Polarization-independent achromatic Huygens’ metalens with large numerical aperture and broad bandwidth

2.1 Huygens’ metasurfaces

We begin with a dual-polarized unit cell design based on the principle of Huygens’ metasurfaces (HMSs) [30, 35], which has advantages of miniaturized size, broad phase response, and wide group delay range. In general, an HMS is composed of a thin surface with co-located and orthogonal electric and magnetic dipoles. When discontinuous fields ( E 1 ⃗ , H 1 ⃗ and E 2 ⃗ , H 2 ⃗ ) in two half-space are separated by the HMS, ( J s ⃗ ) and magnetic ( M s ⃗ ) current densities can be induced to transform the fields from one half-space to the other half-space, as shown in Figure 2(a). From the boundary conditions, the fields on both half regions can be expressed as [30, 35]:

where Y ̄ ̄ es and Z ̄ ̄ ms represent the electric surface admittance and magnetic surface impedance, respectively. E av t ⃗ ∣ s and H av t ⃗ ∣ s are the average tangential fields of the surface. Generally, the surface admittance ( Y ̄ ̄ es ) and impedance ( Z ̄ ̄ ms ) are tensors. However, they can be simplified to scalar quantities (Y_es and Z_ms) for a single polarization surface or polarization-insensitive surfaces [23, 31, 35]. Hereafter we adopt this simplification. These impedance and admittance values can be designed with spatially sampled subwavelength unit cells. According to the boundary condition and the equivalence principle [30, 35], the normalized admittance Y and impedance Z can be extracted from the complex coefficients of reflection (r) and transmission (t):

where η₀ is the wave impedance of free space. When Re Y = Re Z = 0 and Im Y = Im Z , Eq. (2) requires that the surface is reflectionless (r = 0) and a broad transmission phase variation with respect to frequency can be realized [3, 30]. Based on the reflectionless property of the HMS, many interesting and efficient metadevices have been demonstrated for beam steering [36–38], beam generating [39], and beam focusing [22, 23].

Figure 2:

The structure of designed dual-polarized metasurface unit cell. (a) The schematic diagram of designed HMS (cross-section view). (b) The schematic diagram of the unit cell. (c–e) The details of three metallic layers (top-view): (c) the top layer, (d) the middle layer, (e) the bottom layer. The brown lines indicate metallic traces, the black arrows on the metallic traces indicate the direction of current travel at one phase point.

2.2 Unit cell design

We begin with a very brief overview on existing HMSs most relevant to broadband achromatic lensing, their capabilities and their limitations. We then list important properties of a unit cell which will lead to a broadband achromatic metalens. Although existing HMSs show excellent conversion efficiency between the incident and refracted waves with deeply subwavelength thickness, the reported works mostly focus on resonance matching under a single frequency, resulting in limited working bandwidths [22, 23, 36, 37, 39]. Recent works show that broadband HMSs can be realized if the resonances between three metallic layers are controlled independently [24, 27]. However, Ref. [24] ignored phase dispersion within the bandwidth, resulting in a strong focal length shift, while Ref. [27] has a limited phase and group delay coverage, as it only engineers one electric and one magnetic resonance around the operating bandwidth. To construct a highly desirable achromatic metalens with large NA and broad bandwidth, the unit cells should meet the following requirements: (i) high transmission efficiency, (ii) broadly tunable transmission phase at the center frequency, (iii) linear phase response, and (iv) broadly controllable group delay. Here, we propose a dual-polarized Huygens’ unit cell with three metallic layers, which can simultaneously achieve these requirements by engineering two electric and one magnetic resonances within or near the operating bandwidth.

Figure 2 depicts our unit cell structure. Its symmetric pattern distribution ensures polarization insensitivity for x- and y- polarized waves. The outer layers (top and bottom layers) contain four corner strips which vary in length, and in some cases become folded (see Figure 2(c)–(e)). The middle layer contains four straight strips along the four sides of the cell. Although this unit cell is simple, we will show that it provides sufficient control over two electric resonances and one magnetic resonance. Specifically, the outer layer controls the magnetic resonance and one of the electric resonances: the magnetic resonance originates from the antisymmetric surface currents between the outer layers, while the electric resonance originates from the symmetric surface currents in the same layers. The other electric resonance is generated by the four strips on the middle layer. We can tune these three resonances by sweeping three variables: the gap between corner strips of outer layers (g), the folded strip length of outer layers (l₁) and the length of four strips in the middle layer (l).

To demonstrate that the magnetic resonance and electric resonances can be tuned independently, we analyze the unit cell with the surface current method [39]. Assuming the unit cell is illuminated with a y-polarized EM wave, the induced surface currents J ⃗ top , J ⃗ middle and J ⃗ bottom are formed on the top, middle, and bottom layers, respectively. The arrows on the strips indicate the travel direction of currents J ⃗ top , J ⃗ middle and J ⃗ bottom at the same time instance, as shown in Figure 2(c)–(e). The propagation distance separating the top and bottom layers, along with coupling dynamics, make J ⃗ top and J ⃗ bottom generally unequal [39]. J ⃗ top and J ⃗ bottom can be decomposed into the common and differential modes, which give rise to electric and magnetic responses respectively. The induced current of the middle layer mainly comes from the four capacitors, formed by the pairs of metallic strips on the four sides of the unit cell. Thus the surface currents on each layer can be decomposed as:

The equivalent circuit model of our unit cell structure is presented in Figure 3(a). β and h represent the wavenumber and the thickness of the substrate, and Y_s1 and Y_s2 are the equivalent admittances of the outer and middle layers, respectively. Once the input and load admittances, substrate parameters, and reflection and transmission coefficients are specified, the required admittances of different sheets can be calculated by the ABCD matrix method (see Supplementary Note S2). Figure 3(b) and (c) show the transmission amplitude and phase achievable by tuning Y_s1 and Y_s2 at the center frequency of 24 GHz. Particularly, by tuning (Y_s1 and Y_s2) along the dashed blue lines in Figure 3(b) and (c), the transmission amplitude can remain close to unity while the phase varies by more than 360°. This shows that the three-layer isotropic Huygens’ metasurface topology can yield high transmission efficiency and over 360° phase range, provided that Y_s1 and Y_s2 can be tuned over the depicted range. The black “x”s in Figure 3(b) and (c) depict the (Y_s1) and (Y_s2) values we achieved with our metalens unit cells. This shows that our proposed unit cell structure indeed provides sufficient tuning range to achieve near-unity transmission and over 360° phase coverage. We now examine how the ability to flexibly tune Y_s1 and Y_s2 leads to desirable broadband characteristics for the metasurface unit cell. Refs. [27, 37] have shown that Y_s1 and Y_s2 completely determine the surface admittance and impedance Y_es and Z_ms of the metasurface unit cell. The independent control of Z_ms and Y_es allows us to leverage the “Huygens” characteristics of the unit cell and achieve high transmission efficiency over a broad bandwidth. We will also show, by way of example, that by positioning the resonances we can also achieve controllable linear phase changes during the entire operating bandwidth.

Figure 3:

Circuit modelling of the metasurface unit cell. (a) The equivalent circuit model of cascaded three-layer metasurface unit cell. (b) The transmission amplitude as a function of Y_s1 and Y_s2. (c) The transmission phase as a function of Y_s1 and Y_s2.

We simulate the electromagnetic response of the proposed unit cell using Ansys HFSS. We use a Rogers RO4003C dielectric substrate with a relative permittivity of 3.55 and a thickness of 0.813 mm. The patterned copper has a thickness of 9 um and a conductivity of 5.8 × 10⁷ Ω⁻¹ m⁻¹. In our design, the length of the corner strip on the outer layers is a _x = a _y = 4.2 mm and the unit cell size is P _x = P _y = 4.4 mm. A periodic boundary is used in the x- and y- directions to simulate an infinite 2D array of identical cells. Figure 4 gives the imaginary parts of the normalized electric admittance (Y) and magnetic impedance (Z) versus frequency of one unit cell used in our achromatic metalens. The values of Y and Z can be retrieved by incorporating the transmission and reflection coefficients into Eq. (2). In this example, the three variables are g = 0.4 mm, l = 3.3 mm, and l₁ = 0.2 mm. We find that the unit cell has two electric resonances (at f E 1 and f E 2 ) and one magnetic resonance (at f _M ), as shown in Figure 4(a). To analyze the electric and magnetic responses, we separate the three metallic patterns into the outer layers and the middle layer, as shown in Figure 4(b) and (c). The outer layers support one magnetic resonance (f _M ) and one electric resonance ( f E 2 ) , while the middle layer only supports one electric resonance ( f E 1 ) . When we combine the middle and outer layers, we find that the magnetic resonance response (f _M ) remains unshifted, but the two electric resonances ( f E 1 and f E 2 ) are shifted because of mutual couplings among the three layers. The forgoing observations show that the outer layers mainly control the magnetic response while all three layers and their coupling dynamics affect the electric response, which is consistent with previous works reported in Refs. [27, 37, 39], and our insights obtained through the surface current perspective earlier this section.

Figure 4:

Imaginary parts of normalized admittance Y and impedance Z of the unit cell 3. (a) The unit cell with three layers. (b) The unit cell with only the end layers. (c) The unit cell with only the middle layer.

A broad range of group delay and near-zero group delay dispersion within the bandwidth is the key to achieving achromatic metalens with large NA [14, 32]. In this work, we achieve a linear phase variation and broadly controllable group delay by independently controlling two electric and one magnetic resonances. In general, around the resonance frequency, the transmission coefficient changes rapidly in phase and decreases somewhat in magnitude. Making use of this effect, one can achieve a linear phase with controllable dispersion by tuning the spectral positions of resonances respect to the operating bandwidth. We shall illustrate this using examples of three unit cells from our metalens, for which the surface magnetic impedances and surface electric admittances are shown in Figure 5(a)–(c). Unit Cell 12 has two resonances located away from the operation band (22–26 GHz). Correspondingly, Figure 5(e) and (f) (blue, solid) show that this unit cell achieves near-unity transmission amplitude and a slow phase variation over the operation band (i.e. a small group delay). Unit Cell 6 (Figure 5(b), (e), and (f)) achieves a faster phase variation (or a larger group delay) by having two resonances inside the operating band. Finally, Unit Cell 3 (Figure 5(c), (e), and (f)) achieves an even faster phase variation by involving three resonances positioned just outside the operation band. Thus by tuning spectral positions of the resonances, the group delay can be kept mostly constant over the operation band and its slope can be tuned. While the usage of multiple resonances close to, or within, the operation band may reduce the transmission efficiency, we find that even for such cases, absorption and reflection can be kept minimal by using a weakly resonant (low-Q) unit cell design and keeping a similar normalized Y and Z [40,41] . It should be noted that the transmission coefficient reaches its peak when Im Y = Im Z . The frequency of maximum transmission corresponds to the resonant frequency of the complete unit cell, which in general differs from the frequencies of the three individual resonances that make up the cell. Figure 5(d) and (e) show that for all three unit cells, high transmission efficiencies are achieved at the design frequency of 24 GHz, and a transmission amplitude of >0.8 is achieved over the operation band. In summary, we find that our proposed unit cell provides flexible tunability for all three resonances, which is sufficient to design metasurface unit cells with high transmission efficiency, linear phase variation, and broadly controllable group delay over a wide bandwidth. Hence they are attractive for the construction of the high-transmission-efficiency, broadband achromatic metalens reported in this paper. The resonances performance of all unit cells can be found in Supplementary Figure S3.

Figure 5:

Schematic diagrams of three unit cells. (a–c) Imaginary parts of normalized admittances Y and impedances Z of unit cell 12 (a), unit cell 6 (b) and unit cell 3 (c). (d) Reflection coefficients of three unit cells. (e) Transmission coefficients of three unit cells. (f) Transmission phases of three unit cells.

2.3 Achromatic metalens design

To achieve achromatic focusing, the spatial distribution of unit cells should simultaneously perform phase and group delay compensations. Firstly, the phase compensation of the unit cell according to the spatial location can be expressed by the equal wave travel distance as:

where ω, c, F and r represent the angular frequency, speed of light, focal length, and radial coordinate of the unit cell, respectively. Moreover, for the achromatic metalens, the required group delay, as a functional of the radial position, can be expressed as:

where ΔTD(r) is the relative group delay and R is the radius of the metalens. We find that the greatest delay is required at the center of the metalens (i.e. r = 0). A large range of relative group delay is required when the NA of the achromatic lens is large, as the NA can be defined as:

A comparison of required phase distributions and relative group delays of two achromatic metalenses with different NAs is provided in Supplementary Figure S4. It shows that to construct an achromatic metalens with a large NA, unit cells should have a transmission phase coverage of more than 360° and a broad range of group delay. Specifically, some unit cells of the achromatic metalens should have the similar phase response at the center frequency but with different group delays. Most proposed achromatic metalens fail to achieve that, leading to designs with small NAs [14], [15], [16, 18, 21, 27]. In our case, we achieve the required conditions through the independent control of multiple resonances. Figure 6(a) gives the transmission coefficients of two unit cells, which illustrate this point. Both unit cells have high transmission amplitudes over the operating frequency (22–26 GHz) and the same phase response under the center frequency (24 GHz), but they have very different group delays. Figure 6(b) and (c) give the explanation, where for Unit cell 8, the electric and magnetic resonances are closer to the operating bandwidth, resulting in a narrower fractional bandwidth compared to Unit cell 12. Hence by introducing the three-layer Huygens’ metasurface unit cell, and properly tuning the resonances it affords, we can construct a library of unit cells with high transmission efficiency, large phase range (at 24 GHz) and large group delay range, which can then be used to construct a high-performance achromatic metalens.

Figure 6:

Comparison of two constructed unit cells. (a) The transmission coefficients of two unit cells. (b) Imaginary parts of normalized admittance and impedance of unit cell 8 and 12 respect to the frequency.

Based on the unit cell library, we design a polarization-independent achromatic metalens with NA equals to 0.64 by using 12 different unit cells. The transmission amplitudes and phases of these unit cells are given in Figure 7(a) and (b), respectively. It can be found all unit cells can achieve larger than −2 dB transmission amplitude over the operation bandwidth from 22 GHz to 26 GHz, which enables high transmission and focusing efficiencies of our metalens. The group delays of unit cells from the center to the edge gradually decrease, as seen in Figure 7(c). This ensures effective chromatic correction of our metalens. The geometrical parameters and corresponding time delay of unit cells are given in Supplementary Table S1.

Figure 7:

Transmission coefficients of the unit cells of the achromatic metalens (the yellow region is the bandwidth of interest). (a) The transmission amplitude of the used unit cells. (b) The transmission phase response of the used unit cells. (c) The group delay distribution of the used unit cells.

We carry out a full-wave simulation of the achromatic metalens using Ansys HFSS. To verify the performance of the achromatic metalens, we simulated the chromatic metalens and the achromatic metalens. Both metalenses are designed with the same aperture size and focal length, by using the designed unit cell structure but with different group delay distributions. Theoretically, the chromatic metalens should have constant group delay for all unit cells. However, in our case, the unit cells used for the chromatic metalens have a small variation of group delay as the phase is tuned by the resonance. Figure 8 gives the normalized intensity profiles at the XZ-plane for both metalenses and the focal plane for achromatic metalens under different frequencies. The “x” represents the designed focal spot. The focal length has shifted slightly from 80 mm for both metalenses. From Figure 8(a) and (b), it can be seen that the focal length of the chromatic metalens increases from 76 mm to 93 mm with the increase of frequency, while the focal length of the achromatic metalens is at 83 ± 1.5 mm over the whole bandwidth. In addition, the chromatic aberration of a conventional metalens can be described as [42]:

where λ and F are the central wavelength and designed focal length of the metalens, respectively. ΔF represents the corresponding focal shift within the bandwidth. From Eq. (7), we can find that the achromatic metalens has improved the focal length shift from 13.44 mm (in theory) to 1.5 mm (in simulation), achieving an 89 % improvement. Figure 8(c) and (d) show the normalized intensity distributions on the focal plane. We find that the achromatic metalens has a very low sidelobe level (less than 5 % of the peak) within the working bandwidth. The corresponding full-width half maximum (FWHM) of the spot size is close to the diffraction-limited FWHM = 0.5 λ NA . In addition, the metalens has achieved a very high transmission efficiency of 90 %, and a maximum focusing efficiency of 80 % (at 23 GHz). The transmission efficiency is the fraction of power transmitted by the metasurface [24, 27]. The focusing efficiency is defined as the ratio between the integral of the field intensity in the focal plane with a diameter six times the FWHM spot size, to the full plane integral of field intensity in the focal plane [26, 27, 43]. We also simulate the focusing performance of the achromatic lens under oblique incidence and find that the focal length can remain constant over the whole bandwidth upon a moderate oblique incidence of up to 30°. (See Supplementary Note S6 and Supplementary Figures S6 and S7).

Figure 8:

Simulated field distributions of the chromatic and achromatic metalenses. (a) Normalized intensity distribution of chromatic metalens along the propagation plane (XZ-plane). (b) Normalized intensity distribution of achromatic metalens along the propagation plane (XZ-plane). (c) Normalized intensity distribution along the focal plane (XY-plane, 2D). (d) Normalized intensity distribution curves in the x-direction crossing the center of each focal hot spot. (λ = 12.5 mm is the free-space wavelength of 24 GHz EM wave.)