# To What Extent Can Space Be Compressed? Bandwidth Limits of Spaceplates

Kunal Shastri1  Orad Reshef2  Robert W. Boyd2,3,4  Jeff S. Lundeen2  Francesco
Monticone  1  1School of Electrical and Computer Engineering, Cornell
University, Ithaca, NY 14853, USA, 2Department of Physics, University of
Ottawa, Ottawa, ON, Canada, 3School of Electrical Engineering and Computer
Science, University of Ottawa, Ottawa, ON, Canada, 4Institute of Optics and
Department of Physics and Astronomy, University of Rochester, Rochester, NY,
USA  *<EMAIL_ADDRESS>
###### Abstract

Spaceplates are novel flat-optic devices that implement the optical response
of a free-space volume over a smaller length, effectively “compressing space”
for light propagation. Together with flat lenses such as metalenses or
diffractive lenses, spaceplates have the potential to enable a drastic
miniaturization of any free-space optical system. While the fundamental and
practical bounds on the performance metrics of flat lenses have been well
studied in recent years, a similar understanding of the ultimate limits of
spaceplates is lacking, especially regarding the issue of bandwidth, which
remains as a crucial roadblock for the adoption of this platform. In this
work, we derive fundamental bounds on the bandwidth of spaceplates as a
function of their numerical aperture and compression ratio (ratio by which the
free-space pathway is compressed). The general form of these bounds is
universal and can be applied and specialized for different broad classes of
space-compression devices, regardless of their particular implementation. Our
findings also offer relevant insights into the physical mechanism at the
origin of generic space-compression effects, and may guide the design of
higher performance spaceplates, opening new opportunities for ultra-compact,
monolithic, planar optical systems for a variety of applications.

††journal: osac††articletype: Research Article

##  1 Introduction

Optical systems, such as cameras, microscopes, telescopes, etc., typically
have three major components: lenses, detectors, and free space. Metalenses [1,
2, 3, 4, 5, 6] and diffractive lenses [7, 8, 9, 10, 11] have the potential to
replace conventional refractive lenses (within certain limits [12, 13, 14])
and therefore, owing to their light weight and small form factor, 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 volumes 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. While free space obviously does not add to the overall weight, it
can make dramatic contributions to the overall length of the system, 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, another pioneering work,
by 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 line-width 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 numerical aperture of spaceplates cannot be improved
independently but are likely to be constrained by physical bounds and
tradeoffs. This was further corroborated by the results of Ref. [17], where
the three of the present authors observed a distinct tradeoff between
compression ratio and NA in monochromatic multi-layered spaceplates designed
using inverse design techniques, confirming earlier theoretical predictions by
Chen and Monticone in Ref. [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 tradeoffs 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
numerical aperture. We then apply these theoretical results to answer
important questions on the performance of ideal spaceplates, such as how
optimal different spaceplate designs are, how to improve them, how the
bandwidth limits of spaceplates and metalenses compare, and how high the
maximum compression ratio and NA can be for an ideal spaceplate targeting the
entire visible range.

Figure 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
$L_{\textrm{eff}}$ by imparting an angle-dependent transverse shift, $\Delta
s$, to propagating plane waves.

##  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 $L_{\textrm{eff}}$
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
transfer function of free space over a shorter distance. As mentioned in the
Introduction, while this effect can be achieved with homogeneous media in a
higher-index background, a promising way to realize space compression in free
space is by using a suitable nonlocal structure, namely, a structure with an
angle-dependent (i.e., transverse-momentum dependent) but position-independent
response designed to match the angle-dependent phase response of free space.
Structures with a nonlocal response, a property also known as _spatial
dispersion_ – the spatial analogue of the concept of frequency dispersion –
can be implemented, for instance, in the form of photonic-crystal slabs [16,
19] or multilayer thin-film structures [15, 18, 20, 21].

Regardless of the implementation, it is clear that, as illustrated in Fig. 1,
a generic device for space compression needs to impart an angle-dependent
transverse shift to the incident waves, _relative to propagation in the
background_ , equal to $\Delta s=(L_{\textrm{eff}}-L)\tan(\alpha)$, where
$\alpha$ is the incident angle with respect to the surface normal.
(Interestingly, this is the dual process of a lens: whereas a lens is a
position-dependent device that changes the angle of a light ray, a spaceplate
is an angle-dependent device that changes the transverse spatial position of a
light ray). In general, the transverse shift $s$ that a wave undergoes as it
propagates through a material or structure is given by $s=v_{gx}\tau$ [20],
where $v_{gx}$ is the transverse group velocity and $\tau$ is the total time
delay (group delay) imparted by the material or structure, i.e., the
difference, assumed here independent of the transverse position $x_{0}$,
between the time the wave exits the structure at $(x,z)=(x_{0}+s,L)$ and the
time it entered at $(x_{0},0)$. (One could write $\tau=L/v_{gz}$, but it is
important to note that for non-homogeneous or non-periodic structures, the
group velocities $v_{gx}$ and $v_{gz}$ are effective quantities, representing
the total effect of the structure, but are not constant within the structure
itself [22]). Then, with reference to Fig. 1, the additional transverse shift
with respect to the natural transverse displacement experienced by a wave
propagating in the background medium can be written as $\Delta s=v_{gx}\tau-
v_{gx}^{0}\tau^{0}=v_{gx}\tau-c\sin(\alpha)\tau^{0}$, where
$\tau^{0}=L/(c\cos(\alpha))$ is the time delay experienced by the wave in the
background for the same distance $L$ along the $z$-axis, and $c=c_{0}/n_{b}$,
is the wave velocity in the background medium, with refractive index $n_{b}$
assumed dispersionless. Thus, it is clear that in order to create a transverse
displacement greater than the one in the background, and hence a space-
compression effect, there are only two possible options: (i) By increasing the
_transverse group velocity_ $v_{gx}$. This may be possible by simply re-
directing waves away from the surface normal, for example through refraction
if the refractive index of the spaceplate is less than in the surrounding
medium, or through diffraction effects in transversely inhomogenous
structures. The transverse group velocity $v_{gx}$ however can only be
increased at most to approach $c_{0}$ and, in most cases of interest, such as
dielectric stacks, the effective $v_{gx}$ is actually reduced, not increased,
as further discussed below. (ii) By increasing the total _time delay_ with
respect to propagation in the background, which implies that the spaceplate
should act as an angle-dependent slow-light device. However, for a fixed
length $L$ and other general properties of the structure, there are strict
bounds on the maximum “excess” time delay, $\Delta T=\tau-\tau^{0}$ , that the
structure can impart to a signal of certain bandwidth. Indeed, upper bounds on
the product of delay and bandwidth will be crucial to determine the ultimate
bandwidth limitations of a spaceplate.

Starting from the above equation 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,

$\Delta T=\frac{\Delta s+(\sin(\alpha)-v_{gx}/c)L\sec(\alpha)}{v_{gx}},$ (1)

which means that a non-zero positive delay $\Delta T$ may be needed to realize
a specified displacement $\Delta s$. The required $\Delta T$ may actually
vanish for a sufficiently large $v_{gx}$, meaning that an increase in
transverse velocity would be sufficient to achieve the desired displacement
and, hence, space-compression effect. In many cases, however, especially for
large $\alpha$ and small length $L$, it is clear that a positive $\Delta T$
would always be needed for a specified $\Delta s$ even for very large values
of $v_{gx}$. Also note that if the desired $\Delta s$ were zero, and
$v_{gx}>v_{gx}^{0}=c\sin(\alpha)$, Eq. (1) would correctly predict that a
negative $\Delta T$, i.e., a time advance, would be needed to achieve zero
displacement; however, this is not of interest for our purposes and,
therefore, for the spaceplate problem, we only consider positive or zero
values of $\Delta T$, that is, the required time delay should be written as
$\Delta T=\max[0,(\Delta s+(\sin(\alpha)-v_{gx}/c)L\sec(\alpha))/v_{gx}]$.

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. (1) for a lateral
displacement $\Delta s_{m}=(L_{\textrm{eff}}-L)\tan(\alpha_{m})$, which is the
maximum displacement realized by the space-compression device. Using the
definition of compression ratio, $R=L_{\textrm{eff}}/L$, and of numerical
aperture, $\textrm{NA}=n_{b}\sin(\alpha_{m})$, the required spaceplate group
delay can be written as $\Delta
T=L\max[(R\cdot\textrm{NA}/n_{b}-v_{gx}/c),0]/(v_{gx}\sqrt{1-(\textrm{NA}/n_{b})^{2}})$.
As mentioned above, whether this required $\Delta T$ can be implemented
depends on the desired operating bandwidth $\Delta\omega$ of the spaceplate,
namely, the maximum bandwidth of the fields the device is designed to interact
with. In particular, for any linear time-invariant system, there are
fundamental bounds on the delay-bandwidth product that can be achieved by the
system [23], which can generally be written in the form $\Delta
T\Delta\omega\leq\kappa$, where $\kappa$ depends on some general properties of
the structure, as further discussed below. Using the required $\Delta T$
derived above, a general expression for the bandwidth upper bound of any
spaceplate can therefore be written as

$\frac{\Delta\omega}{\omega_{c}}\leq\frac{1}{2\pi}\frac{\kappa}{L/\lambda_{c}}\frac{\sqrt{1-(\textrm{NA}/n_{b})^{2}}~{}v_{gx}/c}{\max[(R\cdot\textrm{NA}/n_{b}-v_{gx}/c),0]},$
(2)

where $\lambda_{c}$ is the center wavelength, in the background medium, and
$\omega_{c}=2\pi f_{c}$ is the corresponding center angular frequency. Eq. (2)
is a key result of this paper, as it represents the general form of the
bandwidth bound for any spaceplate, as a function of its numerical aperture
and compression ratio, and it provides relevant quantitative insight into how
easy or difficult it is to realize a space-compression effect. For example,
the bound predicts the intuitive result that the fractional bandwidth reduces
with increasing $R$ and goes to zero if $R\rightarrow+\infty$. The maximum
bandwidth also typically narrows with increasing NA, but the specific
dependence on NA requires knowledge of $\kappa$, which may depend on the angle
of incidence, as we will see in the following. Moreover, consistent with our
discussion above, whenever the factor $(R\cdot\textrm{NA}/n_{b}-v_{gx}/c)$
vanishes or becomes negative for a certain combination of parameters, it is an
indication that an increase in $v_{gx}$ with respect to the background-
propagation case is sufficient to realize a transverse displacement and a
space-compression effect without the need for the structure to impart an
excess time delay and, therefore, without any bandwidth limitation due to
delay-bandwidth constraints. This observation also offers important insight
towards the design of optimal spaceplates, specifically about whether a
structure acting as a slow-light device is useful or not. Indeed, it can
easily be verified that if a spaceplate needs to be designed with a
compression ratio and numerical aperture such that
$R\cdot\textrm{NA}>n_{b}^{2}$, then a design based purely on increasing
$v_{gx}$ (e.g., refraction in a low-index medium [15]) is insufficient to
achieve the desired space-compression effect, even in a scenario with
$v_{gx}\rightarrow c_{0}$. Instead, a structure imparting an excess time delay
is needed, which implies a finite bandwidth according to Eq. (2).

The case with the highest possible transverse velocity, $v_{gx}\rightarrow
c_{0}$, is interesting as it provides the most general (and loosest) form of a
bandwidth bound, valid for any structure regardless of their implementation.
However, in realistic structures, $v_{gx}$ is usually significantly lower,
angle-dependent, and it may be frequency dispersive. If better estimates of
$v_{gx}$ for specific classes of structures are known, this may be plugged
into Eq. (2) to produce tighter limits on the bandwidth. For example, in
dielectric multilayer thin-film structures, it was demonstrated in Ref. [22]
that $v_{gx}$ is approximately frequency-independent and can be estimated as
$v_{gx}=c\sin(\alpha)/n_{\textrm{eff}}^{2}$, where $n_{\textrm{eff}}^{2}$ is a
weighted average of the square of the refractive indices in the structure. In
the following, we consider and plot two different versions of the bandwidth
bound in Eq. (2): the upper bound for the most optimistic, albeit unrealistic,
scenario with $v_{gx}\rightarrow c_{0}$ (solid lines in Fig. 2), and a tighter
bound assuming that the transverse group velocity in the spaceplate does not
exceed $v_{gx}$ in the background material for the considered maximum
incidence angle, i.e., $v_{gx}=c\sin(\alpha_{m})$ (dashed lines in Fig. 2),
which is relevant for spaceplates made of dielectric slabs/films.

Figure 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 solid lines
representing the most general and loosest form of the bound with
$v_{gx}\rightarrow c_{0}$, and the corresponding dashed lines are for a
tighter version of the bound assuming
$v_{gx}=c\sin(\alpha_{m})=c\textrm{NA}/n_{b}$ (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 more general limit for the delay-bandwidth product of
transversely invariant structures, corresponding to Eq. (3). (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
the published design is the corresponding NA and $n_{b}$ 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 solid
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 $R\cdot\textrm{NA}<n_{b}^{2}$ for any value of NA (hence,
the space-compression effect can be achieved with no excess time delay; see
discussion in the main text).

##  3 Specific cases

The general bandwidth limit in Eq. (2) 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
are those based on structures supporting a guided-mode resonance, such as the
photonic crystal-based spaceplate in Ref. [16, 19] or the spaceplates based on
a single Fabry-Perot cavity in Refs. [18] and [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. (2). The upper bound is plotted in Figs. 2(a) and (b) (for both
scenarios described above, $v_{gx}\rightarrow c_{0}$ and
$v_{gx}=c\sin(\alpha_{m})$) and 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 $v_{gx}$. The
microwave design in [24], marked as $d$, is the closest to the tighter bound
(dashed 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 $v_{gx}$ 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 tradeoffs
between the numerical aperture and the length of compressed space, as
demonstrated in Ref. [18].

The bandwidth bounds in Figs. 2(a) and (b) only apply to structures where the
space-compression effect is based on a single resonance. 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. A more general limit to the delay-
bandwidth product, valid for any linear time-invariant one-dimensional
structure, was derived by Miller in [26, 23] (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 at any point within the structure and at any frequency
within the considered bandwidth [23]. 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 multi-layered structures in [18, 17, 15]. In this case, the problem is
formally 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., [27, 28]). This wave equation applies to the transverse component
of the fields, $U_{y}(x,z)=u(z)e^{-jk_{x}x}$, and, in the non-magnetic case,
can be written in the following general form:
$k_{z,b}^{-2}~{}\partial^{2}u(z)/\partial
z^{2}+u(z)=-\eta(z,\omega,\alpha)u(z)$, where $k_{z,b}$ is the longitudinal
wavenumber in the background medium. Miller’s limit can be derived for any
system that can be described by a wave equation of this form [26], but,
compared to the original case, here the background wavenumber $k_{b}$ is
replaced by the longitudinal wavenumber, $k_{z,b}=k_{b}\cos(\alpha)$ and the
permittivity-contrast factor becomes angle-dependent
$\eta(z,\omega,\alpha)=\eta(z,\omega)/\cos^{2}(\alpha)$. In addition, only for
the TE case we have $\eta(z,\omega)=(\epsilon(z)-\epsilon_{b})/\epsilon_{b}$
as in the original paper, whereas the TM case is significantly more
complicated as this factor also depends on spatial derivatives of the
permittivity distribution [28]. While it can easily be verified that the TM
case is approximately equal to the TE case if the permittivity does not change
too rapidly in space, this dependence on the spatial derivatives of
$\epsilon(z)$ suggests the intriguing possibility of potentially relaxed
bounds for the TM case, which will be the subject of future studies. Here,
instead, considering that ideal spaceplates should work for both
polarizations, we consider the more stringent TE case, for which Miller’s
limit can be rederived with the only difference being that the wavelength
$\lambda$ in the background medium is replaced by the longitudinal wavelength
$\lambda_{z}=2\pi/k_{z,b}=\lambda/\cos(\alpha)$ and the maximum permittivity
contrast becomes $\eta_{\max}(\alpha)=\eta_{\max}/\cos^{2}(\alpha)$. Finally,
the modified upper bound can be written as,
$\kappa=\frac{\pi}{\sqrt{3}}\frac{L}{\lambda_{c}}\eta_{\max}/\cos(\alpha)$,
which converges to the one-dimensional case for normal incidence and increases
with the angle, suggesting that larger time delays may be possible, for the
same bandwidth, at oblique incidence.

Substituting this modified $\kappa$ in Eq. (2), we obtain the following limit
on the fractional bandwidth of the space-compression effect,

$\frac{\Delta\omega}{\omega_{c}}\leq\frac{\eta_{\max}}{2\sqrt{3}}\frac{v_{gx}/c}{\max[(R\cdot\textrm{NA}/n_{b}-v_{gx}/c),0]},$
(3)

which is valid for any spaceplate based on a transversely invariant structure,
such as homogeneous, multilayer, or graded-index slabs, nonlocal metamaterials
composed of thin-film stacks, etc. The bound is plotted in Figs. 2(c) and (d)
(again for both $v_{gx}\rightarrow c_{0}$ and $v_{gx}=c\sin(\alpha_{m})$). As
in the previous case, the maximum achievable fractional bandwidth decreases
significantly for spaceplates with larger compression ratios, whereas the
dependence on NA is significantly reduced (and completely removed in the case
with $v_{gx}=c\sin(\alpha_{m})$) because the requirement of a larger delay at
oblique incidence to achieve a space-compression effect is compensated by the
fact that the delay-bandwidth product limit is larger at oblique incidence for
this type of structures.

All spaceplate designs reported in the literature are below the appropriate
upper bounds. Interestingly, the two implementations with the broadest
bandwidth – the isotropic and uniaxial slabs in a higher-index background [15]
(denoted as $b_{2}$ and $b_{3}$ in the figure) – are permitted to operate over
such a broad bandwidth (one of them exceeding the dashed-line bound for the
case with no increase in $v_{gx}$) because their compression ratio and/or NA
are sufficiently small such that the spaceplate effect can be achieved even
_without_ an excess time delay, but purely through light refraction in the
low-index material. Indeed, if $v_{gx}$ increases sufficiently within the
spaceplate, and $R\cdot\textrm{NA}$ is small, then the bandwidth bound given
by Eq. (3) diverges, meaning that the spaceplate is not limited by delay-
bandwidth constraints. However, low-index refraction effects can be nearly
frequency-independent only if the low-index material forming the spaceplate is
air or an approximately dispersionless dielectric embedded in a higher-index
dielectric background. Instead, if the background medium is free space, a
spaceplate of this type necessitates a material with a refractive index lower
than unity, which is only possible over a limited bandwidth and with non-zero
frequency dispersion due to causality and passivity constraints on the
properties of physical passive materials [29]. In such a case, the spaceplate
bandwidth would be limited by the unavoidable dispersion of the low-index
material, which translates into a dispersive $v_{gx}$ and, in turn, a
frequency-dependent space-compression effect. Finally, as mentioned above, if
the desired $R\cdot\textrm{NA}>n_{b}^{2}$, which is the case for all other
designs reported in the literature (see also Table 1), then an excess time
delay is always necessary, and Eq. (3) always predicts a finite bandwidth
bound even if all materials are perfectly dispersionless.

Figure 3: Illustration of the fractional bandwidth bound of a transversely
homogeneous spaceplate as a function of its NA and compression ratio $R$,
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. (3) (for the case $v_{gx}\rightarrow c_{0}$, leading to the loosest form
of the bound) and with background refractive index set to unity. We considered
two values of the maximum permittivity contrast: (a) $\eta_{\max}=3$ (low-
contrast spaceplate) and (b) $\eta_{\max}=15$ (high-contrast spaceplate). The
optimal trade-off between the NA 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 range: $\Delta
f/f_{c}$=0.65.

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),(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 Ref. [22] to depend on a weighted
average of the refractive indices in the structure,
$v_{gx}=c\sin(\alpha)/n_{\textrm{eff}}^{2}$, 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 Ref. [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 time delay and $v_{gx}$ simultaneously.

Paper | L/$\lambda_{0}$ | $\eta_{\max}$ | $NA$ | R | Rmax(Miller) | Rmax($\kappa=2$) | $\frac{\Delta f}{f_{c}}$ |  $\frac{\Delta f}{f_{c}}\big{\rvert}_{max}$ (Miller) |  $\frac{\Delta f}{f_{c}}\big{\rvert}_{max}$ ($\kappa=2$)
---|---|---|---|---|---|---|---|---|---
$a$[18] | 9.00 | 14.00 | 0.33 | 5.00 | 81.83 | 1.67 | 0.05 | 1.01 | 8.35E-03
$b_{1}$[15](metamaterial) | 6.56 | 10.06 | 0.26 | 5.00 | 146.20 | 3.34 | 0.02 | 0.73 | 0.01
$b_{2}$[15](low index material) | 8.27E+03 | 0.54 | 0.38 | 1.48 | 1.29* | 1.00* | 0.55 | 0.33* | 7.75E-05*
$b_{3}$[15](uniaxial crystal) | 5.60E+04 | 0.25 | 0.85 | 1.12 | 1.13* | 1.00* | 0.55 | 0.62* | 3.98E-05*
$c_{1}$[16](small NA design) | 1.22 | 11.00 | 0.01 | 144.00 | 3.02E+04 | 2.49E+03 | 1.05E-04 | 0.02 | 1.82E-03
$c_{2}$[16](large NA design) | 0.16 | 11.00 | 0.11 | 11.20 | 3.02E+04 | 1.88E+04 | 1.05E-04 | 0.31 | 0.19
$d$[24] | 1.02 | — | 0.22 | 6.60 | — | 13.13 | 0.03 | — | 0.05

Table 1: List of various spaceplate designs reported in the literature and
their thickness (L/$\lambda_{c}$), maximum permittivity contrast
($\eta_{\max}$), numerical aperture (NA), compression ratio ($R$), fractional
bandwidth ($\Delta f/f_{c}$), and the corresponding maximum achievable
compression ratio (for the reported NA) and maximum achievable bandwidth (for
the reported $R$) calculated in this work. The limits are calculated using
both the delay-bandwidth product for a single-mode resonator ($\kappa=2$) and
the more general Miller’s limit for transversely homogenous structures. For
these calculations, we assumed $v_{gx}=c\sin(\alpha_{m})=c\textrm{NA}/n_{b}$,
namely, the transverse group velocity in the spaceplate is assumed not to
exceed its value in the background material. This is true for all spaceplate
designs reported in the literature, except the ones denoted with $b_{2}$ and
$b_{3}$, which indeed approach or exceed this bound (numbers with an asterisk)
since they are based on refraction in a low-index material, resulting in an
increase in $v_{gx}$ (further details in the main text). Miller’s bound is
more relevant for $a$, $b_{1}$, $b_{2}$, $b_{3}$, whereas the single-resonance
bound ($\kappa=2$) is more relevant for $c_{1}$, $c_{2}$, and $d$
($\eta_{\max}$ and Miller’s bound are not listed for $d$ as the corresponding
spaceplate is made of very good conductors at microwave frequencies, hence
$\eta_{\max}\rightarrow\infty$).

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. (3), as a function of
NA and $R$, comparing it to the fractional bandwidth of the entire visible
range. We consider two choices for the maximum permittivity contrast:
$\eta_{\max}=3$ for a low-contrast spaceplate, and $\eta_{\max}=15$ for a
high-contrast spaceplate (the latter is arguably the largest contrast that can
be achieved at visible frequencies with natural materials [30]). As an
example, considering a numerical aperture $\textrm{NA}=0.3$, an ideal
spaceplate with $\eta_{\max}=3$ can achieve a compression ratio $R\approx 8$
over the entire visible spectrum, whereas a spaceplate with $\eta_{\max}=15$
can achieve a compression ratio of 25. These results demonstrate that, at
least in theory, spaceplates with good performance in terms of numerical
aperture and compression ratio can indeed cover the entire visible range.

Figure 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 $F/L_{m}=100$ (horizontal lines, Eq. (4))
and a transversely invariant spaceplate with compression ratio $R$, for
different values of NA. The spaceplate bound is calculated using Eq. (3) (for
the case $v_{gx}\rightarrow c_{0}$, leading to the loosest form of the 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, even in the
most optimistic scenario.

##  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, are 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,
also 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 Ref. [12], where it was shown that the upper bandwidth bound
is given by

$\frac{\Delta\omega}{\omega_{c}}\leq\frac{1}{2\sqrt{3}}\frac{L_{m}\eta_{\max}}{F}\frac{\sqrt{1-(NA/n_{b})^{2}}}{1-\sqrt{1-(NA/n_{b})^{2}}},$
(4)

where $L_{m}$ and $F$ are the thickness and focal length of the metalens. We
stress that, when a (meta)lens is followed by a spaceplate as illustrated in
Fig. 4(a), although focusing is achieved at a shorter distance due to the
presence of a spaceplate, the focal length $F$ is unaltered since it is an
intrinsic property of the lens itself (of its transverse profile) and it
cannot be modified by the presence of a transversely homogeneous structure
behind the lens. In other words, an ideal spaceplate does not change the
imaging functionality or performance of the system, but it simply “shifts” the
entire field distribution closer to the lens [18]. We also note that other
classes of flat lenses, such as achromatic diffractive lenses [11, 31] are not
bound by the same limitations stemming from delay-bandwidth constraints since
they are based on a different mechanism to achieve achromaticity, as nicely
discussed in Ref. [13]; however, they suffer from other performance
constraints, most notably the fact that they can provide near diffraction-
limited performance only at very low Fresnel numbers [13], making them less
appealing for several applications.

By comparing Eqs. (4) and (3), 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 from
ten to around one hundred, 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, 31], 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, 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. 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 a few tens may be
physically achievable over the entire visible range, using spaceplates made of
realistic materials. 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 spaceplate within the same volume [33]. Moreover, a
better understanding of the fundamental limits to the transmission efficiency
of spaceplates, and of the tradeoffs 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.

Funding F.M and K.S. acknowledge support from the National Science Foundation
(NSF) with Grant No. 1741694, and the Air Force Office of Scientific Research
with Grant No. FA9550-19-1-0043 through Dr. Arje Nachman. R.W.B. acknowledges
support from the Natural Sciences and Engineering Research Council of Canada,
the Canada Research Chairs program, and the Canada First Research Excellence
Fund award on Transformative Quantum Technologies, US DARPA award
W911NF-18-1-0369, US ARO award W911NF-18-1-0337, US Office of Naval Research
MURI award N00014-20-1-2558, and DOE award GR530967. J.S.L acknowledges
support from the Canada Research Chairs (CRC) program, the Natural Sciences
and Engineering Research Council (NSERC), and the Canada First Research
Excellence Fund award on Transformative Quantum Technologies.

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

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