1. INTRODUCTION

Optical systems, such as cameras, microscopes, and telescopes, typically have three major components: lenses, detectors, and free space. Metalenses [1–6] and diffractive lenses [7–11] have the potential to replace conventional refractive lenses (within certain limits [12–14]) and, therefore, owing to their light weight and small thickness, contribute to the miniaturization of a multitude of optical systems. However, in this quest toward miniaturization, an often overlooked aspect of optical systems is the large free-space volume between the detector and the lens, or between lenses, which is essential to allow light to acquire a distance-dependent and angle-dependent phase and achieve, for example, focusing at a certain distance. These free-space volumes can dramatically increase the overall length of the system and also determine an increase in weight due to the larger required housing (e.g., lens barrel), thereby rendering the miniaturization of complex optical setups a challenge.

An innovative solution to this problem was recently proposed by Reshef et al. [15] by showing that a low-index isotropic or uniaxial slab, or a nonlocal, i.e., angle-dependent, (meta)material, can implement an optical transfer function approximating that of a volume of free space (or the selected background material) over a smaller distance. Such a device, named a “spaceplate,” can help shorten or even, in theory, fully replace the empty spaces in optical systems by effectively “compressing” space for light propagation. Several of us recently experimentally implemented and tested two broadband proof-of-concept spaceplates with a frequency window spanning the entire visible range, a first one, using a slab made of a uniaxial crystal, and a second one, using a glass cell containing air, both submerged in a higher-index medium (oil). The compression ratio implemented by these homogeneous material spaceplates, defined as the length of replaced free space (effective length of light propagation) over the actual spaceplate thickness, was, however, limited to less than 1.5. In contrast, in another pioneering work, Guo et al. [16] proposed a nonlocal spaceplate design leveraging the dispersion of a guided-mode resonance in a photonic crystal slab to implement the required transfer function, achieving a much larger compression ratio of 144. The sharp linewidth of the guided-mode resonance, however, limited the fractional spectral bandwidth of the device to ${10^{- 4}}$ and the numerical aperture (NA) to 0.01. In the same work, the authors proposed another photonic-crystal-based spaceplate with a moderately larger NA of 0.11; however, this was obtained at the cost of the compression ratio, which was reduced to 11.2.

These early designs and demonstrations suggest that the operating bandwidth, compression ratio, and NA of spaceplates cannot be improved independently but are likely to be constrained by physical bounds and trade-offs. This was further corroborated by the results of [17], where three of the present authors observed a distinct trade-off between the compression ratio and NA in monochromatic multilayered spaceplates designed using inverse design techniques, confirming earlier theoretical predictions by Chen and Monticone in [18]. Within this context, in this paper, we first elucidate the basic physical mechanisms at the origin of generic space-compression effects, and we then derive physical bounds and trade-offs on the performance of spaceplates, with a particular focus on the issue of bandwidth, which has not yet been addressed in the literature and remains as a crucial roadblock for the adoption of this platform. Specifically, starting from well-established limits on the delay-bandwidth product of linear time-invariant systems, we derive general fundamental bounds on the maximum achievable bandwidth of spaceplates, as a function of their compression ratio and NA. We then apply these theoretical results to answer important questions on the performance of ideal spaceplates, such as how optimal different reported spaceplate designs are, how to improve them, how the bandwidth limits of spaceplates and metalenses compare, and how high the maximum compression ratio can be for an ideal spaceplate targeting the entire visible range.

 

Fig. 1. Comparison of a light beam converging at $F^\prime $ in vacuum (or a background material) and at $F$ in the presence of an ideal spaceplate of length $L$. The spaceplate can effectively replace a region of space of length ${L_{{\rm{eff}}}}$ by imparting an angle-dependent transverse shift, $\Delta s$, to propagating plane waves.

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2. PHYSICAL MECHANISM AND FUNDAMENTAL LIMITS

Consider a light beam, perhaps shaped by a (meta)lens, that is converging/focusing at a distance $F^\prime $ in vacuum (or a background material) and at $F$ in the presence of an ideal spaceplate, as illustrated in Fig. 1. Intuitively, to mimic light traversing a distance ${L_{{\rm{eff}}}}$ in a smaller space of length $L$ without distortions, obliquely incident plane waves should bend away from the surface normal inside the spaceplate, with a deflection angle dependent on the incident angle, and then re-emerge from the spaceplate propagating in the original direction. Crucially, to really mimic free space, the spaceplate in Fig. 1 should not reduce the focal distance through a “lensing” mechanism, i.e., deflecting the incident wave due to a position-dependent phase profile. Instead, the device should be transversely homogeneous, not adding any focusing power, but simply implementing the phase transfer function of free space over a shorter distance $L$, i.e.,

Starting from Eq. (2) for $\Delta s$, we can easily find a formula for the excess time delay that is required to implement a desired transverse displacement relative to propagation in the background,

If ${\alpha _m}$ is the maximum angular range of the spaceplate, an ideal spaceplate must be able to impart a time delay given by Eq. (3) for a lateral displacement $\Delta {s_m} = ({L_{{\rm{eff}}}} - L)\tan ({\alpha _m})$, which is the maximum displacement realized by the space-compression device. The fact that, in a spaceplate, maximum $\Delta T$ is required at the maximum angle of incidence can be seen by differentiating Eq. (3) with respect to $\alpha$ [with $\Delta s = ({L_{{\rm{eff}}}} - L)\tan (\alpha)$ assumed positive]. More intuitively, the difference between the required displacement for space compression, which scales as $\tan (\alpha)$, and the displacement obtained through a possible increase in ${v_{\textit{gx}}}$ increases with angle [the second term in the numerator of Eq. (3) is indeed the displacement obtained through a change of ${v_{\textit{gx}}}$ alone]. This difference has to be compensated by an excess time delay, which, therefore, needs to increase with angle. Using the definition of compression ratio, $R = {L_{{\rm{eff}}}}/L$, and of numerical aperture, ${\rm{NA}} = {n_b}\sin ({\alpha _m})$, the required spaceplate group delay can be written as

 

Fig. 2. Bandwidth bounds compared with published spaceplate designs. The lines are the fractional bandwidth bounds derived in the main text (suitably normalized to allow for comparison between different designs) with dashed lines representing the loosest form of the bound with ${v_{\textit{gx}}} \to {c_0}$, whereas the corresponding solid lines are for a tighter version of the bound assuming ${v_{\textit{gx}}} = c\sin ({\alpha _m}) = c{\rm{NA}}/{n_b}$ (details in the main text). The dots in all plots represent the published designs given in Table 1, color-coded to match their respective bound curve. The bounds in (a) and (b) are calculated using the delay-bandwidth product for a single-mode resonator (applicable to spaceplates based on a single guided-mode resonance) and (c), (d) using the delay-bandwidth product of transversely invariant structures, corresponding to Eq. (7). (a), (c) Bandwidth bounds plotted as a function of the compression ratio $R$, for different values of the numerical aperture (NA) and background refractive index ${n_b}$ chosen on the basis of published designs. The number in parentheses next to each dot is the corresponding NA reported. (b), (d) Bandwidth bounds plotted as a function of NA for different values of $R$ and ${n_b}$ chosen on the basis of published designs. The reported compression ratio for each published design is written in parentheses. If not specified, the background material is free space. Note that in panel (d) only one dashed curve is plotted since, for the other two cases (corresponding to designs ${b_2}$ and ${b_3}$), the loosest bound is actually infinite as the reported parameters satisfy $R \cdot {\rm{NA}} \lt {n_b}{v_{\textit{gx}}}/c = n_b^2$ for any value of NA (hence, the space-compression effect can be achieved with no excess time delay; see discussion in the main text).

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3. SPECIFIC CASES

The general bandwidth limit in Eq. (6) can be specialized for any class of structures for which an upper bound $\kappa$ on the delay-bandwidth product is known. In the following, we consider two main types of spaceplates, based on different structures for which different expressions for $\kappa$ should be used, leading to different bandwidth limits. The first class of spaceplates represents those based on structures supporting a single guided-mode resonance, such as the photonic-crystal-based spaceplate in [16,19] or the spaceplates based on a single Fabry–Perot cavity in [18,24]. In this case, assuming that only one resonance is used (as done in [16,24] and the first part of [18]), we can apply the delay-bandwidth product for a single-mode resonator, $\kappa = 2$ (valid for any individual single-mode resonator, reciprocal or nonreciprocal [25]) to calculate the maximum achievable fractional bandwidth from Eq. (6). The upper bound is plotted in Figs. 2(a) and 2(b) and is compared with the spaceplate designs reported in the literature. As mentioned above, it is apparent from these plots that the bandwidth bound is lower for spaceplates with a higher NA or compression ratio. It can also be seen that the guided-mode resonance-based designs from [16], marked as ${c_1}$ and ${c_2}$, obey the bound and, moreover, there is still significant room for improvement especially if a design could be found that not only imparts a delay, but also somewhat increases ${v_{\textit{gx}}}$. The microwave design in [24], marked as $d$, is the closest to the tighter bound (solid line), albeit for a lower compression ratio, which is not surprising since this spaceplate structure is mostly empty, except for thin metallic metamaterial mirrors; hence, the average ${v_{\textit{gx}}}$ in the spaceplate is close to its free-space value. In addition, we note that, even at a single frequency, these resonator-based designs are constrained by other fundamental trade-offs between the NA and the length of compressed space, as demonstrated in [18].

The bandwidth bounds in Figs. 2(a) and 2(b) only apply to structures where the space-compression effect is based on a single resonance at any angle. Better performance can be achieved by including more resonators within the spaceplate thickness, increasing the number of available modes in the bandwidth of operation, as commonly done for slow-light devices [26]. A more general limit to the delay-bandwidth product, valid for any linear time-invariant one-dimensional structure, was derived by Miller in [23,27] (hereafter referred to as Miller’s limit), which is fundamentally based on the idea that the product scales with the number of modes. In this case, the upper bound is given by $\kappa = \frac{\pi}{{\sqrt 3}}\frac{L}{{{\lambda _c}}}{\eta _{{\max}}}$, where ${{\eta }_{\max }}=|({{\epsilon }_{\max }}-{{\epsilon }_{\min }})/{{\epsilon }_{b}}|$ is the maximum permittivity contrast, with ${{\epsilon }_{\max }}$ and ${{\epsilon }_{\min}}$ defined as the maximum and minimum permittivity at any point within the structure and at any frequency within the considered bandwidth [23], and ${{\epsilon }_{b}}=\sqrt n_{b}$ is the relative permittivity of the background. Miller’s limit, however, was derived for one-dimensional structures, while spaceplates are clearly not one-dimensional devices. We circumvent this problem by noting that most spaceplate designs reported in the literature are transversely invariant, such as the homogeneous or multilayered structures in [15,17,18]. In this case, the problem is equivalent to a one-dimensional one, as the transverse wavenumber ${k_x}$ is invariant in such a structure and, for transverse-electric (TE) or transverse-magnetic (TM) incident plane waves, a one-dimensional effective wave equation can be derived that does not depend on the transverse direction (see, e.g., [28,29]).

Table 1. List of Various Spaceplate Designs Reported in the Literature, Their Parameters, and Corresponding Bounds^a

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Since spaceplates based on multilayer structures have a large number of controllable variables such as layer thicknesses and refractive indices, they have the potential to access a wider range of spaceplate parameters. For example, it can be seen in Figs. 2(b) and 2(d) that the multilayer structures denoted as $a$ [18] and ${b_1}$ [15] achieve a good combination of relatively high NA, compression ratio, and bandwidth. At the same time, it is clear that there is still room for improvement, consistent with the fact that these early designs were not fully optimized. We anticipate that the inverse design techniques that have been recently developed to design high-performance multilayered spaceplates at a single frequency [17] may be extended to also widen the operating bandwidth. Moreover, since the effective transverse group velocity in dielectric stacks was shown in [22] to depend on a weighted average of the refractive indices in the structure, we speculate that an ideal spaceplate design would utilize the highest possible refractive index to increase the delay-bandwidth product limit (which depends on the maximum permittivity contrast anywhere within the structure), while confining the high-index material in limited spatial regions (e.g., thin layers) such that the effective transverse group velocity would not decrease too much compared to free space. These qualitative criteria were satisfied for the spaceplates designed in [18], which indeed show the highest fractional bandwidth of all the existing multilayer designs (see Table 1). Other more exotic options could perhaps involve combining both near-zero-index materials and dielectrics to increase the maximum excess time delay and transverse velocity simultaneously.

Finally, for applications in optical imaging, it is interesting to assess the ultimate performance limits of spaceplates that are designed to function over the entire visible range. We do this by plotting in Fig. 3 the bandwidth bound for transversely homogeneous spaceplates, given by Eq. (7), as a function of compression ratio and ${\eta _{{\max}}}$, comparing it to the fractional bandwidth of the entire visible range. We assumed ${v_{\textit{gx}}} = c\sin ({\alpha _m}) = c{\rm{NA}}/{n_b}$; namely, the transverse group velocity in the spaceplate is assumed not to exceed its value in the background material, leading to a NA-independent bandwidth bound. As an example, a low-contrast spaceplate with ${\eta _{{\max}}} = 3$ can achieve a frequency-constant compression ratio no greater than $R \approx 2$ over the entire visible spectrum, whereas a high-contrast spaceplate with ${\eta _{{\max}}} = 15$ can achieve a compression ratio of $\approx 8$ (we note that ${\eta _{{\max}}} = 15$ is arguably the largest contrast that can be achieved at visible frequencies with natural materials [31]). These results demonstrate that, at least in theory, spaceplates with good performance may indeed cover the entire visible range. Even just a twofold reduction in the length of an optical system (e.g., a camera) would be highly beneficial for many real-world applications. We emphasize, however, that there is no guarantee that the derived bounds are achievable by a physical, good-quality spaceplate, as these bounds have been determined from the maximum required transverse shift for the maximum angle of incidence, and not for the specific angle-shift relationship of an ideal spaceplate. In other words, the bandwidth bound may be achievable by spaceplates with strong monochromatic aberrations (i.e., with a distorted angle-shift relationship) as long as they implement the same maximum transverse shift at the maximum angle as an ideal spaceplate. Tighter bounds might exist for the specific angle-shift relationship of an ideal spaceplate.

 

Fig. 3. Fractional bandwidth bound of a transversely homogeneous spaceplate as a function of its compression ratio $R$ and maximum permittivity contrast ${\eta _{{\max}}}$, compared to the fractional bandwidth of the entire visible range (flat plane). The bound is calculated using Miller’s delay-bandwidth product limit given by Eq. (7) [for the case with ${v_{\textit{gx}}} = c\sin ({\alpha _m}) = c{\rm{NA}}/{n_b}$] and with background refractive index set to unity. The optimal trade-off between maximum permittivity contrast and compression ratio for a fixed bandwidth is indicated by the solid lines (level curves), with the thicker line and flat plane corresponding to the fractional bandwidth of the visible frequency range: $\Delta \omega /{\omega _c} = 0.65$.

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Fig. 4. Comparison of bandwidth limits for metalenses and spaceplates. (a) Illustration of a flat optic focusing/imaging system consisting of a dispersion-engineered achromatic metalens followed by a spaceplate shifting the focal plane closer to the lens. (b) Normalized bandwidth bounds for a dispersion-engineered metalens with $F/{L_m} = 100$ for different values of NA [horizontal lines, Eq. (8)] and a transversely invariant spaceplate with compression ratio $R$. The spaceplate bound is calculated using Eq. (7), for the case with ${v_{\textit{gx}}} = c\sin ({\alpha _m}) = c{\rm{NA}}/{n_b}$, leading to a NA-independent bandwidth bound. The devices are assumed to be in free space (${n_b} = 1$) and to have the same maximum permittivity contrast ${\eta _{{\max}}}$. Crossing points between the bounds are denoted by black circles; for larger values of $R$, the spaceplate is the limiting factor to the bandwidth of the combined system.

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4. CASCADED METALENS AND SPACEPLATE

Cascading metalenses, spaceplates, and sensors can result in ultra-thin imaging platforms that are completely planar, ultra-compact, and potentially monolithic, which is one of the main motivations behind research on spaceplates and nonlocal flat optics [15,16,18]. In this context, it is interesting and practically important to study whether the bandwidth would be limited by the spaceplate or by the flat lens itself. This is particularly relevant since one of the most popular classes of broadband flat lenses, namely, achromatic dispersion-engineered metalenses, is constrained by bandwidth limits analogous to the ones studied here, as their operating mechanism is based on implementing, in addition to a standard phase profile, a certain group delay profile to achieve dispersion compensation over a broad bandwidth [4,5]. Bandwidth limits on achromatic metalenses, also originating from delay-bandwidth constraints, have been derived by Presutti and Monticone in [12], where it was shown that the upper bandwidth bound is given by

By comparing Eqs. (8) and (7), we see that the bandwidth of a hybrid system composed of a metalens followed by a spaceplate, as in Fig. 4(a), is limited by the space-compression effect if the compression ratio is sufficiently large. This is further confirmed in Fig. 4(b), where we compare the two bounds, as functions of $R$ and for different values of NA, assuming $F/{L_m} = 100$ and the same maximum permittivity contrast for both devices. We see that, for this example, the crossing point is for compression ratios ranging from values near unity to around 100, depending on the NA, above which, even in the most optimistic scenario, the spaceplate becomes the limiting factor with respect to bandwidth. These results also imply that, even if flat lenses with broader bandwidths are employed, such as multilevel diffractive lenses [32,33], the bandwidth of the combined system cannot improve if large compression ratios are desired.

5. CONCLUSION

In this paper, we have derived, for the first time, fundamental bandwidth limits on spaceplates, novel optical components that implement the optical response of free space over a shorter length. Using relevant upper bounds on the delay-bandwidth product of linear time-invariant structures, we have found that the space-compression effect is fundamentally limited in bandwidth if the necessary transverse spatial shift of a light beam propagating through the spaceplate requires the beam to be temporally delayed, with respect to propagation in the background medium, by relying on resonances. In particular, we have shown that the achievable bandwidth necessarily decreases with an increase in the compression ratio. Importantly, however, we have found that the existing spaceplate designs are still relatively far from their performance limits and that, at least in theory, compression ratios on the order of 10 may be physically achievable over the entire visible range, using spaceplates made of realistic dielectric materials (with the caveat that there is no guarantee the derived bounds are tight). We also stress that our derived bounds indicate over what bandwidth a certain frequency-constant compression ratio can be achieved, corresponding to an achromatic space-compression effect. If a frequency-dependent compression ratio can be tolerated, with a lower compression ratio at some frequencies, broader bandwidths may be possible. In other words, if a bandwidth wider than the one predicted by our bounds is desired, the compression ratio cannot stay constant within that band. Moreover, while the compression ratio is constrained by bandwidth, the absolute length of compressed free space ${L_{{\rm{eff}}}} = R \cdot L$ can be made arbitrarily large, at least in theory, by cascading a large number of ideal spaceplates (i.e., with ideal transmission efficiency), as also discussed in [18]. While our results already apply to broad classes of spaceplates, we believe that future investigations should focus on identifying bandwidth bounds on more complex spaceplate devices, not necessarily transversely homogeneous or periodic, perhaps designed using inverse-design techniques applied to thick, volumetric, planar structures merging local metalenses and nonlocal spaceplates within the same volume [34]. Moreover, a better understanding of the fundamental limits to the transmission efficiency of spaceplates, and of the trade-offs between efficiency, bandwidth, and other metrics, would also be important to assess the potential of spaceplates for practical applications.

In summary, our work sheds new light on the physics, limitations, and potential of space-compression effects in optics and electromagnetics (some of these results and considerations may also be extended to other areas of wave physics). We believe these findings will help guide the design of new spaceplates with better performance, which may pave the way for the realization of ultra-compact, monolithic, planar optical systems for a variety of applications.