## Abstract

Spaceplates have emerged in the context of nonlocal metasurfaces, enabling the compression of optical systems by minimizing the required empty space between their components. In this work, we design and analyze spaceplates that support resonances with opposite symmetries, operating under the so-called Huygens’ condition. Using the temporal coupled-mode theory, we demonstrate that the spatial compression provided by Huygens’ spaceplates is twice that of conventional single-resonance counterparts. Additionally, they can support broader operational bandwidths and numerical apertures, facilitating the reduction of chromatic aberrations. Moreover, Huygens’ spaceplates maintain nearly full transparency over a wide frequency and angular range, allowing their straightforward cascading for multi-frequency broadband operation. Finally, we propose a physical implementation of a Huygens’ spaceplate for optical frequencies based on a photonic crystal slab geometry.

### Similar content being viewed by others

## Introduction

The advent of spaceplates provides new opportunities in the field of non-local metasurfaces^{1,2,3,4,5,6,7}, allowing to decrease unavoidable free-space regions in an arbitrary optical system (e.g., air gaps between two neighboring lenses). This route of minimizing the overall thickness of optical systems drastically differs from the conventional approaches where the reduction is made in terms of the lens or image sensor thickness^{8,9,10,11,12,13,14,15,16,17,18}. For its operation, a spaceplate must have a specific (quadratic) dependence of the complex phase of the transmission coefficient with respect to the incidence angle. At the same time, the magnitude of the transmission coefficient must be close to the unit in a sufficiently broad range of frequencies and incident angles.

Recently, several material platforms were proposed to realize a spaceplate, including photonic crystal slabs^{2,19}, multilayer structures^{1,20,21,22,23}, and Fabry-Pérot cavities^{24,25,26}. The limits of operation of these structures have been studied both numerically^{20} and analytically^{24,25}, showing an intrinsic trade-off relation between the maximum compression factor of the spaceplate and its numerical aperture (NA). As demonstrated in these works, to overcome the trade-off, one needs to increase the number of coupled resonances in the structure. Furthermore, there are stringent limitations on the operational bandwidth of spaceplates providing a given compression ratio^{3}.

Cascading Fabry-Pérot structures have been proposed to increase the number of resonances and surmount the aforementioned limitations^{24}. These resonances have the same symmetries of the electric type. Although this approach increases the compression ratio, the structure becomes less and less transparent outside the operating range. Moreover, cascading many layers leads to a more complex manufacturing process. Therefore, the cascades of Fabry-Pérot structures so far have been implemented only for microwave frequencies^{25,26}.

In this work, we propose and design spaceplates supporting resonances of opposite even (electric) and odd (magnetic) symmetries. In particular, by spectrally overlapping the lowest order electric and magnetic dipolar resonances in a photonic crystal slab, we demonstrate the possibility of reaching the so-called Huygens’ condition that was previously explored in different contexts^{27,28,29,30}. Due to Huygens’ condition, the spaceplate exhibits almost no reflections in a broad frequency range while near the resonance, the phase of transmission varies rapidly with frequency and incidence angle. Such Huygens’ spaceplates exhibit double enhancement of the compression factor compared to the best attainable values for conventional single-resonance spaceplates. Furthermore, we show that Huygens’ spaceplates can provide higher operational bandwidths and remain nearly transparent in a broad frequency range. Using the temporal coupled-mode theory^{31} (CMT), we propose a realistic optical design of a Huygens’ spaceplate. It should be noted that a related idea of double-resonance spaceplates was very recently suggested in ref. ^{32}, however, the structure there was described by a surface susceptibility model assuming zero structural thickness, and no realistic optical design was proposed.

## Results

### Huygens’ spaceplates

The spaceplates allow reducing the size of the optical devices, with the reduction ratio defined as the ratio between the effective distance that the spaceplate emulates and its width itself (*R* = *d*_{eff}/*d*_{sp}, see Fig. 1). The range of angles at which the spaceplate operates (from 0 to \({\theta }_{\max }\) from the normal direction) defines its numerical aperture NA \(=\sin {\theta }_{\max }\).

To mimic light propagation over some distance in free space, the spaceplate must have a specific angular dependence of light transmission. On the one hand, different spatial frequencies of light (plane wave components propagating at different angles *θ* with respect to the normal) must acquire different phases, according to a quadratic law. On the other hand, the transmission magnitude must be angle-independent and close to the unity (to avoid parasitic reflections). Free-space behavior can be expressed as a transmission function (*t*_{fs}) with dependence on the transversal plane wave components (**k**_{t}) using the paraxial approximation as follows^{2}:

being *ϕ*_{0} the global phase, *λ* the wavelength, and *ω* the angular frequency.

To evaluate to what extent a multi-resonant structure with two resonances mimics the response described by Eq. (1), a coupled-mode model has been developed. Considering the resonances to be orthogonal, the first mode with even symmetry and the second with odd symmetry, the transmission coefficient of a structure can be expressed as follows^{31}:

where *i* is the imaginary unit, *t*_{d} and *r*_{d} are the direct transmission and reflection (in the absence of the resonances in the structure), *ω*_{1}(**k**_{t}) and *ω*_{2}(**k**_{t}) are the resonance frequencies for the even and odd modes, respectively, and *γ*_{1} and *γ*_{2} are the widths of these resonances (we assume them being angle-independent). Next, we introduce two auxiliary parameters which denote deviations of the operational frequency *ω* from a given resonance frequency normalized by the resonance width, that is, Ω_{1}(*ω*, **k**_{t}) = [*ω* − *ω*_{1}(**k**_{t})]/*γ*_{1} and Ω_{2}(*ω*, **k**_{t}) = [*ω* − *ω*_{2}(**k**_{t})]/*γ*_{2}. Defining additionally parameter \({q=i{{r}_{{{{\rm{d}}}}}}/{{t}_{{{{\rm{d}}}}}}}(q\in {\mathbb{R}}\;{{{\rm{for}}}}\;{{{\rm{lossless}}}}\;{{{\rm{slabs}}}})\), one can write Eq. (2) as

Since the two resonance frequencies might depend differently on the angle of incidence, we can express the frequency dispersion of each resonance for small incidence angles via *α*_{1} and *α*_{2} parameters as *ω*_{1}(**k**_{t}) ≈ *ω*_{1}(0) + *α*_{1}∣**k**_{t}∣^{2} and *ω*_{2}(**k**_{t}) ≈ *ω*_{2}(0) + *α*_{2}∣**k**_{t}∣^{2}. Introducing the Taylor-series expansion (see Section 1 of the Supplementary Information), the transmission phase for a structure containing two orthogonal even and odd resonances could be expressed as follows

where Ω_{0,1} and Ω_{0,2} represent the frequency detuning factors for the normal incidence as Ω_{0,1} = [*ω* − *ω*_{1}(0)]/*γ*_{1} and Ω_{0,2} = [*ω* − *ω*_{2}(0)]/*γ*_{2}.

Using Eqs. (3) and (4), the transmission produced by the spaceplate with two resonances can be written in the form of transmission through free-space slab described by Eq. (1), that is,

where we introduce compression parameter Φ as

From (1) and (5), one can see that the free-space reduction ratio *R* is directly proportional to the compression parameter Φ as *R* = 4*π*Φ/(*d*_{sp}*λ*). From Eq. (6), we can see that the compression parameter for two-resonance spaceplates can be divided into two terms, one term being related to the even resonance and another to the odd resonance. Due to their arithmetic summation, the total compression parameter can become double that for a single-resonance spaceplate^{32}. This result of doubling of the compression parameter due to combining even and odd resonances in a single geometry is somewhat similar to the doubling of transmission phase span for normal incidence in Huygens’ metasurfaces^{27}. Hence, Huygens’ spaceplates exhibit a significant advantage with a much higher compression parameter (and reduction ratio) than their single-resonance counterparts.

To achieve the full transmission through the Huygens’ spaceplate ∣*t*_{sp}∣ = 1 at a given frequency, one needs to make the following condition hold at that frequency, as derived from Eq. (3):

Let us compare the performance of a single-resonance and Huygens’ spaceplates versus the incident angle (\(| {k}_{{{{\rm{t}}}}}| /{k}_{0}=\sin \theta\)), and the frequency of the incident wave *ω*. For a fair comparison, we design both spaceplates to provide the maximum compression parameter with high transparency (∣*t*∣ ≈ 1). For a spaceplate with a single (even) resonance mode operating at frequency *ω*, parameter Φ is maximized if Ω_{0,1}(*ω*) = 0, while transmission reaches ∣*t*∣ = 1 if Ω_{1}(*ω*, **k**_{t}) = 1/*q*. The former and latter conditions come from Eqs. (6) and (7), respectively, assuming that ∣Ω_{2}∣ → *∞* (the second resonance is far apart). Therefore, to satisfy both these conditions, one needs to maximize the value of *q*, resulting in *t*_{d} → 0 (See Section 2 of the Supplementary Information). In Fig. 2a, we plot the transmission coefficient and compression parameter for the optimized single-resonance spaceplate with *t*_{d} = 0.01. The exact parameters used in this and the following numerical approaches are detailed in Table S1 of the Supplementary Information. The angular behavior of the same spaceplate is depicted in Fig. 2b, c. As observed, while the spaceplate is transparent and maintains a high Φ at normal incidence and frequency *ω* = *ω*_{1}, it forfeits both characteristics outside its narrow operational bandwidth. It should be noted that previously suggested single-resonance spaceplates in Ref. ^{2} were designed with compression parameters that were far away from the highest possible ones, allowing for a broader transparency frequency range. Furthermore, the trade-off between transmission amplitude and compression parameter in such spaceplates was studied in ref. ^{3}.

In sharp contrast to the previous case, Huygens’ spaceplates supporting even and odd resonances provide the unique opportunity to have large compression parameter and high transparency in a wide frequency range. In particular, under Huygens’ condition of degenerated resonances of opposite symmetries *ω*_{1} = *ω*_{2}, we obtain \(\left({\Omega }_{1}={\Omega }_{2}\right)\) that makes the right-hand side of Eq. (7) independent of frequency *ω*. Therefore, the condition of full transparency can be satisfied ideally at each frequency if we design the slab to have *q* = 0 (*r*_{d} → 0). Therefore, a spaceplate that satisfies Huygens’ condition can achieve the total transmission for every frequency, including those with Φ maximized, with the double compression parameter provided by a single-resonance spaceplate (assuming the two resonances to have the same spectral characteristics). Figure 2d–f plot the complex transmission versus incident frequency and angle for the ideal Huygens’ spaceplate whose two resonances have equal spectral properties, that is, *ω*_{1} = *ω*_{2}, *α*_{1} = *α*_{2}, and *γ*_{1} = *γ*_{2}. Here, we assumed *t*_{d} = 1, which in practice can be obtained, e.g., by adding a uniform slab of specific thickness behind and/or in front of the spaceplate^{2}. One can see that this ideal Huygens’ spaceplate achieves total transparency for all angles of incidence and frequencies. The 2*π*-phase span is exactly double of that in the single-resonance scenario (compare Fig. 2c, f), accounting for the twofold increase in free-space compression.

In practice, the bandwidth of the transparency region is always limited by multiple imperfections: other higher-order resonances exist in the structure and perturb the Huygens’ balance, resonances have a mismatch in terms of *α* or *γ* parameters, *γ*, in fact, depends on **k**_{t}, and the bands of the modes become non-parabolic for large **k**_{t}. These limitations will be discussed in more detail in the section related to the implementation of the spaceplate.

Interestingly, for both the considered spaceplates in Fig. 2, operational \({{{\rm{NA}}}}={k}_{{{{\rm{t,max}}}}}/{k}_{0}\) is the same. For the ideal Huygens’ spaceplate, at the degenerate resonance frequency, \(k_{{\rm{t}}, \max}^2=2\pi /\Phi =\pi {\gamma}_{1}/{\alpha}_{1}\) (see Section 1.2 of the Supplementary Information), is the same value as for the single-resonance spaceplates^{24}. Nevertheless, as shown below, NA can be increased in Huygens’ spaceplates under specific small detuning of the two resonances.

### Beyond Huygens’ condition

Spaceplates, similar to lenses^{33}, exhibit intrinsic chromatic aberrations, characterized by the phenomenon where the light of various frequencies undergoes differing compression parameters Φ(*ω*), as illustrated in (see Fig. 2a, d). These aberrations result in image distortions, which are undesirable in most practical applications. Recent work^{21} introduced the theoretical concept of multi-wavelength spaceplates, designed to minimize chromatic aberrations at three specific wavelengths. This was achieved through an elaborate design involving a multilayer structure with several hundred layers. Nevertheless, it worked only for 3 very narrow (nearly *discrete*) frequency regions, remaining highly opaque for light at other frequencies. Another approach to face chromatic aberration is to design the spaceplates via gradient-based freeform optimization^{23}. Unfortunately, the designs proposed in this work could not be fabricated due to their complex grayscale permittivity profile.

Here, we discuss an alternative approach for overcoming chromatic aberrations in spaceplates by partial overlapping of the resonant modes with opposite symmetries. This approach allows us to decrease the aberrations over a *continuous* frequency range. Using the CMT developed in the previous section, we find that by detuning slightly from the Huygens’ condition so that *ω*_{2} − *ω*_{1} = Δ*ω* ≪ *ω*_{1}, it is possible to achieve a nearly flat region of Φ(*ω*) with negligible frequency dispersion of compression parameter. In Fig. 3, we plot the complex transmission and compression parameter for a spaceplate operating slightly away from the Huygens’ condition when Δ*ω* = 4*γ*/*π*. The detuning results in the drop of transmission amplitude between the two resonance frequencies but we mitigate this drop by slightly decreasing the background transmission to *t*_{d} = 0.8 (other parameters stay the same as in Fig. 2d). Although the total transparency is achieved now only at two frequencies, the transmission magnitude is not less than 0.8 for the entire spectrum. We define the compression bandwidth BW(Φ) as the frequency range where \(\Phi (\omega )\ge 0.9{\Phi }_{\max }\) (black vertical lines in Fig. 3a). For the spaceplate analyzed in Fig. 3a, BW(Φ) is increased compared to the spaceplate in Fig. 2d by around 2.5 times.

Figure 4 depicts a numerical study comparing the BW(Φ), \({\Phi }_{\max }\) and NA versus the separation of resonances Δ*ω*. Here two advantages of the double-resonance spaceplates over single-resonance ones can be seen. On the one hand, the achromatic region BW(Φ) is increased (blue curves). On the other hand, the purple curve shows how the NA is also increased with the separation of the resonances. It should also be mentioned that the maximum compression parameter becomes reduced when Δ*ω* ≠ 0. However, it remains higher than that of the single-resonance spaceplate, as illustrated by the red curves. Thus, Huygens’ spaceplates provide the simultaneous enhancement of all three parameters by selecting the appropriate distance between their resonances.

### Implementation

In order to implement the proposed Huygens’ spaceplate, we need to choose a photonic geometry that supports two closely spectrally located resonant modes of opposite symmetries. Although one natural choice for the spaceplate geometry would be a metasurface consisting of dielectric cylinders with quasi-Mie resonances^{27}, we found that the curvature of these resonance bands near **k**_{t} = 0 is negative, meaning that *α*_{1} < 0 and *α*_{2} < 0. Such curvature cannot provide the space compression effect, rendering *d*_{eff} < 0 in Eq. (1).

Therefore, for our spaceplate design, we use a photonic crystal slab of width *d*_{sp} including a square lattice of holes with radius *r*, as shown in Fig. 1. The lattice period is *a*, and the slab permittivity is *ε*_{M}. In contrast to the single-resonance photonic crystal slabs^{2,19}, here we utilize the slab supporting electric and magnetic overlapped resonances recently proposed in a different context^{29}.

As proof of the concept, here we design the spaceplate operating near Huygens’ condition similar to that shown in Fig. 3a. Such design is more straightforward as it operates at ∣*t*_{d}∣ < 1. It should be mentioned that it is also possible to design the spaceplate operating exactly at the Huygens’ condition like that in Fig. 2d. However, in this case, due to the requirement of ∣*t*_{d}∣ ≈ 1, one needs to engineer carefully the background (non-resonant) transmission coefficient, which can be done by adding two uniform material layers around the photonic crystal slab^{2}.

Based on the CMT, we optimized the dimensions of the spaceplate as follows: *d*_{sp} = *λ*_{0}/6, *a* = 0.527*λ*_{0}, *r* = 0.226*λ*_{0}. We selected *ϵ*_{BG} = 1 and *ϵ*_{M} = 11.9 for the permittivities of background and slab, equivalent to those of air and crystalline silicon in the near-infrared regime (see Section 3 of the Supplementary Information for the dimensions of the structure when designed for the frequency of 193.5 THz). Practical realization of such a geometry is overviewed in the Discussion Section. Such a photonic crystal slab supports two resonance modes with opposite symmetries in the considered frequency range, as shown in Fig. 5a. As mentioned above, the modes are slightly spectrally detuned to have a broadband compression parameter with high broadband transmission as the regime in Fig. 3a. Both modes correspond to the same transverse-electric polarization with the electric field along the *y*-axis. The field distributions of the two modes are plotted in Fig. 3b, c, revealing that the lower-frequency mode *ω*_{1} corresponds to the lattice of magnetic dipoles *m* along the *x*-direction and the higher-frequency mode *ω*_{2} to the lattice of electric dipoles *p* along the *y*-direction. The dipoles are excited in the high-permittivity regions between the air voids of the photonic crystal slab. Thanks to the almost overlapping even and odd resonances, we achieve a high transmission close to the unity for a wide range of frequencies and angles of incidence, as shown in Fig. 5d–f. The transmission amplitude curve has a similar shape to the one in Fig. 3a. At the same time, the normalized chromatic-abberation-free bandwidth BW(Φ)/*γ*_{1} = 1.01, which is significantly higher than the maximal bandwidth of single-resonance spaceplates, as seen from Fig. 4. Moreover, the relative bandwidth reaches \(2{{{\rm{BW}}}}(\Phi )/({\omega }_{1}+{\omega }_{2})=5 \%\), is much higher than that of most of the previously proposed narrowband spaceplates^{3}. The reduction ratio *R* of the spaceplate reaches the value of 6.65. It should be noted that this value could be further increased by several orders of magnitude by reaching the Huygens’ condition using higher-order (quadrupoles, octupoles, etc) resonant modes in the photonic crystal slab. Indeed, their quality factors are significantly higher than those of the dipolar modes, allowing higher reduction ratios.

Next, we fit the transmission coefficient and reduction ratio spectra to the expressions derived from the coupled-mode model. We chose the resonance frequencies *ω*_{1} = 0.5834 × 2*π**c*/*a* and *ω*_{2} = 0.6085 × 2*π**c*/*a* from Fig. 5a. The background transmission was estimated to be *t*_{d} = 0.818 + 0.147*i* from the transmission coefficient for a continuous slab of thickness *d*_{sp} and permittivity *ε*_{M} at the frequency (*ω*_{1} + *ω*_{2})/2. Furthermore, we chose *γ*_{1} = 0.05*ω*_{1}, *γ*_{2} = 0.047*ω*_{2}, *α*_{1} = 0.18 × *c**a*/2*π*, *α*_{2} = 0.1856 × *c**a*/2*π*. The transmission and compression parameter spectra are shown in Fig. 5d with dashed lines. One can see a noticeable deviation between the spectra from the full-wave simulations and the coupled-mode model. The deviation can be explained by the fact that the considered photonic crystal slab also possesses high-order resonance modes that are not taken into account in the CMT. Moreover, the background transmission coefficient was calculated merely approximately and assumed frequency-independent. Nevertheless, the CMT provides a good qualitative description of the considered process.

According to the simulated results in Fig. 5d, the highest reduction ratio of *R* = 6.65 is achieved at *ω* = 0.6094 × 2*π**c*/*a*. We have also tested the operation of the spaceplate when illuminated by a 2D Gaussian beam via full-wave simulations and obtained a similar reduction ratio (see Section 4.1 of the Supplementary Information). For this frequency, the NA reaches as high as 0.32 (\({\theta }_{\max }=18.{5}^{\circ \,}\)) (see Section 4 of the Supplementary Information). Such a high value of NA became possible due to the slightly shifting from Huygens’ condition (see Fig. 4). As shown in Supplementary Fig. S1 of the Suppemental Information, beyond the operational NA, the PCS phase behavior starts to differ from that of free space, while the transmission remains relatively high. Moreover, for the same spaceplate geometry but at a frequency of *ω* = 0.6262 × 2*π**c*/*a*, we achieve NA = 0.54 \(({\theta }_{\max }=3{3}^{\circ } \,)\) with slightly reduced *R* = 4.89 (Supplementary Fig. S1 in the Supplementary Information). This result is far better than that in known single-resonance spaceplates with similar values of *R*, including those based on multilayer structures^{1} (NA = 0.29) or Fabry-Perot resonators^{24} (NA = 0.33). Higher NA are important for many realistic applications, such as cameras in smartphones or professional imaging equipment^{34}.

It is important to make a comparison of the presented spaceplate with the previous structures based on the performance factor derived in ref. ^{24}. This factor, expressed as PF = (NA)^{2}*d*_{eff}/*λ*, has a theoretical bound, rendering a trade-off between achievable NA and the reduction ratio. In particular, for single-resonance spaceplates it is limited by PF < 1^{24}, while for the double-resonance scenario, PF ≤ 2^{32}. In our case, due to the small width of the spaceplate *d*_{sp}, PF = 0.11 for the frequency *ω* = 0.6094 × 2*π**c*/*a* where *R* is maximized. At the frequency *ω* = 0.6262 × 2*π**c*/*a* where the NA is maximized, the factor reaches PF = 0.24. On the one hand, these values are much higher than those obtained in single-resonance spaceplates based on photonic crystal slabs, where PF < 0.01^{2}. On the other hand, these factors are below the values obtained in Fabry-Perot spaceplates (PF ≈ 0.5^{24}). However, in contrast to them, our spaceplate supports broadband transparency both in frequency and angles of incidence, which, combined with the high NA, makes our design more suitable for applications where larger bandwidths are preferable.

## Discussion

Using the CMT, we have demonstrated that Huygens’ spaceplates supporting two resonance modes with opposite symmetries provide important significant enhancement of several characteristics compared to single-resonance counterparts: operational bandwidth, reduction ratio, and NA. The first proof-of-principle design confirms the predicted performance. It was shown that departing from Huygens’ condition for the two resonances in the spaceplate provides additional advantages, such as the possibility of reducing chromatic aberrations.

Another important advantage of Huygens’ spaceplates is their transparency in a wide range of angles and frequencies. In this aspect, they are similar to Huygens’ metasurfaces^{35,36}. This feature allows one to straightforwardly cascade several different Huygens’ spaceplates one after another. Through this cascading, it would be possible to either further extend the operational range of the spaceplate or even to create multi-band spaceplates.

It is worth mentioning that the results obtained in this paper can be extended to any frequency, especially in the optical domain. To this aim, we proposed an example of a silicon-based photonic crystal slab that can operate as a spaceplate in the near-infrared region. We suggest using crystalline silicon to have a lossless and mechanically robust membrane, similar to the previously fabricated and characterized membrane in ref. ^{29}. Moreover, the proposed spaceplate can be easily redesigned for the case when it is located on a substrate with permittivity *ϵ*_{BG} ≠ 1. To eliminate the wave impedance contrast of the materials below and above the spaceplate which could lead to unwanted reflections, one can use the matching layer as proposed in ref. ^{37}. In this case, the spaceplate is covered with a thin layer of material with the same refractive index as the substrate covered additionally by a layer with high permittivity and a specific thickness. The proposed structure is monolayer, leading to the simplicity of its fabrication. It could be fabricated in integrated CMOS-compatible systems. The high NA and straightforward integration possibilities pave the way for the proposed spaceplates to be used in real-world applications, from reducing invasive medical devices to cameras in professional environments or smartphones.

## Methods

### Analytical development

Along with this work, the analytical development of the CMT has been performed (see Section 1 of the Supplementary Information for the complete development). The authors have used Wolfram Mathematica to support the performed development. The numerical results presented in this work have been calculated and presented via Matlab software.

### Full-wave simulations

The eigenmode analysis was performed in COMSOL Multiphysics. Periodical boundary conditions were applied on the in-plane faces of the structure in conjunction with the Floquet condition along the *x*-axis to obtain the Γ − X band diagram. Scattering boundary conditions were applied in out-of-plane faces. The system is, therefore, open, and eigenfrequencies have non-zero imaginary parts, even in the absence of material losses.

To study the behavior of the proposed spaceplate (transmission coefficient), the numerical simulations of the photonic crystal slabs were performed using the electromagnetic wave frequency domain (emw) solver in COMSOL Multiphysics. Two periodic boundary conditions were assigned, one in the *x*-direction and another along the *y*-direction, with the Floquet ports defined as normal to *z-*axis. The angle of incidence *θ* has been swept in the *x**z*-plane, with the electric field polarized along the *y*-direction. The study for different values of *k*_{x}, *k*_{y}, and both TE-TM polarizations can be seen in Section 4.2 of the Supplementary Information. The diffraction orders were evanescent for the entire range of considered incident angles. We used Matlab to fit the phase behavior of the photonic crystal slab with the empty space of an effective distance *d*_{eff} = *R**d*_{SP} with the tolerance of 2% (see Section 4 of the Supplementary Information).

## Data availability

Data used in this study are available from the corresponding author upon request.

## References

Reshef, O. et al. An optic to replace space and its application towards ultra-thin imaging systems.

*Nat. Commun.***12**, 3512 (2021).Guo, C., Wang, H. & Fan, S. Squeeze free space with nonlocal flat optics.

*Optica***7**, 1133 (2020).Shastri, K., Reshef, O., Boyd, R. W., Lundeen, J. S. & Monticone, F. To what extent can space be compressed? Bandwidth limits of spaceplates.

*Optica***9**, 738 (2022).Overvig, A. & Alù, A. Diffractive nonlocal metasurfaces.

*Laser Photonics Rev.***16**, 2100633 (2022).Kolkowski, R., Hakala, T. K., Shevchenko, A. & Huttunen, M. J. Nonlinear nonlocal metasurfaces.

*Appl. Phys. Lett.***122**, 160502 (2023).Liu, W. et al. Imaging with an ultrathin reciprocal lens.

*Phys. Rev. X***13**, 031039 (2023).Long, O. Y., Guo, C. & Fan, S. Topological nature of non-Hermitian degenerate bands in structural parameter space.

*Phys. Rev. Appl.***20**, l051001 (2023).Pan, M. et al. Dielectric metalens for miniaturized imaging systems: Progress and challenges.

*Light.: Sci. Appl.***11**, 195 (2022).Miller, D. A. B. Why optics needs thickness.

*Science***379**, 41–45 (2023).Chen, H.-T., Taylor, A. J. & Yu, N. A review of metasurfaces: Physics and applications.

*Rep. Prog. Phys.***79**, 076401 (2016).Lalanne, P. & Chavel, P. Metalenses at visible wavelengths: Past, present, perspectives.

*Laser Photonics Rev.***11**, 1600295 (2017).Khorasaninejad, M. et al. Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging.

*Science***352**, 1190–1194 (2016).Banerji, S. et al. Imaging with flat optics: metalenses or diffractive lenses?

*Optica***6**, 805 (2019).Yu, N. & Capasso, F. Flat optics with designer metasurfaces.

*Nat. Mater.***13**, 139–150 (2014).Monticone, F., Estakhri, N. M. & Alù, A. Full control of nanoscale optical transmission with a composite metascreen.

*Phys. Rev. Lett.***110**, 203903 (2013).Khorasaninejad, M. & Capasso, F. Metalenses: Versatile multifunctional photonic components.

*Science***358**, eaam8100 (2017).Arbabi, A., Horie, Y., Bagheri, M. & Faraon, A. Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission.

*Nat. Nanotechnol.***10**, 937–943 (2015).Zhou, Y., Guo, S., Overvig, A. C. & Alù, A. Multiresonant nonlocal metasurfaces.

*Nano Lett.***23**, 6768–6775 (2023).Long, O. Y., Guo, C., Jin, W. & Fan, S. Polarization-independent isotropic nonlocal metasurfaces with wavelength-controlled functionality.

*Phys. Rev. Appl.***17**, 024029 (2022).Pagé, J. T. R., Reshef, O., Boyd, R. W. & Lundeen, J. S. Designing high-performance propagation-compressing spaceplates using thin-film multilayer stacks.

*Opt. Express***30**, 2197 (2022).Pahlevaninezhad, M. & Monticone, F. Multi-color spaceplates in the visible. Preprint at https://arxiv.org/abs/2312.02378v1 (2023).

Sorensen, N. J., Weil, M. T. & Lundeen, J. S. Large-scale optical compression of free-space using an experimental three-lens spaceplate.

*Opt. Express***31**, 19766 (2023).Shao, Y. et al. Multifunctional spaceplates for optical aberration correction.

*ACS Photonics***11**, 1753-1760 (2024).Chen, A. & Monticone, F. Dielectric nonlocal metasurfaces for fully solid-state ultrathin optical systems.

*ACS Photonics***8**, 1439–1447 (2021).Mrnka, M. et al. Space squeezing optics: Performance limits and implementation at microwave frequencies.

*APL Photonics***7**, 076105 (2022).Mrnka, M., Hooper, I. R., Penketh, H., Phillips, D. B. & Hendry, E. A dual-band spaceplate: Contracting the volume of quasi-optical systems.

*IEEE Trans. Microw. Theory Tech.***72**, 3279-3287 (2024).Decker, M. et al. High-efficiency dielectric huygens’ surfaces.

*Adv. Opt. Mater.***3**, 813–820 (2015).Epstein, A. & Eleftheriades, G. V. Huygens’ metasurfaces via the equivalence principle: design and applications.

*J. Opt. Soc. Am. B***33**, A31 (2016).Yang, Q. et al. Mie-resonant membrane huygens’ metasurfaces.

*Adv. Funct. Mater.***30**, 1906851 (2019).Pfeiffer, C. & Grbic, A. Metamaterial huygens’ surfaces: Tailoring wave fronts with reflectionless sheets.

*Phys. Rev. Lett.***110**, 197401 (2013).Suh, W., Wang, Z. & Fan, S. Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities.

*IEEE J. Quantum Electron.***40**, 1511–1518 (2004).Dugan, J., Smy, T. J., Monticone, F. & Gupta, S. Surface susceptibility synthesis of spatially dispersive metasurfaces for space compression and spatial signal processing. Preprint at https://doi.org/10.36227/techrxiv.24174513.v1 (2023)

Yu, H. et al. Dispersion engineering of metalenses.

*Appl. Phys. Lett.***123**, 240503 (2023).DelMastro, M. Spaceplates: The final frontier in compressing optical systems.

*MSc degree Thesis*at https://doi.org/10.20381/ruor-27334 (2022).Shaham, A. & Epstein, A. Generalized Huygens’ condition as the fulcrum of planar nonlocal omnidirectional transparency: from meta-atoms to metasurfaces. Preprint at https://arxiv.org/abs/2309.07294v1 (2023).

Ra’di, Y. & Tretyakov, S. A. Angularly-independent huygens’ metasurfaces. In

*Proc. Int. Symp. Antennas Propag. USNC-URSI Natl. Radio Sci. Meet.*874-875 (2015)Bai, H., Shevchenko, A. & Kolkowski, R. Recovery of topologically robust merging bound states in the continuum in photonic structures with broken symmetry. Preprint at https://arxiv.org/abs/2401.16105v1 (2024).

## Acknowledgements

The authors would like to thank Prof. Sergei Tretyakov, Prof. Constantin Simovski, and Dr. Francisco S. Cuesta for the fruitful discussions. F.J.D.-F. acknowledges the Next Generation EU program, Spanish National Research Council (Ayuda Margarita Salas), and Universitat Politècnica de València (PAID-06-23). L.M.M-E. acknowledges Universitat Politècnica de València (PAID-01-23). A.D.-R. acknowledges the Beatriz Galindo excellence grant (grant No. BG-00024) and Generalitat Valenciana PROMETEO Program (CIPROM/2022/14). V.A. acknowledges the Research Council of Finland (Project No. 356797), the Finnish Foundation for Technology Promotion, and Research Council of Finland Flagship Programme, Photonics Research and Innovation (PREIN), decision number 346529, Aalto University.

## Author information

### Authors and Affiliations

### Contributions

A.D.-R. and V.A. conceived the idea. F.J.D.-F., L.M.M-E., and V.A. carried out the theoretical development. F.J.D-F. performed the numerical results, designed and carried out the simulations of the implemented design. L.M.M-E. performed the eigenvalues study. F.J.D-F., A.D-R., and V.A. co-wrote the paper. V.A. supervised the project.

### Corresponding author

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Additional information

**Publisher’s note** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Supplementary information

## Rights and permissions

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## About this article

### Cite this article

Díaz-Fernández, F.J., Máñez-Espina, L.M., Díaz-Rubio, A. *et al.* Broadband transparent Huygens' spaceplates.
*npj Nanophoton.* **1**, 30 (2024). https://doi.org/10.1038/s44310-024-00025-6

Received:

Accepted:

Published:

DOI: https://doi.org/10.1038/s44310-024-00025-6