, which causes the average focal length to decrease roughly linearly as a result. And the plus CD bias has a greater impact on FLFC than the minus CD bias. As the diameter changes, the transmission of a single pillar will also change, which will affect the average focusing efficiency of the metalens shown in Fig. 4(f). The impact on the average focusing efficiency is larger for the minus CD bias. When the diameter of pillars changes with ΔD as a whole, it is challenging to maintain the phase difference with the design wavelength, which is disadvantageous for an achromatic metalens. It can be seen that the small or large diameter of the pillar has a different effect on the performance of the achromatic metalens.
The influence of pillar height variation on the performance of the metalens is shown in the Fig. 4(g), (h). The FLFC remains almost unchanged when the height of the pillar is less than the design value but the average focal length is larger. However, the FLFC increases when the height of pillars is higher than the design value. It can be noticed that the higher pillar height, the smaller the effective focal length of the metalens, while there is not much effect on the average efficiency. When the height of pillars deviates, the phase response and transmission of a fixed diameter pillar for different wavelengths will also change. Here, the pillar height has a small effect on the average efficiency.
Non-ideal etching process normally gives rise to inclined sidewalls which lead to deviations in the phase response of nanopillars, resulting in changes in Chromatic aberration. As shown in Fig. 4(i), FLFC increases with the deviation of angled sidewall from the design angle (90°). The average focal length of the metalens ramps up while the average focusing efficiency drops with increasing of the SA. The performance deterioration may be caused by larger phase errors induced by slanted sidewall.

III. CONCLUSION
In summary, we propose and numerically demonstrate an achromatic metalens in a LWIR regime. Through exploiting the spectral degrees of freedom, the proposed metalens with a diameter of 200 μm and NA of 0.24 can achieve achromatic focusing over the wavelength of 9∼12 μm with a FLFC of 4.1%. Our approach provides some advantages like reduced design complexity of unit structures to eliminate the achromatic aberration, compatibility of CMOS process and the realization of polarization insensitive metalens as the proposed nano-structures have fourfold rotational symmetry. Furthermore, we also analyze the influence of fabrication process deviation on the performance of the metalens. Our work provides a new way to design achromatic metalenses and may be promising for applications in infrared thermal imaging and miniaturized vision systems.