To address the simultaneous elimination of chromatic and off-axis aberrations, it is crucial to initially consider chromatic aberration, which represents the derivative of the expected performance parameter relative to frequency or wavelength. For metalenses, the dispersion is assessed by deriving the focal length (*f*) with respect to frequency (ω). Traditional bulky refractive dielectric lenses exhibit normal dispersion (d*f*/dω < 0), as illustrated in Fig. 2a, indicating that the deflection angle of high-frequency electromagnetic waves (EM) is greater than that of lower-frequency EM waves. Conversely, diffractive lenses, such as Fresnel lenses and metalenses, exhibit abnormal dispersion (d*f*/dω > 0), as depicted in Fig. 2a, leading to the corresponding focal length of low-frequency EM waves becoming smaller than that of the higher frequency waves.

To achieve a perfect achromatic lens, the dispersion curve of the designed metalens must be flat, implying that d*f*/dω = 0. In other words, EM waves within the operating frequency range will converge to the same focal plane and share the same focal length. Most prior achromatic metalenses are predominantly composed of resonant nanopillar structures with a high aspect ratio, which cannot meet the requirement for non-dispersive behavior. An exceptional case, capable of achieving effective achromatic refractive index distribution from 430 to 780 nm, is constructed using a single layer of silicon nitride nanopillars47. However, the associated NA is merely 0.086, and the imaging resolution is relatively low. In this context, all metalenses can be considered equivalent to a slab with uniform thickness, as illustrated in Fig. 2b. Inspired by this equivalence, a novel method for designing achromatic metalenses from the perspective of RI is proposed. The corresponding phase distribution to frequency and a reference coordinate at a specific position r is expressed as follows:

$$\varphi {\text{(}}r,\omega {\text{)}}=\frac{\omega }{c}{\text{(}} - \sqrt {{r^2}+{f^2}} +f+\alpha {\text{)}}$$

1

where *c* symbolizes the light speed in the air, *f* denotes the focal length of the metalens, and α determines the reference phase at the center of the lens. Assuming that reflection is negligible and EM waves propagate in the vertical direction within the lens, the phase distribution could be translated into the corresponding RI profile with a fixed thickness *d*. In consequence, Eq. (1) is displayed as

$$\begin{gathered} \varphi {\text{(}}r,\omega {\text{)}}=\frac{\omega }{c}n{\text{(}}r,\omega {\text{)}}d \hfill \\ {\text{in which }}n{\text{(}}r,\omega {\text{)}}=\frac{1}{d}{\text{(}} - \sqrt {{r^2}+{f^2}} +f+\alpha {\text{)}} \hfill \\ \end{gathered}$$

2

To obtain nondispersive metalens (d*f*/dω), the derivative of *n*(*r*,*ω*) with respect to the frequency ω must be zero

$$\frac{\partial }{{\partial \omega }}n{\text{(}}r,\omega {\text{)}}=0,{\text{ }}\omega \in {\text{[}}{\omega _{\hbox{min} }},{\omega _{\hbox{max} }}{\text{]}}$$

3

Therefore, the challenge of realizing a perfect achromatic lens is redefined as the task of designing a non-dispersive, ultra-broadband GRIN lens. The effective local refractive index (*n*eff) at each position should ideally be independent of the frequency. This requirement necessitates that the reflection at the lens/air interface is minimized, leading to an emphasis on prioritizing and considering a low refractive index distribution.

After comprehensive consideration, the parameters in Eq. 2 that govern the dimensions of the terahertz achromatic metalens are selected as follows: As shown in Fig. 2b, the lens radius is set to R = 3mm, the focal length to *f* = 4.5mm, α to 5.4, and the uniform thickness to d = 1.5mm. Evaluating the pure RI profile of the GRIN lens as defined in Eq. 2, we have determined, as demonstrated in Fig. S1, that it is theoretically free from aberrations when analyzed from the perspectives of geometric and wave optics. This feature aligns it with the characteristics of the Luneburg lens48, the Maxwell-fisheye lens49, and the Mikaelian lens50, thereby introducing a novel and unique optical device. However, because the thickness of the metalens (d = 1.5mm) is comparable to its radius (R = 3mm), the assumption of vertical propagation within the lens becomes inaccurate. Consequently, the actual focal length of the lens is reduced to 4mm, deviating from the expected 4.5mm. To compensate for this deviation, a correction factor of 0.9 is introduced to modify the radius coefficient. Thus, the RI distribution presented in Eq. 2 is reformulated as

$$n{\text{(}}r{\text{)}}=\frac{{f+\alpha - \sqrt {{\text{0}}{\text{.9}}{r^2}+{f^2}} }}{d}$$

4

and displayed in Fig. 2b. The RI ranges from 1.0 to 1.6, ensuring high transmission efficiency and a broad working bandwidth. The calculated results of electric field distributions in Fig. 2d attest to the continued achromatic property upheld by this altered RI profile from 0.2 to 0.9 THz. The resultant focal length precisely measures 4.5mm, aligning with the anticipated value. The extracted FWHM and focusing efficiency are depicted in Fig. 2e, with the corresponding electric field profiles in the x-y plane shown in Fig. S2a. The focusing efficiency is defined as the proportion of concentrated power at the focal plane within a 3×FWHM diameter. It is evident that focusing efficiencies hovers around 80%, and FWHM values remain below the diffraction limit, except for the FWHM(y) at 0.2 THz. Moreover, due to our proposed metalens possessing a thickness comparable to its radius, its capability to manipulate oblique waves is significantly stronger than that of an ultra-thin achromatic metasurface.

The off axis focusing characteristics are illustrated in Fig. 2c, showing the corresponding ray traces and electric field distributions as the incident angle varies from 0° to 45° at a frequency of 0.8THz. Additionally, the corresponding electric field distributions and extracted FWHMs from 0.5-0.9THz are presented in Fig. S2b-d. As the incident angle increases from 0° to 45°, the focal spot undergoes a shift along the x-axis. Notably, the maximum angle that sustains focal spots under oblique incidence, independent of frequency, is 45°. The specific distance along the x-axis is given by dx = 5sin(θ), where θ represents the incident angle. Consequently, the curves of different colors depicted in Fig. 2f, illustrating the diverse trends between focal shift along the x-axis and incident angle at varying frequencies, exhibit uniform characteristics. It is evident that super-resolution focusing maintains efficacy across a wide FOV of 90°. Therefore, it can be inferred that the simultaneous elimination of chromatic and off-axis aberrations has been achieved. In comparison with previously published achromatic metalenses, the trade-off between large bandwidth and large NA is overcome, as evidenced by our work's unprecedented accomplishments in NA, relative bandwidth, and efficiency, as demonstrated in Fig. 2g. This innovative lens holds significant potential for facilitating achromatic super-resolution imaging across a wide FOV, a crucial capability for practical terahertz imaging applications.