Submitted to Optics Express (Feb. 7, 2023)

# Large-scale optical compression of free-space using an experimental three-
lens spaceplate

Nicholas J. Sorensen  Michael T. Weil  and Jeff S. Lundeen

(Physics Dept. and Nexus for Quantum Technologies, University of Ottawa, 25
Templeton Street, Ottawa ON K1N 6N5, Canada)

##  1 Introduction

The miniaturization of optical systems, such as microscopes, telescopes, and
cameras, has long been a topic of interest for physicists. By designing
smaller optical systems, we can increase their versatility and reduce their
cost. Most work has aimed at making the lenses thinner, e.g., metalenses [1,
2, 3, 4, 5, 6, 7, 8], diffractive lenses [9, 10, 11, 12, 13, 14], and Fresnel
lenses [15]. Instead, our aim is to miniaturize the space between lenses, an
often neglected element in optical systems. Since this is usually the largest
contributor to the total length of an imaging system, its miniaturization
promises the largest impact.

Recently, the "spaceplate" was introduced as an optical element that
compresses the effect of propagation on light into a plate [16]. Such a
device, by definition, must replace a slab of free-space with thickness
$d_{\text{eff}}$ while only occupying a physical thickness $d$ less than
$d_{\text{eff}}$. The compression ratio, $\mathcal{R}=d_{\text{eff}}/{d}$, is
used to characterize spaceplate designs [16]. Like free-space propagation, a
spaceplate changes the height of a ray but leaves its angle unchanged. Thus,
unlike a lens, which does change ray angle, a spaceplate does not alter the
magnification of an imaging system. From the view of physical optics, the
spaceplate changes the phase of an incident plane wave without altering its
angle.

A variety of optical devices have been proposed to compress space. The work of
Reshef et al. [16] first introduced and experimentally demonstrated the
spaceplate. They experimentally tested two designs, one being a uniaxial
crystal and the other a low-index medium. The designs were found to have
compression ratios of $\mathcal{R}=1.12$ and $1.48$, respectively, which
resulted in each spaceplate saving less than $4\text{\,}\mathrm{m}\mathrm{m}$
of space. Remarkably, the uniaxial crystal had a large numerical aperture
($\text{NA}=\sin\theta$) and functioned at high incident angles
$(\text{NA}=0.85)$ while remaining broadband, i.e., highly transmissive across
the visible spectrum. In the same paper, the group also proposed and modeled a
multilayer thin-film design. This had a compression ratio of
$\mathcal{R}=5.4$, but was only capable of saving
$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ of space and was
narrowband. In another, almost simultaneous work, Guo et al. [17] numerically
demonstrated a much higher compression ratio of $\mathcal{R}=144$ using the
Fano resonances of a photonic crystal slab at wavelength $\lambda$. Unlike the
uniaxial crystal, however, this design was significantly limited in NA
$(\text{NA}=0.01)$ and the actual space saved was limited to approximately
$360\lambda$ [17]. Subsequent designs, including those made of homogeneous
materials [16], improved multilayer stacks [18], repeated Fabry-Perot cavities
[19, 20], and others [21, 22, 23], made incremental improvements upon
compression ratio and NA. None, however, can be scaled up to save space on the
order of meters. Additionally, most of the previously demonstrated designs are
limited in either polarization or bandwidth [16, 17, 18, 19]. In this paper,
we present a broadband, polarization-independent three-lens spaceplate capable
of meter-scale compression.

Figure 1: Propagation of a light beam in a three-lens spaceplate. a) Light
travels through a spaceplate of thickness $d$ with an equivalent propagation
through free-space a distance $d_{\text{eff}}>d$. b) Rays passing through a
spaceplate will maintain their angle $\theta$, but are displaced by walkoff
$\Delta w$, just as if they were passing through free space. c) Inverting
spaceplates do the same while also inverting the rays about the
$\mathbf{z}$-axis. The transverse direction is $\mathbf{x}$, and the
longitudinal direction is $\mathbf{z}$. In this figure, $\beta$ is the angle
of the boundary ray of the given input field, $\gamma$ is the maximum angle of
the boundary ray within the spaceplate, $f_{\text{ext}}$ is the focal length
of the exterior lenses, $f_{\text{mid}}$ is the focal length of the middle
lens, $x$ and $x_{\text{SP}}$ are the transverse displacements of the original
and space-compressed rays entering at angle $\theta$, respectively, and
$\Delta z$ is the longitudinal shift of an output light field.

The three-lens spaceplate [Fig. (1)] utilizes the Fourier plane of a 4-$f$
lens system to impart the same angle-dependent phase that free-space
propagation imparts. Our 4-$f$ lens system consists of two positive lenses of
focal length $f_{\text{ext}}$, the "exterior lenses". The first lens maps the
complex amplitude distribution of an incident field across the front focal
plane to the back focal plane via a Fourier transform [24]. Thus, there is a
one-to-one mapping between the angle of an incoming plane wave and the
position in the back focal plane, known as the "Fourier plane", which is
located a distance $f_{\text{ext}}$ after the lens. The second lens is a
distance $2f_{\text{ext}}$ after the first lens and performs another Fourier
transform, undoing the action of the first lens. With a spatially-varying
phase element (i.e., a "phase mask") in the Fourier plane, the 4-$f$ system
phase shifts each incoming plane wave but otherwise leaves it unchanged. The
angle-dependent phase shift required to emulate free-space propagation happens
to correspond to the phase mask created by a thin positive lens. Consequently,
such a lens placed in the Fourier plane (i.e., a middle lens with focal length
$f_{\text{mid}}$) will cause the field transmitted by the 4-$f$ system to
appear as though it had propagated through a slab of free-space, potentially
longer than the 4-$f$ system itself. Note that while this three-lens device is
neither monolithic nor plate-like, we still refer to it as a spaceplate due to
its similar effect on light. Note as well that the use of a 4-$f$ system
results in inversion of the light about the $\mathbf{z}$-axis ($x=0$); while
this does not affect the compression or absolute magnification of the device,
it does differentiate the device from other spaceplates.

In the next section, we theoretically demonstrate how this three-lens design
can be used to attain free-space compression on a much larger scale than
previous spaceplate designs. In Section 3, we experimentally demonstrate the
three-lens spaceplate and characterize its performance for a number of lens
choices. We show how it can be straightforwardly inserted into the design of
long imaging systems, even if they are broadband. In Section 4, we outline
some fundamental theoretical limits on the performance of the three-lens
spaceplate. We focus on compression ratio, NA, and bandwidth, which recent
work has identified trade-offs between [19, 25] for other spaceplate designs.
In the last two sections, we compare and contrast the three-lens design to
related optical systems and discuss prospects for its application.

Figure 2: Measurement of the focus advance due to the three-lens spaceplate.
At each $z$, we plot the intensity along $x$ (after selecting the $x$ row with
the highest intensity) without the spaceplate - just air - (in greyscale,
bottom) and for each spaceplate middle lens $f_{\text{mid}}$ (in colorscale,
associated with nominal compression ratio $\mathcal{R}$). The vertical dotted
lines mark the peak intensity along $z$, i.e. the position of the focus, and
the diamond scatter points denote the nominal focus positions. With the air-
only focus taken as $z=0$, the vertical lines indicate the focal advance for
each spaceplate. Negative advances correspond to a spaceplate that replaces
more space than it occupies, i.e., $\mathcal{R}>1$. For more experimental
details and descriptions see Sections 3.1 and 3.2.

##  2 Fourier optics description of a three-lens spaceplate

To motivate the design of the three-lens spaceplate, we follow the Fourier
optics description of the spaceplate effect introduced by Reshef et al. [16].
Suppose we have a light field propagating through free-space centered on the
$\mathbf{z}$-axis. We then consider how each transverse spatial Fourier
component of this initial field is transformed by this propagation. Each of
these components, or plane waves, is described by its momentum vector,
$\mathbf{k}=\left(k_{x},k_{z}\right)=k(\sin\theta,\cos\theta)$, where $\theta$
is the angle between the $\mathbf{z}$-axis and the direction of the plane
wave. When propagated over a distance $z_{0}$ in free-space, the amplitude and
direction of each plane wave is conserved, but the phase is not. Between two
points on the $\mathbf{z}$-axis separated by $z_{0}$, each plane wave
accumulates phase $\varphi=k_{z}z_{0}=|\mathbf{k}|\cos(\theta)z_{0}$.

A spaceplate must produce this same phase response, but in a shorter distance
$d$ than $z_{0}$. An ideal spaceplate, if inserted into the light path, will
produce the phase response
$\varphi_{\text{SP}}=k_{z}z_{0}=z_{0}(k^{2}-k_{x}^{2})^{1/2}$. In the small
angle approximation, $\varphi_{\text{SP}}=z_{0}k(1-k_{x}^{2}/2k^{2})$. Then,
by neglecting global phase terms, we arrive at our target phase,
$\varphi_{\text{SP}}=-z_{0}k_{x}^{2}/2k$. An approach for considering larger
incident angles is given in Appendix A. In the Fourier plane of the 4-$f$
system in Fig. (1a), the field at position $r$ (i.e., axial distance) is the
incoming field component with transverse momentum $k_{x}=rk/f_{\text{ext}}$.
Substitution of this relation into the target phase gives the spatial phase
mask required to emulate propagation through space,

$\displaystyle\varphi_{\text{SP}}=-\frac{z_{0}{kr^{2}}}{2f_{\text{ext}}^{2}}.$
(1)

The phase added by a spherical, thin lens of focal length $f_{\text{mid}}$ in
the paraxial approximation is given by
$\varphi_{\text{mid}}=-kr^{2}/2f_{\text{mid}}$, which is of the same form as
Eq. (1). By setting $\varphi_{\text{SP}}=\varphi_{\text{mid}}$ we find that
the spaceplate effectively propagates light a distance
$z_{0}=f_{\text{ext}}^{2}/f_{\text{mid}}$ . In this way, we can compress space
by placing a positive lens at the Fourier plane.

A slight complication is that, as part of the 4-$f$ system, there already is
free-space propagation of distance $f_{\text{ext}}$ before the first lens and
$f_{\text{ext}}$ after the last lens. We do not include those regions in the
length of space replaced and thus, $d_{\text{eff}}=z_{0}-2f_{\text{ext}}$.
Similarly, we define the spaceplate thickness to solely be the distance
between the external lenses, $d=2f_{\text{ext}}$. Therefore, the compression
ratio is

$\displaystyle\mathcal{R}=\frac{d_{\text{eff}}}{d}=\frac{f_{\text{ext}}}{2f_{\text{mid}}}-1,$
(2)

which only depends on the focal lengths used. If $\mathcal{R}>1$ or,
equivalently, if $f_{\text{mid}}<f_{\text{ext}}/4$, more space is replaced
than is occupied by the three-lens spaceplate. Further discussion of system
length and compression ratio is given in Appendix B.1.

Figure 3: Experimental setup used to test the spaceplate focus advancement,
the walkoff, as well as the full-color imaging. The lens labeled
$f_{\text{image}}$ is removed for walkoff experiments. Figure 4: Measurement
of the transverse walkoff and compression ratio as a function of middle focal
length. a) The measured transverse walkoffs ($\Delta w$) of each of the tested
spaceplates are plotted as a function of angle with fitted linear regressions.
The theoretical limit is derived from the aperture sizes of the lenses used
and the apparatus limit is produced by the bounds of the translation stage and
image sensor. b) Here, the compression ratios measured from the corresponding
walkoffs in (a) are plotted against the focal lengths of the middle lens
($f_{\text{mid}}$). The point associated with least compression,
$f_{\text{mid}}=\infty$, does not appear in b). All points have their
uncertainties plotted, though most are obscured by the symbol for the point.

##  3 Experimental demonstration of three-lens space compression

###  3.1 Design of the focus advancement, walkoff, and imaging experiments

The setup used to test the three-lens spaceplate is shown in Fig. (3). We
measure the advance of the focus of a beam, the walk-off of a beam incident at
an angle, as well as imaging properties. These three types of measurements
require minor setup variations that will be detailed in the following three
subsections.

We start by describing elements in the setup common to all three measurements.
The external lenses, $f_{\text{ext}}=$150\text{\,}\mathrm{m}\mathrm{m}$$, are
the same throughout all the tests and have $50.8\text{\,}\mathrm{m}\mathrm{m}$
diameters and $15.0\text{\,}\mathrm{m}\mathrm{m}$ thicknesses. We test the
effect of the following series of middle lenses:
$f_{\text{mid}}=\\{40,30,25,19,18,11,10,4.51\\}\,$\text{\,}\mathrm{m}\mathrm{m}$$
whose respective diameters and thicknesses were
$\\{25.4,25.4,12.7,12.7,6.5,8.0,5.5\\}\,$\text{\,}\mathrm{m}\mathrm{m}$$ and
$\\{12.5,14.0,7.0,6.0,2.2,5.0,6.5,2.94\\}\,$\text{\,}\mathrm{m}\mathrm{m}$$.
With the exception of the
$f_{\text{mid}}=\\{18,11,4.51\\}$\text{\,}\mathrm{m}\mathrm{m}$$ aspheric
lenses, all lenses used throughout these tests were achromatic doublets,
designed and anti-reflective (AR) coated for the visible range
(400-$700\text{\,}\mathrm{n}\mathrm{m}$ light). The
$f_{\text{mid}}=$18\text{\,}\mathrm{m}\mathrm{m}$$ aspheric lens was designed
for $780\text{\,}\mathrm{n}\mathrm{m}$ light, and the
$f_{\text{mid}}=\\{11,4.51\\}\,$\text{\,}\mathrm{m}\mathrm{m}$$ aspheric
lenses were designed for $633\text{\,}\mathrm{n}\mathrm{m}$ light. All
aspheric lenses were AR-coated for 600-$1100\text{\,}\mathrm{n}\mathrm{m}$.
After the spaceplate, an image sensor
($$5.86\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$\times$5.86\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$
pixel size, $1936\times 1216$ pixels, color) was mounted on a motorized
translation stage capable of moving both in the transverse ($x$) and
longitudinal ($z$) directions. With this sensor, we recorded $x\times y$
spatial intensity distributions at chosen $z$ positions. The electronic
settings of the image sensor were constant throughout the various tests.

###  3.2 Focus advancement of various three-lens spaceplates

We measure the focus advancement of a $633\text{\,}\mathrm{n}\mathrm{m}$
wavelength Helium-Neon laser that is collimated so that its beam has an
intensity full-width-half-maximum (FWHM) of $3.5\text{\,}\mathrm{m}\mathrm{m}$
using three different middle lens focal lengths corresponding to three
distinct compression ratios. To create a focusing beam, the laser travels
through a $50.8\text{\,}\mathrm{m}\mathrm{m}$ diameter positive lens with a
thickness of $11.3\text{\,}\mathrm{m}\mathrm{m}$ and a focal length of
$f_{\text{image}}=$1000\text{\,}\mathrm{m}\mathrm{m}$$ [see Fig. (1a)]. The
three-lens spaceplate is placed between this positive lens and the nominal
focus $z$-position, then the image sensor records intensity distributions over
a range of $500\text{\,}\mathrm{m}\mathrm{m}$ with a
$0.25\text{\,}\mathrm{m}\mathrm{m}$ step size. Each $x$ pixel row of the
recorded distribution is summed to determine the row with the maximal
intensity. This row becomes the 1D intensity profile at each $z$, which is
plotted in Fig. (2). For each middle lens, the peak intensity along $z$ is
taken as the new focal position.

We measure longitudinal focal shifts of $\Delta
z=$94\text{\,}\mathrm{m}\mathrm{m}$$, $-49\text{\,}\mathrm{m}\mathrm{m}$, and
$-221\text{\,}\mathrm{m}\mathrm{m}$ corresponding to observed compression
ratios $\mathcal{R_{\text{obs}}}=0.69$, $1.16$, and $1.74$, respectively. The
respective nominal compression ratios are $\mathcal{R}=0.9,1.5,$ and $2.0$.
While these are significantly different from the measured values, they
demonstrate that the three-lens spaceplate can successfully replace space.

Deviations between $\mathcal{R}$ and $\mathcal{R_{\text{obs}}}$ result because
the three-lens spaceplate relies on a lens located directly at the Fourier
plane of the 4-$f$ system, and any deviation of the thick middle lens from
this position results in a reduction of compression and non-unity
magnification. Specifically, we observe that upon transmitting an expanded
collimated beam with a $\text{FWHM}=$3.5\text{\,}\mathrm{m}\mathrm{m}$$
through the three-lens spaceplate, the resultant beam width varies
($1.7-$4.4\text{\,}\mathrm{m}\mathrm{m}$$) depending on small adjustments
($0-$5\text{\,}\mathrm{m}\mathrm{m}$$) to the longitudinal position of the
middle and rear lenses in the spaceplates. The effect of these sensitivities
are reduced in subsequent walkoff and imaging experiments.

Nonetheless, the observed negative longitudinal shift demonstrates that this
design is capable of replacing free-space on a scale of three orders of
magnitude greater than previous spaceplate designs. This experiment is a
qualitative demonstration of space compression - the following experiments are
used to characterize the three-lens spaceplate with greater accuracy.

Figure 5: Full-color imaging with a three-lens spaceplate. a) Top: imaging
system; Bottom: imaging system with the three-lens spaceplate. b) The imaging
object, a stamp; c) An image of the stamp. Right: without the spaceplate;
Left: with a $f_{\text{mid}}=10$ mm spaceplate with nominal compression ratio
$\mathcal{R}=6.5$ (image is rotated by $180^{\degree}$ to compensate for the
three-lens spaceplate inversion). With the spaceplate, a sharp full-color
image is formed $1.65\text{\,}\mathrm{m}$ closer to the imaging lens,
compressing the size of the imaging system. The stamp is a photo of the French
River in Prince Edward Island by Barrett and Mackay Photography. Reproduction
is permitted under the Canada Post Corporation Act [26].

###  3.3 Transverse beam walkoff of a three-lens spaceplate

We next measure the transverse walk-off $\Delta
w=|x_{\text{SP}}|-|x|=d(\mathcal{R}-1)\sin{\theta}$ [see Fig. (1b) for a
diagram] of a beam transmitted through the spaceplate, incident at angle
$\theta$ to the $z$-axis, rotating about the $y$-axis. For each spaceplate,
the walk-off as function of angle characterizes the NA and aberrations, and
determines the compression ratio with greater accuracy than in the focal
advance measurement. Specifically, before conducting walkoff experiments, the
longitudinal positions of the middle and rear lenses are incrementally
adjusted until a beam ($\text{FWHM}=$3.5\text{\,}\mathrm{m}\mathrm{m}$$)
transmitted through the spaceplate becomes collimated and has unity
magnification. That is, until the exiting beam has diameter of
$\text{FWHM}=(3.5\pm 0.1)\,$\text{\,}\mathrm{m}\mathrm{m}$$ both immediately
after the rear lens of the spaceplate, and one meter after it.

To test the beam walkoff, the imaging lens and beam-expander are absent,
leaving the laser beam collimated. The rotation stage in Fig. (3) varied
$\theta$ from 0 to 2 degrees in 0.02 degree increments. The stage is located
$f_{\text{ext}}$ before the first lens, and we measure the transverse
displacement $z=4f_{\text{ext}}=$600\text{\,}\mathrm{m}\mathrm{m}$$ from the
rotation axis of the stage. The rotation stage is placed here due to our
definition of $\text{NA}=\sin\theta$, where $\theta$ is the half-angle of a
maximum cone accepted by the system, originating $f_{\text{ext}}$ before the
first lens. By putting the stage at this position, we are able to
contextualize the experimental angular limitations of the system with an
analysis of its numerical aperture. The position at which we measure the
transverse displacement, however, does not matter in general. One will measure
the same $\Delta w$ regardless of the position of the imaging plane.

First, we measure the transverse displacement $x(\theta)$ of a beam from the
$z$-axis in free space, and then perform analogous measurements
$x_{\text{SP}}(\theta)$ of beams transmitted through each spaceplate. Due to
inversion, the beams are displaced in opposite directions, but we can still
compute the walkoff as: $\Delta w=|x_{\text{SP}}|-|x|$. We measure $x$ and
$x_{\text{SP}}$ by fitting Gaussians to the measured transverse intensity
profiles.

Fig. (4a) presents this series of walkoff measurements for each of the eight
middle lenses given in Section 3.1, with corresponding observed compression
ratios in Fig. (4b). Each of the configurations produced compression ratios
within error of their nominal compression ratios over the entire measured
range. For the highest experimental compression ratio, $\mathcal{R}=15.6$,
$f_{\text{mid}}=$4.51\text{\,}\mathrm{m}\mathrm{m}$$, the spaceplate replaced
4.39 meters of free-space, which is more than one-thousand times greater than
that demonstrated by other spaceplate designs.

###  3.4 Broadband imaging with a three-lens spaceplate

In our final measurement, we investigate the performance of the three-lens
spaceplate for imaging with full-visible-spectrum light. We first illuminate a
postage stamp by white-light and use the $f_{\text{image}}$ lens to form a
sharp image on the image sensor. The object and image distances are
approximately $1.9\text{\,}\mathrm{m}$ and $2.1\text{\,}\mathrm{m}$,
respectively, and we measure a magnification of 1.1. We record images with and
without a $f_{\text{mid}}=$10\text{\,}\mathrm{m}\mathrm{m}$$ spaceplate [Fig.
(5c), left and right panes, respectively] in the imaging path. Before imaging,
the middle and rear lenses of the spaceplate are calibrated using an expanded
collimated beam, as was done in the walkoff experiment. The camera is then
placed at the nominal longitudinal position relative to the focus position
seen without a spaceplate in the imaging path. Then, the middle and rear lens
longitudinal positions are adjusted slightly to achieve the same magnification
seen without the spaceplate. To reduce aberration and edge-effects, a large
iris was positioned immediately before the third lens in the spaceplate and
was adjusted until the highest contrast and resolution were seen. This was at
an aperture size of $(12\pm 2)\,$\text{\,}\mathrm{m}\mathrm{m}$$. While there
still is a reduction in contrast and brightness, which we discuss later, the
images show that the three-lens spaceplate can advance the image plane of a
full-visible-spectrum imaging system by 1.65 m.

Figure 6: The effect of the three-lens spaceplate on resolution and contrast.
a) The three-lens spaceplate is used to compress an imaging system consisting
of a $1000\text{\,}\mathrm{m}\mathrm{m}$ focal length doublet. b) Images taken
using different spaceplates. c) Images taken with a spaceplate and an iris. d)
Measured (background, filled plots) and fitted Modulation Transfer Functions
(MTF) for three full-color stamp images; one was taken with only the imaging
lens, another with the imaging lens and a spaceplate ($\mathcal{R}=6.5$,
$f_{\text{mid}}=$10\text{\,}\mathrm{m}\mathrm{m}$$), and the third with an
iris just before the third spaceplate lens. The dotted curves represent the
standard error resulting from the method used to approximate the MTFs. See the
text for discussion and comparison of results.

Next, we image an NBS 1963a resolution target in order to characterize the
reduction in resolution and contrast caused by the addition of four different
three-lens spaceplates [see Fig. (6)]. The addition of the spaceplates does
not change the magnification, as expected. However, Fig. (6b) shows that every
spaceplate reduces image contrast and resolution in comparison to the image
formed in the absence of a spaceplate. We quantify this reduction with the
modulation transfer function (MTF) of the imaging system following the method
of Saiga et al. [27] in Fig. (6d). For the case of
$f_{\text{mid}}=$10\text{\,}\mathrm{m}\mathrm{m}$$ ($\mathcal{R}=6.5$) and the
stamp imaged without an iris, the spaceplate lowers the spatial frequency at
half-max of the MTF by 45% (i.e. $10.9\,\text{lp/mm}$ is reduced to
$6.0\,\text{lp/mm}$). Three potential causes for this reduction are
aberrations, scattering from lens edges, and diffraction by the lens
apertures. The first two would be mitigated by a reduced aperture, whereas the
effect of diffraction would be worsened. Adjusting the diameter of an iris
$[(12\pm 2)\,$\text{\,}\mathrm{m}\mathrm{m}$]$ just before the third lens, we
take the images shown in Fig. (6c), which are qualitatively improved in
contrast and resolution. This improvement is reflected in the MTF measured
with an iris in place, which increases by 38% (i.e. $6.0\,\text{lp/mm}$ to
$8.3\,\text{lp/mm}$) suggesting that diffraction is not the main cause of the
reduction. Equivalently stated, addition of the spaceplate and iris causes a
24% reduction in the MTF (i.e. $10.9\,\text{lp/mm}$ to $8.3\,\text{lp/mm}$).

##  4 Fundamental limits on performance of the three-lens spaceplate

In this section, we briefly discuss limits on the NA and compression inherent
to the three-lens design of the spaceplate. In most physically realizable
setups, the NA of the three-lens spaceplate will be limited by the diameter of
the third lens, $D_{3}$. For this common case, the NA can be expressed in
terms of the f-number of the third lens, $N_{3}=f_{\text{ext}}/D_{3}$, and
$\mathcal{R}$:

$\displaystyle\text{NA}=\left[1+4N_{3}^{2}\left(1+2\mathcal{R}\right)^{2}\right]^{-\frac{1}{2}}.$
(3)

Eq. (3) shows that there is trade-off between compression ratio and NA. The
lower the f-number of the third lens, the less strict this trade-off is. The
third lens used in our experiment has $N_{3}=3$, which limits the NA of the
three-lens spaceplate be less than 0.06, worsening as $\mathcal{R}$ increases.
This represents the greatest shortcoming of the three-lens spaceplate design:
even when using a relatively large third lens diameter
($D_{3}=$50\text{\,}\mathrm{m}\mathrm{m}$$), the maximum incident angle is
three degrees from the optical axis. For information on the less common cases
where the first or second lenses limit the NA, see Appendix C.1.

We have considered two additional physical mechanisms inherent to the three-
lens spaceplate that lead to limitations. However, these turn out to be much
less restrictive than the above limitation in practical applications with
standard lenses. Firstly, the Abbe Sine Condition limits the NA independent of
the lens diameters. Secondly, diffraction from the first lens in the
spaceplate limits the achievable compression ratio. These two limitations may
be relevant for new types of lenses, such as metalenses, and are therefore
explored in depth in Appendices (C.2) and (C.3), respectively.

##  5 Comparison to other compressive optics

A number of distinctions must be made between the three-lens spaceplate and
other methods to conserve space in imaging systems. We start by comparing it
to other spaceplate designs. All of these exhibit some degree of transverse
translation invariance [25], whereas the three-lens spaceplate does not, since
lenses are not translationally invariant. Another difference is that the
three-lens spaceplate inverts the transmitted field through the axis of the
lenses (i.e., $x\rightarrow-x$ and $y\rightarrow-y$), though this is avoided
simply by using a pair of spaceplates.

Many conventional imaging systems are already designed to minimize the space
they occupy. Examples include the telephoto lens and the Cassegrain reflector
whose goal is to create an imaging system shorter than its effective focal
length $f_{\text{eff}}$. Thus the compression of space is intertwined with the
main purpose of the imaging system - to provide optical magnification. In
practice, telephoto lenses are limited to a length $>0.8f_{\text{eff}}$ [28,
29, 30]. This is equivalent to a non-telephoto lens followed by a three-lens
spaceplate with $\mathcal{R}\geq 1/0.8=1.25$, much lower than the highest
value reported here for three-lens spaceplates $\mathcal{R}=15.7$, albeit with
a higher NA.

The nature of the components of these three-lens systems means that they are
large. In this way, they differentiate themselves from thin spaceplates.
Because they use lenses, implementable three-lens spaceplates have limitations
on thickness, aperture, and, consequentially, compression ratio. Using off-
the-shelf components and manageable system lengths
($2f_{\text{ext}}\lesssim$1\text{\,}\mathrm{m}$$), compression ratios are
limited ($\mathcal{R}\lesssim 20$); however, due to the large size of these
systems, they can save large absolute quantities of space (on the order of
meters). This is in contrast to existing spaceplate designs, which only save
space on the order of microns or millimeters [16, 17, 18, 19].

One advantage of using standard optical components in three-lens spaceplates
is that the resulting spaceplate is broadband, polarization-independent, and
highly configurable in compression ratio. The compression ratio of the three-
lens spaceplate can easily be changed by replacing the middle lens, whereas
the compression ratio of other spaceplate designs is fixed upon manufacturing.

Similar systems have been used in microscopy to rapidly tune imaging planes
[31, 32, 33, 34] in transport-of-intensity phase microscopy. However, such
implementations do not compress space and are instead intended to artificially
increase the depth of field.

##  6 Conclusion

In this paper, we have successfully demonstrated the compression of free-space
using a device that we call a "three-lens spaceplate", a positive lens placed
at the Fourier plane of a 4-$f$ optical setup. We have shown that our
spaceplate design can be used for meter-scale space compression and the
miniaturization of broadband imaging systems. Using the three-lens spaceplate,
we have achieved experimental compression ratios up to
$\mathcal{R}_{\text{obs}}=15.7$ that replace 4.4 meters of free-space. Until
now, replication of the transfer function of free-space over lengths of this
scale had not been demonstrated: previously, space compression of only one or
two millimeters had been shown [16]. The three-lens spaceplate has not only
shown that this is possible, it has accomplished this using off-the-shelf
optical components. Moreover, the system was shown to reduce the length of
both monochromatic and full-color imaging systems with a 24% loss of
resolution as quantified using its modulation transfer function. Finally, we
discussed the trade-off between compression ratio and numerical aperture.
Specifically, we found that the numerical aperture is highly limited for the
three-lens spaceplate and represents its greatest shortcoming.

The simplicity and versatility of this spaceplate design motivate its
application and future development, though there are certain limitations which
differentiate its applications from other spaceplates. The three-lens
spaceplate represents a simple, cost-efficient method for reducing the size of
axially-symmetric, long focal length systems such as telescopes. Because these
three-lens spaceplates are so large and have such limited numerical apertures,
however, their application in smaller devices such as smartphone cameras or
virtual reality headsets would be difficult to implement. That said, the
inherent use of free-space within the system itself implies the potential for
even higher compression ratios through the use of metalenses and other space-
saving devices. Metalenses could potentially also reduce the numerical
aperture limitations of the three-lens spaceplate by reducing lens thickness,
and increasing aperture size. Future work includes using such hybrid designs
to reduce overall system length and increase numerical aperture, as this could
make the implementation of three-lens spaceplates in small devices more
realistic.

### Funding

The authors acknowledge support from the Canada Research Chairs Program, the
Natural Sciences and Engineering Research Council of Canada, as well as the
Canada First Research Excellence Fund.

### Acknowledgments

The authors would like to thank Orad Reshef, Yaryna Mamchur, and Ryan Hogan
for encouragement and discussions, as well as Cheng Li and Aldo Martinez for
experimental lendings and accommodations.

### Disclosures

The authors declare no conflicts of interest.

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##  Appendix A Performance beyond the small-angle regime

In motivating the design of the three-lens spaceplate (see Section 2), we used
the small-angle approximation to draw similarities between the ideal
spaceplate phase, $\varphi_{\text{SP}}$, and the phase of a spherical, thin
lens $\varphi_{\text{mid}}$. In reality, this approximation is more
restrictive than necessary for such a comparison and thus the possibility of
extending beyond this regime merits further discussion.

The general spaceplate phase is
$\varphi_{\text{SP}}=kd_{\text{eff}}(1-k_{x}^{2}/k^{2})^{1/2}$, while the
spherical lens phase is
$\varphi_{\text{mid}}=k(f_{\text{mid}}^{2}-x^{2})^{1/2}$. Assuming the first
lens is capable of performing a perfect Fourier transform of the incident
field such that $k_{x}=rk/f_{\text{ext}}$ holds for large angles, we can
directly compare the spaceplate phase to that of a spherical lens outside the
small-angle approximation. The maximum value of $k_{x}/k=\sin{\theta}$ such
that the phase difference is less than some fraction of a period (i.e.
$\sim\pi/5$), disregarding any global phase, determines the maximum NA of the
three-lens spaceplate. Geometrically, it represents the angle at which the
circular phase profile of a spherical lens deviates a fraction of a cycle from
the elliptical phase profile of an ideal spaceplate. This method can be
generalized to other middle lens forms as long as the first lens linearly maps
each incident transverse wavevector component $k_{x}$ to a single position $x$
in the back focal plane.

It must be said, however, that conventional lenses produce limits on NA that
make this method of no practical use. As we have shown for our setup, this is
because the limits on NA will always be more restrictive than those predicted
by the method above. There are likely lens designs that are more suited for
matching the phase profile of an ideal spaceplate in which case this method
may prove useful.

##  Appendix B Defining the compression ratio and system length

###  B.1 Ray-matrix approach to deriving the compression ratio

One can also assess the three-lens spaceplate using a ray matrix method [24,
35, 36, 37]. The ray-transfer matrix for the system is given by

$\displaystyle M=-\begin{bmatrix}\hskip
3.0pt1&\frac{f_{\text{ext}}^{2}}{f_{\text{mid}}}-2f_{\text{ext}}\hskip
4.0pt\\\ \hskip 3.0pt0&1\hskip 3.0pt\end{bmatrix}$

which is equivalent to the ray-matrix for inversion and propagation over an
effective distance
$d_{\text{eff}}={f_{\text{ext}}^{2}}/{f_{\text{mid}}}-2f_{\text{ext}}$. The
compression ratio can then be expressed as

$\displaystyle\mathcal{R}=\left(\frac{f_{\text{ext}}^{2}}{f_{\text{mid}}}-2f_{\text{ext}}\right)\frac{1}{d}=\frac{f_{\text{ext}}}{2f_{\text{mid}}}-1$

which is identical to Eq. (2).

###  B.2 The effect of the defined system length

In the initial Fourier optics derivation of the spaceplate compression ratio
(Section 2), we consider a 4-$f$ system without a thickness $f_{\text{ext}}$
of free-space before or after the external lenses $(d=2f_{\text{ext}})$. It is
mathematically simpler to consider a system with these preceding and
succeeding lengths of free space $(d=4f_{\text{ext}})$; this is because they
are required to perform a Fourier transform without any additional phase
factors [24]. In such a system, we can directly equate the spaceplate phase
[Eq. (1)] with the lens phase $(\varphi_{\text{SP}}=\varphi_{\text{mid}})$ and
solve for the compression ratio $\mathcal{R}_{4f_{\text{ext}}}$, where we use
a subscript to indicate the redefinition of system length:

$\displaystyle\mathcal{R}_{4f_{\text{ext}}}=\frac{f_{\text{ext}}}{4f_{\text{mid}}}.$
(4)

This is similar to the initial definition of compression ratio [Eq. (2)] but
it predicts less compression due to the added free space.

In comparing the two definitions of compression ratios, one may notice that
Eq. (2) can evaluate to be negative, but this is simply a demonstration that
the three-lens spaceplate can extend the length of systems, not just compress
them. Recall that the compression ratio is defined as
$\mathcal{R}=d_{\text{eff}}/d$ and that negative compression ratios simply
define a system for which $d_{\text{eff}}<0$. Effectively, such systems
produce an output field equivalent to a field seen a distance $d_{\text{eff}}$
before the first lens. This is practically irrelevant to our exploration as
only compression ratios greater than unity are of interest, though an example
can be used to illustrate this discussion as well as demonstrate that Eqs. (2)
and (4) are physically equivalent.

Suppose we have a three-lens spaceplate without a middle lens
($f_{\text{mid}}\rightarrow\infty$) such that $\mathcal{R}=-1$ and
$\mathcal{R}_{4f_{\text{ext}}}=0$, as given by Eqs. (2) and (4). Eq. (2)
predicts that the light field immediately after the final lens will equal -
ignoring inversion - the field a distance $2f_{\text{ext}}$ before the first
lens in the spaceplate. This is a separation of $4f_{\text{ext}}$ between
equivalent fields. Further, Eq. (4) predicts that the field at the output (a
distance $f_{\text{ext}}$ past the final lens in the spaceplate) will equal
the field a distance $f_{\text{ext}}$ before the first lens. This is also a
separation of $4f_{\text{ext}}$ between equivalent fields. This demonstrates
that both compression ratio definitions [Eqs. (2) and (4)] are contextually
equivalent.

##  Appendix C Fundamental Limits

###  C.1 NA limitations due to lens diameter

Figure 7: Maximal light cone used to define the numerical aperture (NA) of the
three-lens spaceplate and contextualize the Abbe Sine Condition (ASC). We
define $\text{NA}=\sin\beta$ for a boundary ray at angle $\beta$ from the
$z$-axis, crossing the $z$-axis a distance $f_{\text{ext}}$ before the first
lens. See text for discussion of NA and the ASC limitations. Figure 8: Various
fundamental limitations of three-lens spaceplates. a) These three plots show
the three cases in Eq. (5) labeled by the aperture which limits the system NA.
b) A plot of the relationship between $\mathcal{R}$ and the maximum input
angle as given by the Abbe Sine Condition: each scatter point gives the limit
for each experimentally tested spaceplate. c) This plot compares the
compression ratio as a function of focal lengths $f_{\text{ext}}$ and
$f_{\text{mid}}$ as predicted by Eq. (4) to that as predicted by Eq. (7) which
considers diffractive effects. In order to make the effect more apparent, the
diameter of the first aperture is made artificially small
($\sim$10\text{\,}\mathrm{m}\mathrm{m}$$). d) This plot shows a line cut of
both plots in (c) for our experimentally tested focal lengths. The solid line
denotes the original expression, and the dashed line denotes the diffraction-
limited expression.

The use of finite-sized lenses results in a number of limitations which must
be considered. The first limitation we discuss is that of NA which depends on
the system parameters including compression ratio and the f-numbers of the
three-lenses used, $N_{1}=f_{\text{ext}}/D_{1}$, $N_{2}=f_{\text{mid}}/D_{1}$,
and $N_{3}=f_{\text{ext}}/D_{3}$, where $D_{i}$ is the diameter of the $i$-th
lens. As before, we define $\text{NA}=\sin\beta$ for a boundary ray at angle
$\beta$ from the $z$-axis, crossing the $z$-axis a distance $f_{\text{ext}}$
before the first lens [see Fig. (7)]. The angular range of the spaceplate is
defined as the largest angle $\beta$ at which a ray can successfully be
transmitted through the spaceplate. In most cases, the third lens limits the
angular range of the spaceplate as evident in Fig. (7), where we assume
$f_{\text{ext}}>f_{\text{mid}}$ such that the resulting compression ratio is
greater than one. This is because as long as the system compresses space, the
third lens must act on a range of angles greater than that incident on the
system. However, if the leading or middle lens diameter is reduced enough, the
overall limitation on NA will no longer be set by the third lens and instead
will be set by the first or second lens. Considering these different cases, we
arrive at the following expressions for the NA:

$\displaystyle\text{NA}=\begin{cases}\left(1+4N_{1}^{2}\right)^{-\frac{1}{2}},&D_{1}\leq
D_{2}\leq\frac{f_{\text{mid}}D_{3}}{f_{\text{ext}}-f_{\text{mid}}}\\\
\left[1+16N_{2}^{2}\left(1+\mathcal{R}\right)^{2}\right]^{-\frac{1}{2}},&D_{2}\leq
D_{1}\leq\frac{f_{\text{mid}}D_{3}}{f_{\text{ext}}-f_{\text{mid}}}\\\
\left[1+4N_{3}^{2}\left(1+2\mathcal{R}\right)^{2}\right]^{-\frac{1}{2}},&D_{1},D_{2}\geq\frac{f_{\text{mid}}D_{3}}{f_{\text{ext}}-f_{\text{mid}}}.\end{cases}$
(5)

Eq. (5) is plotted in Fig. (8a) where each piece-wise function is plotted
independently. The third case, where the system is $N_{3}$-limited, is most
physically realizable. The right-most figure in Fig. (8a) shows that for
compression ratios greater than one, and $N_{3}=3$ as used in our experiment,
the achievable NA is less than 0.06.

###  C.2 NA limitations due to the Abbe Sine Condition

In designing high-quality imaging systems using common refractive optics, the
minimization of aberrations is of great importance. Aplanatism, or the absence
of spherical aberration and coma, may be achieved by satisfying two
conditions: the system must be axially stigmatic (fully corrected for on-axis
imaging) and the system must satisfy the Abbe Sine Condition (ASC) [38]. For a
system to produce sharp images of extended flat objects, which was a central
objective of this work, the latter is of particular interest and thus merits
further discussion.

As we have shown, the three-lens spaceplate does not provide any magnification
or lensing effects and hence is not an imaging system, however, the ASC is
still relevant. We can see from Fig. (7) that the spaceplate can be divided
into two subsystems: a leading imaging system of magnification $\mathcal{M}$
and an auxiliary imaging system of reciprocal magnification
${\mathcal{M}}^{-1}$. For simplicity, we assume an object placed a distance
$f_{\text{ext}}$ from the first lens and that each imaging system is free of
spherical aberration. In other words, we will only examine limitations on NA
due to satisfying the ASC.

We begin by defining the magnification of each imaging system. According to
geometrical optics, the transverse magnification constant for the leading
imaging system is given by:

$\displaystyle\mathcal{M}=-\frac{f_{\text{mid}}}{f_{\text{ext}}}=-\frac{1}{2(\mathcal{R}+1)}.$
(6)

As we are only interested in systems with compression ratios greater than
unity, $\mathcal{R}>1$, the magnification constant for the leading imaging
system will always be less than ${1}/{4}$. In contrast, the magnification of
the auxiliary imaging system will always be greater than four. Therefore, the
leading imaging system is responsible for creating the largest internal ray
angles [set by $\gamma$ in Fig. (7)] and, hence will limit the incident angles
according to the ASC: $\mathcal{|M|}={\sin{\beta}}/{\sin{\gamma}}$ [39]. This
condition states that the ratio of the sine of the angle made by a ray exiting
the object of an imaging system to that of the ray at the imaging plane must
be equal to the system’s magnification constant - where either angle is
measured from the optical axis as in Fig. (7). Using Eq. (6), the ASC gives
the maximum incident angle the three-lens spaceplate can act on irrespective
of aperture size limitations:
$\sin{\beta}_{\text{max}}=[2(\mathcal{R}+1)]^{-1}$. For practical lenses, this
limit is not attainable since the effects of finite lens diameter are more
restrictive.

###  C.3 Compression ratio limitations due to diffractive effects

Another limitation set by finite aperture size is the diffractive limit on the
maximum compression ratio achievable by the spaceplate. For simplicity, we
consider diffraction from only the first lens the phase imparted by the middle
lens. This assumption is valid for most realistic cases in which the first and
third lens are approximately the same size such that the additional
diffraction effects attributable to the third lens are negligible. The
aperture function of the first lens is Fourier transformed by the first lens,
and the resultant point spread function is optically convolved with the middle
lens phase [Eq. (1)] at the Fourier plane. We approximate the point spread
function of the first aperture by a Gaussian with standard deviation
$\sigma=g\lambda f_{\text{ext}}/2D_{1}$ where $g=1.22$ is the Airy first node
scaling, $D_{1}$ is the diameter of the first lens, and $\lambda$ is the
wavelength:

$\displaystyle
H_{1}=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left(\frac{-x^{2}}{2\sigma^{2}}\right).$

The phase distribution across the range of input $k_{x}$ is then given by the
convolution of the point spread function and the phase of the middle lens.

$\displaystyle\begin{split}H\ast
l&=\frac{1}{2\pi\sigma^{2}}\frac{1}{\sqrt{\frac{2a_{1}^{2}}{f_{\text{ext}}^{2}g^{2}\lambda^{2}\pi}+\frac{1}{f_{\text{ext}}\lambda}}}\exp\left(\frac{2a_{1}^{2}\pi
x^{2}}{j2a_{1}^{2}f_{\text{mid}}\lambda-
f_{\text{ext}}^{2}g^{2}\pi\lambda}\right)\\\
&=\underbrace{(1+\alpha^{2})^{-1/4}\exp\left(-\frac{\alpha}{1+\alpha^{2}}\frac{\pi
x^{2}}{f_{\text{mid}}\lambda}\right)}_{\text{attenuation}}\\\
&\qquad\times\underbrace{\exp\left(j\frac{\arctan\alpha}{2}\right)}_{\text{global
phase}}\underbrace{\exp\left(-j\frac{1}{1+\alpha^{2}}\frac{\pi
x^{2}}{f_{\text{mid}}\lambda}\right)}_{\text{phase}}\end{split}$

where $\alpha=2\pi\sigma^{2}/f_{\text{mid}}\lambda=g^{2}\pi
f_{\text{ext}}^{2}\lambda/2f_{\text{mid}}a_{1}^{2}$. Here, we can ignore the
effects of the global phase and radial attenuation, and extract the
compression ratios from this new phase. Recognizing the spaceplate phase as
$\varphi_{\text{SP}}=-\pi x^{2}/f_{\text{mid}}\lambda(1+\alpha^{2})$, we can
equate this to Eq. (1) to solve for the compression ratio:

$\displaystyle\mathcal{R}_{4f_{\text{ext}}}=\frac{1}{1+\alpha^{2}}\frac{f_{\text{ext}}}{4f_{\text{mid}}}\Rightarrow\mathcal{R}=\frac{2f_{\text{ext}}f_{\text{mid}}}{4f_{\text{mid}}^{2}+\alpha^{2}f_{\text{ext}}^{4}}-1.$
(7)

Note that as the aperture of the first lens gets larger, $\alpha$ goes to
zero, and the compression limit approaches our original expression [Eq. (4)].
Eq. (7) is plotted in Fig. (8c) and (8d): in c), 2D color plots compare the
compression ratios achieved given a pair of focal lengths $f_{\text{ext}}$ and
$f_{\text{mid}}$, and it is clear that diffraction results in a maximally
achievable compression ratio. For example, in d) we set
$f_{\text{ext}}=$150\text{\,}\mathrm{m}\mathrm{m}$$ to match our experimental
parameters and then sweep $f_{\text{mid}}^{-1}$. As expected, Eq. (4) goes
linearly with $f_{\text{mid}}^{-1}$ where as the diffractive considerations
result in a global compression ratio maximum. Due to the achievable focal
lengths and aperture sizes, however, one would be hard pressed to observe this
effect in a real system: Fig. (8c) assumes an initial lens diameter of
$10\text{\,}\mathrm{m}\mathrm{m}$ which is considerably smaller than
achievable lens diameters. Even then, one would need middle lenses with focal
lengths smaller than two or three millimeters to observe this limitation in
compression ratio. This effect is easily designed around.