Asymptotic dispersion engineering for ultra-broadband meta-optics

Dispersion decomposes compound light into its monochromatic components, which is detrimental to broadband imaging but advantageous for spectroscopic applications. Metasurfaces provide a unique path to modulate the dispersion by adjusting structural parameters on a two-dimensional plane. However, conventional linear phase compensation does not adequately match the meta-unit’s dispersion characteristics with required complex dispersion, hindering at-will dispersion engineering over a very wide bandwidth particularly. Here, we propose an asymptotic phase compensation strategy for ultra-broadband dispersion-controlled metalenses. Metasurfaces with extraordinarily high aspect ratio nanostructures have been fabricated for arbitrary dispersion control in ultra-broad bandwidth, and we experimentally demonstrate the single-layer achromatic metalenses in the visible to infrared spectrum (400 nm~1000 nm, NA = 0.164). Our proposed scheme provides a comprehensive theoretical framework for single-layer meta-optics, allowing for arbitrary dispersion manipulation without bandwidth restrictions. This development is expected to have significant applications in ultra-broadband imaging and chromatography detection, among others.

The formula S1 expresses the relationship between the wavenumber and the effective refractive index (ERI) in the one-dimensional equivalent medium theory.
where  is the duty cycle. eff is the effective refractive index. air is the refractive index of air. Tio 2 is the experimentally measured refractive index of titanium dioxide (TiO2).Figure S2 shows the relationship between the wavenumber and the effective refractive index derived from the formula S1 when =0. 5.The relationship between the wavenumber and phase of four meta-units is shown in Figure S4, which proves that the nanostructures have nonlinear phase dispersion.First, the ERIs of 14 kinds of meta-units are simulated by the Lumerical MODE Solutions.The period of meta-units was set as 500 nm.Considering the constraints of experimental conditions and period size, we set the minimum and maximum size constraints of the nanostructures to be 50 nm and 450 nm, respectively.Six nanostructures with different cross-sectional shapes are shown in Figure S5.For the simulation, the boundary conditions of the simulation was set to periodic boundary conditions.The refractive index of the TiO2 was the measurement result by ellipsometer.
We used eigenmodes to analyze and calculate nanostructures with different wavelengths and different cross-sectional shapes, and obtain the equivalent refractive index   .
Then we calculated the phase at different wavelengths (400 nm, 450 nm, 550 nm, 650 nm, 750 nm, 850 nm, 900 nm, 1000 nm) when the height is 600 nm according to formula 1.We calculated the phase  0 , at the largest wavelength (λ=1000 nm) and dispersion, ∆ =  −  0 , for the chosen ∆ (i.e., λ=400~1000 nm) of each metaunits.The calculation results is shown in Figure S6.In the process of matching the structure, we found that it is difficult to achieve ultra-broadband achromatic with these meta-units libraries.In order to obtain better matching results, we changed the structure height to 1000 nm to obtain greater phase compensation.The calculation results is shown in Figure S7.Section 6: The transmission of six meta-units.
Figure S8 shows the transmission of six meta-units, it can be seen from the figure that the transmission of all meta-units is high.Figure S9 shows the transmission simulation of the metalens (NA=0.164).Therefore, the achromatic metalens designed in this work can have high focusing efficiency.First, we designed ultra-broadband achromatic metalenses with different NAs (0.164, 0.243) in the 400 nm to 1000 nm band by linear dispersive phase compensation methods.The first step is to calculate the expected phase  construct_0 , at the largest wavelength (λ=1000 nm) and expected dispersion, ∆ construct =  construct −  construct_0 ,, for the chosen ∆ construct (i.e., λ=400~1000 nm) of each meta-units.Then we matched the structures at different  0 , and chosen the  0 with the smallest error.The error here can be expressed by formula S2, where ∆ construct is the expected dispersion, ∆ metaunit is the dispersion of the chosen meta-units. max is the farthest position of the meta-unit on the layout,   is the different wavelength.
Figure S10a-S10c shows the matching results of the linear dispersion phase compensation method.In order to obtain a smaller matching error, we have proposed an asymptotic dispersion phase compensation method (   is different at different wavelength) in the design of ultra-broadband achromatic metalens.Figure S10d-S10f shows the matching results of the asymptotic dispersion phase compensation method.In this section we show the simulation results of several metalenses matched by linear and asymptotic compensation approach.The fabrication process of the device is shown in Figure S14, the details are described in the methods section.In order to verify the performance of ultra-broadband achromatic and arbitrary dispersion controlled metalenses, we designed two optical setups to characterize the focal length and imaging effect, respectively.
The optical setup used to verify the ultra-broadband achromatic metalens and the arbitrary dispersion control metalenses is shown in Figure S15.The laser beam is emitted by a supercontinuum laser source, and we can obtain the target wavelength after the laser beam passes through the color filter.The laser beam is imaged on the CCD through the metalens and the objective lens.Finally, the translation stage moved from the position where the sample surface is imaged, and the camera acquired an imaging pattern every 1 μm until it stops at 800 μm from the starting position.
In the imaging experiment, we replaced the super continuum laser source with a halogen lamp.We placed the resolution target in front of the ultra-broadband achromatic metalens and obtained the imaging result at the focal point of the lens.
Finally, we judged the imaging quality of the ultra-broadband achromatic metalens by comparing images of different wavelengths.Figure S16 shows the measured focusing efficiency of the asymptotic matching metalens (NA=0.164),where the focusing efficiency is calculated by the intensity within three times the full width half height (FWHM) at the focus plane divided by the intensity of incident light within the range of the metalens.It can be seen that the designed metalens achieves achromatic focusing while also having high focusing efficiency in the operation band.Figure S17 shows the performance of the single-layer achromatic metalens achieved in this work compared to previous ones in experiments.
This work.
Section 12: The effects of processing errors.
In fact, some errors will inevitably be introduced in every step of sample preparation.Therefore, for the processing plan used in this article, we had selected two locations where errors are most likely to occur and performed corresponding simulations.First, we simulated the overexposure caused by the large dose in the electron beam exposure process.The left side of Figure S18 is the simulation result with normal structure size, and the result on the right is the simulation result after all the structure sizes are increased by 40 nm.We also simulated the loss of the structure height during the ion beam etching process.The left side of Figure S19 is the simulation result when the height of the nanostructure is 1000 nm, and the right side is the simulation result when the structure height is reduced to 800 nm.
In the process of comparing these two sets of simulation results, it can be found that whether the size of the nanostructure increases or the height decreases, the focal length will have a larger deviation at a short wavelength, while the impact will be less at a long wavelength.Part of the reason is that nanostructures need to provide greater phase compensation at shorter wavelengths, and the deviation of the structure will cause a bigger phase change, which will cause a change in the focal length.[6] 0.06 0.24 12.3 1400 1400 250 [7] 0.18 0.16 8.925 800 500 200 [8] 0.36 0.2879 5 600 450 350 [9] 0.14 0.384 4 1200 300 100 [10] 0.59 0.165 9.9 400 400 340 [11]  As a conventional criterion for focusing performance, the Strehl ratio (SR), which is defined as the peak intensity normalized to that of the Airy disk, increases with decreasing  rms .As described in article, the  rms we calculated of the achromatic metalenses are very small.According to the Maréchal Criterion, when the RMS is less than 0.071, the Strehl ratio is greater than 0.8, resulting in the theoretically diffraction-limited focusing.Our calculated values of the Strehl ratio of the focused spot from the FDTD simulation also confirm the conclusion in Figure 2c.However, due to the existence of errors in the actual fabrication, the final wavefront error will be further enlarged, so the Strehl ratio of the actual fabricated metalens is decreased shown in in Figure S20.The design challenge for large area achromatic metalens is the inability to compensate for large dispersion phases using a single structure.Figures S21a and S20b show the phase distribution for three wavelengths with the same NA and different areas.
It can be seen that increasing the radius of the metalens from 50μm to 500μm also increases the range of phase compensation required from the nanostructure by a factor of ten to a maximum of 24p, which is well beyond the range of phase compensation that can be provided by the nanostructure.The result of the phase dispersion matching is shown in Figure S21c, which can only satisfy the phase requirements for one wavelength.Accordingly, this scheme folds all phases to the 2p range for matching the structure.Figures S21d and S21e show the matching results of the linear and asymptotic dispersive phase compensation schemes after collapsing the phases, respectively, where the first scheme suffers from a large matching error, while the second scheme achieves a change in the shape and position of the constructed phase curve by modulating   at different wavelengths, and minimize the matching error.Subsequently, we designed an achromatic metalens with R=500μm, f=100μm, NA=0.98 and the working wavelengths are three primary colors R(633nm), G(531nm), and B(450nm) by combining the above scheme and the particle swarm algorithm.
Figure S22 shows the results of the scalar diffraction far-field simulation using MATLAB, from which it can be seen that the position of the focal point at 100μm which is in perfect agreement with the designed focal length.

Fig. S1
Fig. S1 Schematic of the linear phase compensation method.a, c The schematic wavefronts

Section 2 :
Theoretical derivation of wavenumber and effective refractive index.

Fig. S2 The relationship between wavenumber and effective refractive index is derived from the one-dimensional equivalent medium theory.Section 3 :
Fig. S2 The relationship between wavenumber and effective refractive index is derived

Figure
FigureS3shows the experimentally measured n and k values of TiO2 in the 210

Fig. S4 The intrinsic phase dispersion response of nanostructures.Section 5 :
Fig. S4 The intrinsic phase dispersion response of nanostructures.

Fig. S5 Schematic of six nanostructures with different cross-sectional shapes. 85 86 Fig. 89 Fig.
Fig. S5 Schematic of six nanostructures with different cross-sectional shapes.

FigSection 8 :
Fig. S10 The linear and asymptotic dispersion phase compensation method matching FigureS11-S13 shows the simulated intensity distribution along the propagation direction of the metalenses with NA=0.243,R=25 μm, NA=0.164,R=25 μm, NA=0.164,R=50 μm, respectively.In the simulation results of each figure, with the same NA, radius, and matching method, the bandwidth is 350nm, 600nm, and 1100nm, respectively.For the matching of metalenses in the 400~1500 nm band, a large phase compensation is required, so we have increased the height of the nano-structures to 1500 nm.
Fig. S11 The simulated intensity distribution along the propagation direction of the

Fig. S12
Fig. S12 The simulated intensity distribution along the propagation direction of the Fig. S13 The simulated intensity distribution along the propagation direction of the

Fig. S14 Section 10 :
Fig. S14 The fabrication process of TiO2 structure based on the conformal filling process.

Fig. S15 Section 11 :
Fig. S15 The optical characterization system.a Schematic diagram of optical setup for

Fig
Fig. S16 The measured focusing efficiency of the asymptotic matching metalens

Fig. S18
Fig. S18 Simulation of the impact of overexposure on the focusing effect.a-d Comparison

Fig. S19
Fig. S19 Simulation of the impact of overetching on the focusing effect.a-d Comparison of

Fig
Fig. S20 Simulated and experiment Strehl ratios of metalenses.

Fig. S21
Fig. S21 Schematic diagram of the design principle of a large-area achromatic metalens

Figure S22
Figure S22 Scalar diffraction simulation of the metalens along the propagation direction

Section 13: Comparison of data for single-layer achromatic metalenses. 223
By comparison with the previous works of single-layer broadband achromatic 224 metalenses in TableS1, in both simulations and experiments, we stay well ahead in 225 terms of bandwidth and LPF while maintaining comparable lens size and NA.226