Depth of focus (DOF) and transverse resolution, which determine longitudinal range and the definition of a focusing lens, are two important technical parameters of imaging systems [1–3]. Deep DOF lenses have received continuously growing interest due to the increasing practical demands for high resolution focusing capabilities in an extended lateral range. However, traditional solutions, such as light-sword optical elements, forward logarithmic axicons, and inverse quartic axicons, achieve desired phase profiles by propagation phase accumulation in the dielectric materials [4–6]. The continuous phase contour places critical demand on the surface curvature of lenses, resulting in tremendous fabrication challenges [7–9]. Also, bulky and nonplanar shapes may increase the difficulty in system integration, which limits their further practical applications [10–13]. Alternatively, metasurfaces and the two-dimensional modality of metamaterials, have produced considerable advances in compact spatial-light-modulation devices, such as functionality multiplexing, owing to their flexible regulation of electromagnetic wavefronts and lower complexity in fabrication [14–20]. With the capability of engineering amplitude, phase, and polarization of electromagnetic waves over subwavelength thickness [21–30], metalenses exhibit significant superiorities over traditional lenses. Novel deep DOF metalenses are mainly developed by ring segmentation of the aperture, radial modulation (RM), and angular modulation (AM) [31,32]. The ring segmentation method divides the metalens aperture into multiple concentric rings so that the illuminating beam impinging on each ring region is focused at a different focal position. It is often used to implement nondiffractive beams such as Bessel beams [33–37]. However, segmentation leads to low aperture utilization efficiency, resulting in relatively large aperture size and thus limiting practical applications of the metalenses. The RM and AM obtain phase profiles of the metalens analytically based on quasi-optical technology and ideally require a continuous phase variation profile on the metalens. Nevertheless, only under the condition of a small numerical aperture (NA), can the discrete pixels on the metalens be approximately treated as the ideal point source, and the phase profile is thus roughly continuous. Otherwise, these aforementioned design methodologies will generate large error in terms of the generated focal length and DOF. Therefore, these methods are suitable for deep DOF metalens design in a small NA scenario. An entirely different route for the DOF extension is to adopt the meta-atom supporting polarization multiplexing, e.g., Y-shaped nanoantennas and interleaved nanorods [26]. These metalenses enable simultaneous focusing of two orthogonally polarized beams, which thereby doubles the DOF achieved in a single polarization case [38–40]. However, the Y-shaped nanoantenna operates in the reflection mode and is thus not a favorable solution for practical metalens implementation. Moreover, their asymmetric structure may result in focal spot deformation when the constituent metalens is offset-illuminated. As for the meta-atom composed of interleaved nanorods, which regulate the illuminating beams of different polarizations, nonnegligible cross talk may be encountered and complicate the design process. The desired phase profiles of the corresponding metalens are usually obtained by the RM only suitable for a small NA scenario or the time-consuming genetic algorithm, making it impossible to achieve the deepest possible DOF. Indeed, metalenses of small NAs can construct more uniform focal spot sizes within the entire DOF. However, this inevitably leads to an imaging system of relatively low resolution.

Here, we demonstrate an ultracompact high transverse-resolution metalens with ultradeep DOF in the microwave region. The metalens is composed of the birefringent cross I-shaped meta-atoms, enabling flexible and independent phase regulation of two orthogonal linearly polarized electromagnetic waves. Unlike Y-shaped nanoantennas and interleaved nanorods, the meta-atom operates in the transmission mode and can realize independent modulation on two orthogonal polarization modes. In addition, we apply a modified weighted Gerchberg–Saxton method (GSWm) to retrieve phase profiles of the metalens, conquering the precision issue encountered by RM and AM for a large NA scenario [41–45]. Benefiting from the high efficiency and flexibility of GSWm, the metalens can perform large NA high-resolution focusing, and meanwhile realize the deepest possible DOF for each single polarization state. The proof-of-concept metalens experimentally enables a DOF of near 41λ in a record-breaking wide NA range from 0.14 to 0.7. In addition, we combine the generalized scattering matrix with the multipole expansion theory for the first time to elucidate the physical mechanism of the stratified meta-atom.

First of all, we decompose an arbitrary single-layered metal–dielectric square meta-atom into a metal sheet and a dielectric substrate, whose scattering matrices can be separately modeled, as shown in detail in Appendix A.1. Thus, for a stratified structure, its generalized scattering matrix can be formulated by successively cascading the scattering matrices of every single-layered meta-atom and interlayer air gap. For the dielectric substrate, the scattering parameters can be written as S11=S22=Γ(1−e−j2βLd)1−Γ2e−j2βLd,S12=S21=(1−Γ2)e−jβLd1−Γ2e−j2βLd,(1)where Γ, β, and Ld correspond to reflection coefficient, phase-shift constant, and thickness of the dielectric substrate. With the symmetric (S11=S22), reciprocal (S12=S21), and lossless (∑j=12|Sji|2=1, i=1,2 and S11S12*+S21S22*=0) assumptions, one can finally express all scattering parameters of the square metal lattice as functions of the transmission phase, which read S11=sin(∠S21)ej(∠S21±π2),S21=cos(∠S21)ej(∠S21).(2)Therefore, we can analytically study the monochromatic coherent relation between transmission amplitude and phase. The red solid line in Fig. 2(d) illustrates the theoretically predicted transmission amplitude variation with the phase for single-, double-, and triple-layered meta-atoms composed of the arbitrary metal–dielectric square lattice, similar to that shown in Fig. 2(a). The simulation results are provided as validations. It should be noted that every small blue circle represents the simulated transmission amplitude and phase of a certain cross I-shaped meta-atom with lx of a fixed value. In general, good agreement can be observed, although for double- and triple-layered meta-atoms, while lx or ly is relatively large, the simulation results slightly deviate from theoretical analysis due to the cross talk of orthogonally polarized beams, resulting from the close distance of slots. For a single-layered meta-atom, the phase covers about 100°, with the transmission amplitude larger than −2dB. By adding extra layers, the phase coverage gradually grows, and a triple-layered meta-atom can extend the phase coverage to a complete 360°. This monochromatic analysis thus confirms that, in theory, a stratified meta-atom can effectively broaden the phase coverage while maintaining a high transmittance level. It further implies that if one can maintain the transmission amplitude while varying the geometrical details of the meta-atom, the phase coverage can be effectively extended.

The extinction spectra of a scatterer with near-zero absorption can indirectly measure its transmittance due to the absence of absorbance [48–50]. They are determined by the multipole moments that the electromagnetic illumination can excite in the scatterers, which reads σext=−πk2∑l=1∞∑m=−1,1(2l+1)Re[maE(l,m)+aM(l,m)],(3)where the integer l represents the order of the multipole, and the integer m ranges from −l to l.

The utilized meta-atom provides a powerful physical platform for the realization of a polarization multiplexing metalens with an ultradeep DOF. We will further show how to determine accurately the phase profiles of the metalens in a large NA scenario. As aforementioned, we adopted the GSWm to obtain the desired phase profiles. We discretize the preset fields as a series of field concentration points with customized intensities along the optical axis, as shown in Fig. 1. Corresponding phase profiles of the metalens can then be defined as ϕn=arg(∑m=1MejkrmnrmnwmVm|Vm|),(4)where rmn denotes the distance between the nth out of N pixels in metalens and the mth out of M target points, and Vm denotes the vector superposition of the scattering field emitted from each pixel of the metalens to the mth target point of preset fields. The weighting factor wm is applied to adjust the reconstruction accuracy of different preset field points. To improve the robustness of GSW, the relaxation factor p is introduced to scale dynamically wm for the ith step in an iteration procedure, which reads wmi=wmi−1(⟨|Vmi−1|⟩|Vmi−1|)p,(5)ϕni=arg(∑m=1MejkrmnrmnwmiVmi−1|Vmi−1|).(6)It is worth mentioning that GSWm reduces to GS and GSW when p is equal to 0 and 1, respectively. To investigate the influence of the index p on the robustness of GSWm, we calculate the sum-squared error (SSE) when p takes different values. The SSE is defined by SSE=∑(|Vm|−⟨Vm⟩)2∑|Vm|2.(7)

In summary, we have theoretically and experimentally proposed a high transverse-resolution transmissive metalens with an ultradeep DOF based on the birefringent stratified cross I-shaped meta-atom and the robust GSWm method. An ultradeep DOF of 1230 mm (41λ) ranging from 220 mm (7.3λ) to 1450 mm (48.3λ), corresponding to a wide NA range from 0.14 to 0.7, is presented under the illumination of a normally incident beam with a linear polarization angle of 45° at the wavelength of 30 mm. More importantly, the focal spot size is close to the corresponding diffraction limit in a wide NA range, which provides high-resolution characteristics for application, overcoming the limitation of previous deep DOF lenses that only adopted small NA focusing. The underlying operational mechanism of stratified meta-atoms is elucidated by the generalized scattering matrix and the multipole expansion theory. Further work will focus on extending the capability of the GSWm to finely shape the three-dimensional near-field distribution. The marriage of such an advanced iterative method with the birefringent stratified meta-atom in this paper will make possible a metalens with an ultra-deep DOF and more uniform focal spot size within it. Due to the size restriction in available microwave anechoic chambers, practical imaging experiments are numerically performed, which further validates our proposed methodology.

Acknowledgment. The paper was written through contributions of J. Qi, Y. Mu, and S. Wang. All authors have given approval to the final version of the paper. J. Qi and J. Qiu conceived the idea for this paper. J. Qi, and Y. Mu together formulated and programed the GSWm, elucidated the underlying operational mechanism of the stratified cross I-shaped meta-atom, fabricated the prototypes, and designed the experiments. Y. Mu and S. Wang conducted the measurements.