Multi-zone taylor expansion method for broadband achromatic polarization-insensitive metalens design

Broadband achromatic polarization-insensitive metalens with large numerical aperture (NA), large diameter and high efficiency are in demand. Existing methods can dramatically improve one of the above performances at the expense of the others. In this work, the multi-zone Taylor expansion method (MZTEM) is proposed to reduce these trade-offs between different key figures of merit. In this method, a metalens is devided into odd ring and even ring regions, then the odd (even) ring metalens focus peak is shifted forward with tailored functions of even (odd) ring region, and finally the arrangement of the meta-units is optimized to get higher efficiency. Based on the MZTEM, an achromatic polarization-insensitive lithium niobate on insulator (LNOI) metalens is designed working in the visible from wavelength λ = 430 nm to 750 nm, which exhibits a focusing efficiency of 35%–63% and NA of 0.255. Besides, the full width at half maxima (FWHM) variation is less than 0.14 μm and the focusing efficiency variation is less than 2% within 0°–10° incidence angle.

In recent years, metalenses with large numerical aperture (NA) [21,22], wide field of view [23], large broadband [24] and high efficiency [25] have been demonstrated separately.However, it is challenging to realize a metalens with all these four features.Up to now, several methods have been developed to design achromatic metalens with high efficiency.The multi-wavelength method eliminates chromatic aberrations at several discrete wavelengths [26][27][28][29], but it cannot achieve broadband achromatic imaging.The Taylor expansion phase profile method [30] mitigates this by satisfying the phase, group delay (GD) and group delay dispersion (GDD) profiles, but it's hard to get high NA while maintaining the bandwidth and efficiency because of the undersampling of the phase profile when increasing the deflection angle of metalens [12].In this work, we propose a new method named multi-zone Taylor expansion method (MZTEM).Compared with Taylor expansion phase profile method, the MZTEM can be used to get metalens with higher NA and bigger size without increasing the phase gradient, at no expense of efficiency and bandwidth.For the proof of concept, a broadband achromatic polarization-insensitive lithium niobate on insulator (LNOI) metalens is designed in the visible using MZTEM.

Principle of multi-zone taylor expansion method (MZTEM)
The phase, GD and GDD conditions should be satisfied in the Taylor expansion phase profile method [30] for a broadband achromatic metalens, which can be described by equations ((1)-( 3)): where r x y = + is the distance between an arbitrary point and the center of the metalens, 0 l is the center wavelength of incident light, f is the focal length, w is the angular frequency of incident light, C is light speed, and A and B are constants.
The MZTEM starts from dividing the metalens into a set of concentric rings (centre circle is regarded as a ring with an infinitely small inner diameter) at π phase interval according to the phase profile (equation ( 1)), as shown in figure 1(a).The odd ring and even ring area are designed separately with different phase, GD and GDD profiles, which is different from the Taylor expansion phase profile method [30] in which the whole metalens has a single set of phase, GD and GDD profiles.Next, we consider a metalens that has meta-units configured only in odd ring area guided by tailored phase, GD and GDD profiles, and the even ring area is opaque.Similar to Fresnel zone plate in terms of focusing, this metalens has more than one focus point.Then the phase, GD and GDD profiles for even ring area are ingeniously designed to ensure that the phase and amplitude of the refracting light from the even ring area at a certain focal point f1, closer to the metalens, can match with those of the odd ring area.This step makes the maximum intensity peak value move from point f2 to f1, resulting in a higher NA and smaller FWHM.Since all rings satisfy their respective phase, GD and GDD functions, the broadband focal length consistency of the metalens designed by MZTEM is close to that of metalens designed by Taylor expansion phase profile method.It is worth noting that the MZTEM is suitable for both polarization-dependent and polarization-insensitive metalens designs.
To better understand the principle of Multi-zone Taylor expansion method, we take 3π metalens working in transmission mode as an example in the discussion below whose maximum phase difference is 3π.So this 3π metalens is divided into odd ring area: central circular area (2π−3π), outer ring area (0−π) and even ring area: inner ring area (π−2π), as shown in figure 1(b).

Design of broadband achromatic polarization-insensitive LNOI metalens in the visible
Lithium niobate (LiNbO3, LN) has a large refractive index and low dispersion, the former can reduce the influence of adjacent nanostructures, and the latter helps to reduce dispersion in broadband focusing.Meanwhile, Lithium niobate has a large transparency window from 350 nm to 5000 nm [27], and broadband achromatic LNOI metalens in infrared band can also be designed using MZTEM.Furthermore, recent advances in LN thin film fabrication techniques have enabled the creation of dense nanorod arrays with sub 30 nm ultrasmall gaps and more than 2.5 μm etching depth using focused ion beam (FIB) milling [31], and nearly 90°w aveguide sidewalls using thin oxide masks and high-selectivity wet-etching [32], which make LNOI metalens possible.In this work, we use LNOI to construct a broadband achromatic polarization-insensitive metalens.LN has a refractive index of 2.3995-2.2624 in the visible (λ = 430 nm to 750 nm) [33], and the substrate is a 500 μm thick quartz.
Isotropic nanostructures, such as circular and square shapes, enable polarization-independent metalens [34].Nanoposts with a square cross-section, shown in figure 2(a) and the inverse nanoposts, shown in figure 2(b) are selected as meta-units to construct the metalens.The inverse nanoposts can expand the phase delay and GD range of nanoposts to obtain larger diameter achromatic metalens.
Firstly, we fix a wavelength λ 0 at 500 nm and obtain each meta-unit' phase delay, as well as its GD.There are three principles in building a basic meta-units group: (1) The transmission of meta-units should be as high as possible, which determines the focusing efficiency of the metalens.(2) The width w should be varied to cover a range of 3π phase delays, and the GD should be as wide as possible.(3) In the visible wavelength range, phase delay should be linear with angular frequency when w varies within a lattice constant [35].This linear condition ensures achromatic focusing within a given bandwidth.
The basic meta-units group we select is shown in figure 2, the height of the nanoposts is 500 nm and the lattice constant p is 230 nm.The phase delays of nanoposts range from 0 to 2.12π while the transmission is > 90% (figure 2(a)).Together with 600 nm high inverse nanoposts, full 3π phase coverage (figure 2(b)) and      and (f) illustrate that angular frequency is in a linear relation with phase.Metalens of different heights can be fabricated by nesting.It is worth noting that the inverse nanoposts can be replaced by other 500 nm high isotropic nanostructures with the same phase shift and GD, and the focusing characteristics will remain the same.
Secondly, an odd ring metalens containing central circular (2π−3π) area and outer ring (0−π) area is simulated with a focal length of 100 μm and a diameter of 24.52 μm. Figure 3(a) shows the intensity profiles along z-axis for different wavelengths in the visible.As predicted, the odd ring metalens has more than one intensity focus peak along z-axis in which f1 and f2 are the two largest intensity peaks.Both f1 and f2 peaks are close to the diffraction limited value (figure 3(b)).However, the FWHM of f1 peaks is significantly smaller than that of f2 peaks, indicating better focusing property.
As mentioned above, we are going to increase the intensities of the f1 peaks and decrease that of f2 peaks as well as other peaks, and decrease the focal length shift of f1 at the same time.The focal length f is set as variable for the phase and GD functions of inner ring (π−2π) area, which are combined with those of the odd ring area to form many groups of phase and GD profiles of 3π metalens.To fulfill these profiles, the meta-units are selected with their phase delay and GD closest to the required values.Because only the relative values of phase and GD parameters are important, the basic meta-units group can be shifted in 2-dimensional space to better fit the required values [30], and the shift corresponds to the different constants in equations (1) and (2).
In the process of matching the meta-units width with phase and GD profiles, the lattice constant p is an additional control in inner ring area functions optimization besides diameter and focal length.On one hand, the meta-units in metalens can be treated as a set of sampling points, so a smaller lattice constant can smooth the phase curve at the plane immediately after passing through the metalens and reduce the difference between the actual phase and the required value.Figure 4(a) shows the target phase and actual phase at the output plane of  a The efficiency is defined as the power contained within a focal spot, whose diameter is equal to twice the diameter of an ideal Airy disk, divided by the power of incident light.
metalens with p = 225 nm and 270 nm at 0−π area, and the phase curve of p = 225 nm is smoother than that of p = 270 nm.On the other hand, figure 4(b) illustrates that the change in lattice constant leads to a change in the weight of f1 and f2 peaks.However, the lattice constant cannot be too small due to the interaction between adjacent meta-units and the increasing difficulty in fabrication.

Results and discussion
There are many combinations of odd rings' meta-units and even rings' meta-units distribution.The final distribution of meta-units is chosen by the particle swarm algorithm and the corresponding intensities along zaxis of LNOI 3π metalens are shown in figures 5(a) and (b).As predicted, the f1 peak increase from 77.15 V 2 m −2 to 301.39 V 2 m −2 (λ = 500 nm) while f2 and other peaks decrease to an extremely low level.The focusing efficiency of the metalens is defined as the power contained within a focal spot, whose diameter is equal to twice the diameter of an ideal Airy disk, divided by the power of the incident light.The relative efficiency is defined as the ratio of the optical power in the focal plane to the source power [36].Figure 5(c) shows that the focusing efficiency remains at a high level when λ < 500 nm, and drops while the wavelength gets longer.This is attributed to the stronger coupling between adjacent nanostructures for the longer incident wavelength.Figure 5(d) illustrates that the FWHMs at different wavelengths are at or close to the diffraction limit.Moreover, incident light at different wavelengths can focus on nearly the same point (figure 5(e)).
The field of view of metalens is important in photography and projection.The incident light discussed above is all at normal incidence.As for the oblique incidence condition, 500 nm wavelength light is incident obliquely along the z-axis onto the metalens.There are nearly no differences in FWHM, efficiency within 6°and peak value (figures 6(a)-(c)).Efficiency drops and FWHM increases when the incident angle is larger than 6°, but there is still no overlap between FWHM ranges of the adjacent peaks.As the angle continues to increase to greater than 10°, the sidebands and aberrations will appear, as shown in figure 6(d).

Conclusion
In summary, we have proposed a new method to design metalens.Unlike the Taylor expansion phase profile method, which achieves a larger NA by increasing the phase gradient, the METEM uses the coordination between odd and even rings to achieve the forward shift of the focus peak, resulting in a smaller focal length, larger NA and higher focusing efficiency.The chromatic aberrations are eliminated by fulfilling the phase, GD and GDD profiles in each ring of metalens.In addition, the METEM is suitable for full wave band design and can be used in conjunction with the Pancharatnam−Berry (PB) phase to realize polarization-dependent metalens.Furthermore, we have demonstrated the feasibility of METEM by designing a broadband achromatic LNOI polarization-insensitive metalens in the visible.This LNOI metalens (NA = 0.255) is capable of focusing light to the diffraction limit with focusing efficiency of 35%-63% (λ = 430 nm-750 nm).Considering its angle characteristic, the FWHM variation is less than 0.14 μm and the focusing efficiency variation is less than 2% within 10°incidence angle.Some key performance metrics of broadband achromatic polarization-insensitive metalens in the visible are summarised in table 1.Compared with TiO 2 metalens designed with Taylor expansion phase profile method in [30,37], the proposed LNOI metalens has higher NA, wider bandwidth and higher peak efficiency.In addition, the Si 3 N 4 metalens designed with a method similar to Taylor expansion phase profile method in [38] has almost the same bandwidth and peak efficiency as the Lithium Niobate metalens, but its diameter is nearly half that of LNOI metalens, and the NA is less than half that of LNOI metalens.Finally, by further increasing the number of rings and optimizing the arrangement of the structures, the achievement of the large aperture, large field of view, wide bandwidth and high focusing efficiency metalens is possible, which can be used in industry and scientific research, such as in lithography, microscopy, endoscopy and virtual and augmented reality.

Figure 1 .
Figure 1.(a) The schematic of the METEM and intensity profiles along z-axis for odd ring metalens and the final combination metalens, and the pink area in metalens is opaque.(b) Schematic structure of 3π metalens.There are central circular area, inner ring area and outer ring area from inside to outside.

Figure 2 .
Figure 2. Simulated phase (red dots) and transmission (blue dots) for nanoposts (a) and inverse nanoposts (b), the insets show the structure of meta-unit.The GD as a function of width of nanoposts (c) and inverse nanoposts (d).Simulated phase and fitted curves for different width of nanoposts (e) and inverse nanoposts (f).

Figure 3 .
Figure 3. (a) Intensity profiles along z-axis for wavelengths in the visible.(b) The FWHM of f1 peaks and f2 peaks for incident light in the visible.

Figure 4 .
Figure 4. (a) The target phase and actual phase at the output plane of metalens with p = 225 nm and 270 nm for outer ring (0−π) area.(b) Intensity profiles along z-axis for odd ring metalens with p = 184 nm, 203 nm and 230 nm.

Figure 5 .
Figure 5. (a) The meta-units width distribution.The width inside R = 7.13 μm refers to the width of inverse nanoposts and the others refer to the width of nanoposts.(b) Intensity profiles along z-axis.(c) Focusing efficiency and relative efficiency, (d) FWHM of focal spot as a function of wavelength in the visible.(e) Normalized light intensity profiles for LNOI 3π metalens at various incident wavelengths.The white dashed line indicates the position of the focal plane.

Table 1 .
| summary of performance metrics for broadband achromatic polarization-insensitive metalenses in the visible.