single=true te short = TE, long = transverse electric, foreign-plural = tm
short = TM, long = transverse magnetic, foreign-plural = tod short = TOD, long
= third order dispersion, foreign-plural = pcf short = PCF, long = photonic
crystal fibre, foreign-plural = sr-pcf short = SR-PCF, long = single-ring
photonic crystal fibre, foreign-plural = hcf short = HCF, long = hollow core
fibre, foreign-plural = wgm short = WGM, long = whispering gallery mode,
foreign-plural = nir short = NIR, long = near infrared, foreign-plural = cd
short = CD, long = core diameter, foreign-plural = fft short = FFT, long =
Fourier transform, foreign-plural = od short = OD, long = outer diameter,
foreign-plural = sem short = SEM, long = scanning electron micrograph,
foreign-plural = rs short = RS, short-plural = , long = rotation stage,
foreign-plural = ts short = TS, long = translation stage, foreign-plural = mmf
short = MMF, long = multimode fibre, foreign-plural = zdw short = ZDW, long =
zero dispersion wavelength, foreign-plural = uv short = UV, long =
ultraviolet, foreign-plural = euv short = EUV, long = extreme ultraviolet,
foreign-plural = mir short = mid-IR, long = mid-infrared, foreign-plural = gvd
short = GVD, long = group velocity dispersion, foreign-plural = spm short =
SPM, long = self-phase modulation, foreign-plural = hhg short = HHG, long =
high harmonic generation, foreign-plural = arr-pcf short = ARR-PCF, long =
anti-resonant-reflection photonic crystal fibres, foreign-plural = PDMS short
= PDMS, long = polydimethylsiloxane, foreign-plural = MoO short = -MoO3, long
= -molybdenum trioxide, foreign-plural = VO short = -V2O5, long = -vanadium
oxide, foreign-plural = AFM short = AFM, long = atomic force microscopy,
foreign-plural = FTIR short = FTIR, long = Fourier transform infrared,
foreign-plural = MCT short = MCT, long = mercury cadmium telluride, foreign-
plural = TMM short = TMM, long = transfer matrix method, foreign-plural = FP
short = FP, long = Fabry–Pérot, foreign-plural = TO short = TO, long =
transverse optical, foreign-plural = LO short = LO, long = longitudinal
optical, foreign-plural = ER short = ER, long = extinction ratio, foreign-
plural = IL short = IL, long = insertion loss, foreign-plural = vdW short =
vdW, long = van der Waals, foreign-plural =

# Deep-subwavelength Phase Retarders at Mid-Infrared Frequencies
with van der Waals Flakes

Michael T. Enders  Mitradeep Sarkar  Aleksandra Deeva  Maxime Giteau  Hanan
Herzig Sheinfux  ICFO — Institut de Ciencies Fotoniques, The Barcelona
Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
Mehrdad Shokooh-Saremi  Department of Electrical Engineering, Ferdowsi
University of Mashhad, Mashhad 91779-48944, Iran  Frank H.L. Koppens  Georgia
T. Papadakis<EMAIL_ADDRESS>ICFO — Institut de Ciencies
Fotoniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain

(August 28, 2024)

###### Abstract

Phase retardation is a cornerstone of modern optics, yet, at mir frequencies,
it remains a major challenge due to the scarcity of simultaneously transparent
and birefringent crystals. Most materials resonantly absorb due to lattice
vibrations occurring at mir frequencies, and natural birefringence is weak,
calling for hundreds of microns to millimeters-thick phase retarders for
sufficient polarization rotation. We demonstrate mir phase retardation with
flakes of MoO that are more than ten times thinner than the operational
wavelength, achieving $90$ degrees polarization rotation within one micrometer
of material. We report conversion ratios above $50\text{\,}\mathrm{\char
37\relax}$ in reflection and transmission mode, and wavelength tunability by
several micrometers. Our results showcase that exfoliated flakes of low-
dimensional crystals can serve as a platform for mir miniaturized integrated
polarization control.

##  I Introduction

From understanding the early universe [1] and mapping our galaxy [2] to
encoding quantum information in photons [3], the polarization of an
electromagnetic field is an important internal property of light. Controlling
light’s polarization is also an indispensable part of modern life [4],
spanning applications in sensing [5], imaging [6], microscopy [7], and
telecommunications [8]. Phase retardation in particular, the process of
introducing a phase difference between two orthogonal polarization states, is
a crucial building block of modern optics. This phase difference is typically
introduced by a phase retarder (or wave plate), which converts between linear
and circular polarization and can achieve any elliptically polarized state in-
between [9]. Despite the central role phase retardation plays in manipulating
photons, for a large portion of the mir spectral range, namely at wavelengths
above the central wavelength of the CO2 laser of
$10.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$, commercial wave plates are
extremely scarce.

The lack of phase retardation schemes at mid-IR frequencies is an important
roadblock in photonics, as this spectral range is fundamental to both science
and technology, and currently experiencing rapid advancements [17]. Plank’s
law of thermal radiation predicts a peak of thermal emission at wavelengths
between $10\text{\,}\mathrm{\SIUnitSymbolMicro
m}12\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for near-room temperatures, thus
enabling a variety of light-harvesting applications including contactless
temperature regulation, renewable energy [18, 19, 20], and thermal camouflage
[21]. The mir range is also crucial in astronomy, for example mir polarimetry
enables probing hot interstellar matter in our galaxy [22, 23]. Additionally,
the earth’s atmosphere becomes transparent in this range, making mir night
vision [24] and daytime radiative cooling [25] feasible and relevant, as well
as paving the way for mir communications [26]. Molecular vibrations also occur
in the mir range, thereby mir sensing [27], detection [28, 29], and medical
microscopy [30] depend on controlling mir photons.

Like molecular vibrations, however, crystal lattice vibrations (phonons) in
solids also occur at the mid-IR frequencies, yielding strong resonant
absorption in all polar dielectrics. This greatly limits the range of
available materials for polarization control elements and explains the
technological gap in mid-IR phase retarders. In particular, for efficient
phase retardation, transparent materials with strong intrinsic material
birefringence, defined as $\Delta n=\left|n_{\text{o}}-n_{\text{e}}\right|$,
where $n_{\text{o}}$ and $n_{\text{e}}$ are the ordinary and extraordinary
refractive indices of a uniaxial material, are required. Most materials that
retain some degree of birefringence at mir frequencies resonantly absorb.
Furthermore, birefringence is a weak effect and remains below unity ($\Delta
n\ll 1$) in bulk crystals, introducing major scalability challenges in
integrated mir photonics. To understand this, let us consider the principle of
operation of a phase retarder composed of a lossless uniaxial crystal. Its
optimal thickness is given by:

$d_{\lambda/\rho}=\frac{\lambda}{\rho\Delta n},$ (1)

where $\lambda$ is the wavelength of operation. Eq. 1 gives the ideal
thickness ($d_{\lambda/\rho}$) for half-wave plate ($\rho=2$) or quarter-wave
plate ($\rho=4$) operation, rotating linear polarization and converting it to
circular polarization, respectively. To compensate for weak natural
birefringence at mir frequencies, wave plates with thicknesses ranging from
hundreds of microns to millimeters are required for sufficient phase
retardation [31]. This is shown in LABEL:fig1:material-comparison, which
presents a map of available birefringent materials that can operate as half-
wave plates and the relevant spectral range within which they remain
transparent.

Conventional materials like CaCO3 in the near-IR or AgGaSe2 in the mid-IR,
shown in LABEL:fig1:material-comparison, ought to be hundreds of wavelengths
thick to perform as half-wave plates and fall within the class of “optically
thick” components (cyan highlighted region). The only available substitute for
mid-IR wave plates are Fresnel rhomb retarders. These are even bulkier,
approaching in size the centimeter-scale. To reduce size, one can engineer
strong birefringence ($\Delta n>1$) for selected wavelengths using form
birefringence, for example in liquid crystals [33, 34], or by design, using
topology-optimized metasurfaces [35]. Nonetheless, these methods often require
complex lithographic steps, and remain limited in their scalability. Reducing
the dimensions of mir phase retardation schemes will resolve major practical
challenges including mechanical instabilities, beam incoherence, overheating,
thus enabling miniaturization for on-chip and integrated optics applications,
where device compactness is required [36, 37].

In this article, instead of conventional phase retardation schemes with
lossless anisotropic crystals, we consider a different class of materials as a
platform for mid-IR phase retardation: vdW, low-dimensional, in-plane
anisotropic, highly dispersive materials. Several low-dimensional materials,
such as MoO [38, 39, 40] and VO [41], have been recently considered for their
in-plane hyperbolic dispersion, leading to a plethora of intriguing phenomena.
Examples include negative refraction [42, 43] and reflection [44], reversed
Cherenkov radiation [45], and topological [46, 47], configurable and tunable
[48, 43], steerable [49] phonon polaritons. By contrast, here, we utilize the
hyperbolic response of MoO as an extreme case of birefringence to demonstrate
a practical mechanism: phase retardation, for which a hyperbolic response is
beneficial but not a requirement.

In particular, we measure experimentally the colossal resonant birefringence,
approaching $\Delta n=8$ (LABEL:fig1:permittivity), owing to the ultra-long
lifetimes of phonon polaritons in MoO along orthogonal crystallographic
directions. This huge $\Delta n$ is ideal for polarization control, as Folland
and Caldwell commented in 2018 [50]. By Eq. 1, in the lossless-limit,
increasing birefringence $\Delta n$ can reduce the thickness of a phase
retarder. Considering the measured birefringence of MoO and its frequency
dispersion, we show in LABEL:fig1:material-comparison that it is placed near
the top-left corner of the map (purple highlighted region), in the extreme
sub-wavelength regime, where the notation “$x$” and “$y$” corresponds to a
phonon resonance along the $x$\- and $y$-crystal direction, respectively. This
map shows that by leveraging the birefringence of MoO, the required thickness
for sufficient phase retardation reduces by up to three orders of magnitude
with respect to the operation wavelength, as compared to conventional bulk
crystals. Recently, thin-film polarizers composed of MoO were demonstrated
that reflect one linear polarization and dissipate the other [51]. Dereshgi et
al. experimentally demonstrated a mir phase retarder employing MoO [52];
however these results pertain to heterostructure, requiring multiple steps of
deposition and transfer. These devices, only working in reflection, exhibited
high resonant absorption due to the optical phonons, and could only reflect a
small portion ($15\text{\,}\mathrm{\char 37\relax}$) of the incoming light
into the cross-polarized state.

In contrast, we probe experimentally the polarization rotation as well as the
ellipticity of mid-IR photons, and showcase that single vdW flakes of
anisotropic crystals suffice and can serve a practical material platform for
miniaturized mir phase retardation in either reflection or transmission, with
improved performance metrics. Despite the strong frequency dispersion and
resonant absorption of MoO (LABEL:fig1:permittivityc) near the wavelength
range of operation, insertion losses remain small due to the negligible
thickness of the flakes as compared to the wavelength. With respect to the
canvas of available materials, as a final remark, we note that a class of
quasi one-dimensional chalcogenides has recently been introduced [16, 53],
having birefringence comparable to MoO in the mir, as shown in
LABEL:fig1:material-comparison for BaTiS3. Nonetheless, no phase retardation
devices operating in the mir have been demonstrated so far, even with such
material systems.

To our knowledge, our results mark the first reported phase retarders in the
mid-IR above $11\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Our results also
demonstrate the thinnest reported phase retarders relative to the operation
wavelength, for all spectral ranges. Finally, these do not require any
lithography, thus reducing simultaneously design complexity and material
volume with respect to existing phase retardation schemes. The concepts shown
in this work with MoO for the sake of demonstration also apply to VO (see
previous numerical results in [54]), which has the same type of anisotropy as
MoO at larger wavelengths [41].

##  II Results

We mechanically exfoliate flakes of MoO from bulk crystals and transfer them
to silicon and gold-coated glass substrates in order to realize phase
retarders that operate in transmission and reflection mode, respectively.
Exfoliated flakes of MoO typically possess a rectangular shape, due to the
crystal’s orthorhombic structure [40] (LABEL:fig1:permittivitya). We refer to
the crystal direction $[100]$ as the $x$-axis, to the $[001]$ as the $y$-axis,
and to the $[010]$ direction as the $z$-axis. We measure the dielectric
permittivity of MoO using FTIR spectroscopy, implementing the method described
in [32]. The results we obtain, shown in LABEL:fig1:permittivityb and
LABEL:fig1:permittivityc, are in agreement with previous reports [38, 39, 40].
As shown in panel b, near $12\text{\,}\mathrm{\SIUnitSymbolMicro m}$, the
dielectric function along the $x$-axis ($\epsilon_{\mathrm{x}}$) resonates due
to a TO phonon occurring at the corresponding photon energy, while
$\epsilon_{\mathrm{y}}$ remains very small. Thus, the birefringence of MoO in
this spectral region reaches values near 15.

Aiming to achieve $90$ degrees polarization rotation, we note that, while Eq.
1 appropriately describes the operation of optically thick and lossless
dielectric wave plates, the class of vdW materials considered here requires
taking into account their strong frequency dispersion due to phonon resonances
as well as the non-negligible dichroism near-resonance. The latter makes
orthogonal components of the electromagnetic fields experience different level
of losses, whereas the former yields strong sensitivity to the optimal
operational wavelength with respect to small changes in a flake’s thickness.
Another important factor is the extremely small thickness of the phase
retarders as compared to the wavelength. In this deeply subwavelength region,
interference effects take precedence in describing the conversion process. It
is thus essential to consider not only the phase acquired upon propagation
within the slab, but also the reflection and transmission at its boundaries
[55]. In the following sections, we utilize dichroism to effectively leverage
constructive interference so that light maximally experiences the anisotropy
of the crystal. Through a combination of simulations and experiments, we
illustrate our approach to achieve significant polarization conversion while
maintaining low insertion losses.

The topography of exfoliated flakes is characterized with AFM, while their
spectral transmittance and reflectance is measured with FTIR micro-
spectroscopy. To characterize the flakes, we control the input beam’s
polarization and read the output beam’s polarization using two wire-grid
polarizers, in both transmission and reflection modes, as shown in
LABEL:fig2:setup and LABEL:fig3:setup, respectively. The input polarizer is
oriented at $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ with respect to the
crystal’s axes, as shown with the blue straight line in the sketches next to
the setup in LABEL:fig2:setup and LABEL:fig3:setup. This is a sketch of the
polarization ellipse of the input beam with respect to the two
crystallographic axes. As a result of the input polarizer, the input
polarization ellipse reduces to a straight line, oriented at an angle of
$45\text{\,}\mathrm{\SIUnitSymbolDegree}$ with respect to the $x$-axis.
Correspondingly, with the orange shape in the sketch next to the output
polarizer in LABEL:fig2:setup and LABEL:fig3:setup, we represent a general
elliptical polarization state with arbitrary ellipticity, defined by the
quantity $e=\sqrt{1-b^{2}/a^{2}}$, with $a$ and $b$ being the half-lengths of
the major and minor axes of the ellipse, respectively. For linearly polarized
output light, the ellipse collapses into a line $e=1$, whereas circularly
polarized light traces a circle $e=0$, corresponding to circular polarization.
In the same schematics of LABEL:fig2:setup and LABEL:fig3:setup, we denote
$\psi$, which is the rotation angle of the principal axis of the output beam’s
polarization ellipse, with respect to the input polarization, termed
henceforth polarization rotation. In the case of a lossless dielectric
crystal, the representation of a phase retarder via the pair ($e$, $\psi$)
reduces to the standard description via the retardance $\delta$ ($\delta=\pi$
and $\delta=\pi/2$ correspond to a half-wave plate and quarter-wave plate
operation, respectively), quoted in commercial wave plates.

We use simulations of reflectance and transmittance, performed using an in-
house implementation of the TMM [56, 57], to estimate the optimal flake
thickness for which the polarization rotation approaches $90$ degrees in the
vicinity of $12\text{\,}\mathrm{\SIUnitSymbolMicro m}$, where birefringence is
maximal. As shown in LABEL:fig2:trans-heatmap and LABEL:fig3:ref-heatmap, for
both transmission and reflection, this occurs for thicknesses near
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$. This thickness yields constructive
interference within the flake, thus light maximally experiences the anisotropy
of the crystal resulting in large linear polarization rotation while
suppressing multiple reflections [58]. In transmission mode (LABEL:fig2:full),
we define the polarization conversion ratio as
$T_{\perp}/\left(T_{\parallel}+T_{\perp}\right)$ and the phase retarder’s IL
as $-10\log(T_{\perp}+T_{\parallel})$, where $T_{\perp}$ and $T_{\parallel}$
are the measured transmittance with the output polarizer perpendicular and
parallel to the input one, respectively. Both $T_{\perp}$ and $T_{\parallel}$
are normalized to $T_{\parallel,sub}$, taken on the bare silicon substrate. In
this notation, therefore, $T_{\perp}$ denotes the absolute conversion.
LABEL:fig2:trans-spectrum shows the transmittance spectra measured for a flake
of thickness $1.39\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (inset shows the AFM
data). As can be seen from $T_{\perp}$, maximum polarization rotation occurs
at $12.95\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and the absolute conversion
is approximately $50\text{\,}\mathrm{\char 37\relax}$.

Using the TMM, we show in LABEL:fig2:trans-heatmap a color map representing
the calculated polarization conversion ratio as a function of flake’s
thickness and wavelength. To gain a better understanding of the position of
the maximum conversion ratio, we extract the ellipticity $e$ and the
polarization rotation angle $\psi$ for each wavelength and thickness. The
solid blue curve in LABEL:fig2:trans-heatmap shows the points where the output
polarization is purely linear, meaning $e=1$. Evidently, the maximum
conversion ratio is achieved when
$\psi=$90\text{\,}\mathrm{\SIUnitSymbolDegree}$$ along this line. The dashed
black contours mark the areas where $\psi$ is within
$\pm$5\text{\,}\mathrm{\SIUnitSymbolDegree}$$ (inner contour line) and
$\pm$10\text{\,}\mathrm{\SIUnitSymbolDegree}$$ (outer contour line) from
$\psi=$90\text{\,}\mathrm{\SIUnitSymbolDegree}$$, hence the intersection
between the blue curve and the dashed contours represents near-ideal quarter-
wave plate operation. Experimental results are superimposed and shown as green
dots, each corresponding to a particular flake with given thickness
($x$-axis). The wavelength at which each measured data point is placed
corresponds to the wavelength at which optimal polarization conversion
($y$-axis) was measured. We observe good agreement between the experimental
results and the solid line representing purely linear polarization (half-wave
plate operation). Notably, three of our measurements, corresponding to three
different flakes with different thicknesses, fall within a polarization
rotation angle range between
$80\text{\,}\mathrm{\SIUnitSymbolDegree}100\text{\,}\mathrm{\SIUnitSymbolDegree}$
with high conversion ratios. From LABEL:fig2:trans-conversion, the maximum
experimentally measured polarization conversion ratio is $0.67$, and is
achieved for $d\approx$1.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ for the
wavelength of $12.95\text{\,}\mathrm{\SIUnitSymbolMicro m}$ as shown by the
vertical arrow. For all the experimental data points, the IL remains below
$1.3\text{\,}\mathrm{dB}$. This was expected from the discussion above, owning
to the deeply subwavelength thickness of the flakes, compensating for the
large $\mathrm{Im}(\epsilon_{\mathrm{x}})$ (LABEL:fig1:permittivityc).

We repeat the phase retardation characterization in reflection mode
(LABEL:fig3:full), where $R_{\parallel}$ and $R_{\perp}$ correspond to the
output polarizer parallel and perpendicular to the input polarizer,
respectively, as shown in LABEL:fig3:setup, normalized to the gold-coated
substrate. We present in LABEL:fig3:ref-spectrum the measured reflectance
spectra for a flake of thickness $920\text{\,}\mathrm{nm}$ (AFM measurement in
inset). For this flake, the maximum absolute conversion ($R_{\perp}$) is
$64\text{\,}\mathrm{\char 37\relax}$ near
$13.9\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The maximum conversion is
determined for each flake in the same manner as in transmission mode
(LABEL:fig2:full), and varies for different flakes and different wavelengths
depending on thickness. The green data points in LABEL:fig3:ref-heatmap
represent these experimental results, and are superimposed with the
corresponding TMM numerical calculations. The solid blue curve corresponds to
pure linear polarization with $e=1$. Unlike the transmission mode, in
reflection mode, due to additional losses in the Au, the maximum predicted
polarization rotation $\psi$ deviates from
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$, as also verified experimentally
(Fig. 5). The maximum polarization rotation angle predicted is
$75\text{\,}\mathrm{\SIUnitSymbolDegree}$. The regions where $\psi$ exceeds
$70\text{\,}\mathrm{\SIUnitSymbolDegree}$ and
$60\text{\,}\mathrm{\SIUnitSymbolDegree}$ are depicted by the inner and outer
black dashed contour lines in white, respectively. The intersection between
the blue curve and the dashed contours represents operation as close as
possible to quarter-wave plate in reflection mode. Experimentally, although we
do not expect maximal polarization rotation of
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$, we still measure considerably high
conversion ratio from one polarization state to its orthogonal counterpart. In
reflection mode, light traverses the flake twice, thus experiencing the strong
birefringence twice. This yields high conversion for a wider range of
wavelengths than in transmission mode, as expected. This is shown in
LABEL:fig3:ref-heatmap, where one can observe the potential to passively tune
the operational wavelength and achieve linear polarization conversion by
controlling the thickness of the flake within the range of
$0.45\text{\,}\mathrm{\SIUnitSymbolMicro
m}1.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$. This enables operation within
the wavelength range of $12.6\text{\,}\mathrm{\SIUnitSymbolMicro
m}15\text{\,}\mathrm{\SIUnitSymbolMicro m}$, covering the entire Reststrahlen
band of the MoO (LABEL:fig1:permittivityb), demonstrating that this effect
does not stem from the hyperbolic behavior of MoO but rather from the resonant
nature of its birefringence. Finally, as shown in LABEL:fig3:ref-conversion,
we obtain conversion ratios up to $0.82$, with IL remaining below
$1.7\text{\,}\mathrm{dB}$ for all measured flakes.

(b)

(d)

Fig. 5: Polarization state characterization. The polarization rotation angle
$\psi$ of the transmitted and reflected beam in each mode of operation are
reconstructed in (a) transmission (flake thickness:
$1.39\text{\,}\mathrm{\SIUnitSymbolMicro m}$, silicon substrate) and (c)
reflection (flake thickness: $0.92\text{\,}\mathrm{\SIUnitSymbolMicro m}$, on
Au). The ellipticity $e$ for transmission and reflection is shown in (b) and
(d), respectively. The ribbon around the lines shows the standard deviation of
the numerically fitted parameters. The orange shaded region shows the spectral
range where the rotation $\psi$ is close to
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$.

The results in LABEL:fig2:full and LABEL:fig3:full describe the intensity of
the output beam’s component that is $90$ degrees rotated with respect to the
input beam’s polarization. Nonetheless, these measurements do not suffice in
characterizing a phase retarder. In particular in the case of considerable
frequency dispersion and absorption (LABEL:fig1:permittivityb and
LABEL:fig1:permittivityc), one ought to measure the ellipticity of the output
beam, as linear polarization is not necessarily preserved. To do this
explicitly (while probing depolarization and handedness), the mere use of two
polarizers is insufficient, as one requires an additional phase retarder.
Nonetheless, as mentioned in the introduction, commercial phase retarders are
not available in the mir range. Hence, instead, following basic principles of
ellipsometry [59], we illuminate the samples with linearly polarized light and
measure the outgoing light in two configurations while rotating the sample:
with the two linear polarizers parallel and orthogonal to each other. Assuming
absence of depolarization, as expected in the deep-subwavelength range that
these MoO flakes operate, via a numerical fit to the experimental data, we
determine $\psi$ and $e$ as a function of the wavelength [60]. These are shown
in Fig. 5b and Fig. 5d for transmission and reflection mode, respectively,
whereas the corresponding ellipticity is shown in Fig. 5b and Fig. 5d.

As shown in Fig. 5b, in transmission mode, the polarization rotation angle
reaches $90$ degrees for the spectral range from
$12.8\text{\,}\mathrm{\SIUnitSymbolMicro
m}13.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$, as was initially designed. The
ellipticity of the transmitted beam (Fig. 5b) deviates from the ideal value of
unity for linear polarization, as expected due to losses. Nonetheless, this
can be easily corrected with another polarization element on top of the flake
or by directly etching a wire grid structure into the sample. In reflection
mode, in Fig. 5d, we show that the rotation angle $\psi$ is approximately
$50\text{\,}\mathrm{\SIUnitSymbolDegree}$, as expected from our simulations
shown in LABEL:fig3:ref-heatmap. In Fig. 5d the ellipticity $e$ of the
reflected beam, however, is close to unity over the entire wavelength range
for which it was designed, as highlighted in the orange shaded region from
$13.6\text{\,}\mathrm{\SIUnitSymbolMicro
m}14.15\text{\,}\mathrm{\SIUnitSymbolMicro m}$. This shows that linear
polarization is preserved while being rotated. The deviations from the ideal
behavior of a half-wave plate operation both both flakes operating in
transmission and reflection mode are easilly corrected with polarizers that do
exist in the mid-IR range.

##  III Discussion

We design and characterize a wide range of phase retarders composed solely by
exfoliated flakes of MoO. By controlling the flakes’ thicknesses, we
demonstrate polarization rotation in both transmission and reflection mode,
with conversion ratios as high as $82\text{\,}\mathrm{\char 37\relax}$. Phase
retardation is demonstrated over the spectral range from
$12\text{\,}\mathrm{\SIUnitSymbolMicro m}15\text{\,}\mathrm{\SIUnitSymbolMicro
m}$, where bulk crystals perform poorly and commercial phase retarders are
scarce. Due to the deeply subwavelength thickness of the flakes, the
absorption of MoO does not considerably affect the performance of the phase
retarders, and insertion losses remain extremely low. Nonetheless, the losses
introduce a degree of circular polarization upon transmission, which, however,
can be corrected by further adjusting the flakes’ thickness and using
commercial polarizers.

Our experimental results demonstrate the potential of naturally in-plane
anisotropic vdW materials as highly efficient phase retarders. This phase
retardation is not associated with the hyperbolic dispersion of MoO, but
rather the result of the resonant features in its dielectric permittivity due
to crystal vibrations that occur at different frequencies along different
lattice directions. Our results open new possibilities for practical
applications in the mir spectral range, particularly beyond wavelengths of
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ where conventional, where
conventional bulk materials perform poorly. The principle of operation of the
phase retardation scheme discussed in this work is general and applies to any
polar dielectric low-dimensional anisotropic medium. Thus, these results can
be generalized to other spectral ranges, where other materials exhibit
Reststrahlen bands, for example near $18\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ using VO [41] or even to MoO itself for the resonance along the
$y$-direction (see LABEL:fig1:material-comparison). As crystal growth
techniques continue to improve, emerging vdW materials like MoO offer a new
paradigm as polarization control elements, beyond the reach of conventional
materials for integrated on-chip photonic devices.

##  IV Acknowledgments

This work is dedicated to the memory of John S. Papadakis. The authors declare
no competing financial interest. The authors would like to thank V. Pruneri
for generously granting access to his laboratory and equipment, as well as
engaging in fruitful discussions. G.T.P. acknowledges financial support from
the la Caixa Foundation (ID 100010434). M.E. acknowledges financial support
from MCIN/AEI/ 10.13039/501100011033 (PRE2020-094401) and FSE “El FSE invierte
en tu futuro”. M. G. acknowledges financial support from the Severo Ochoa
Excellence Fellowship. This work was supported by the Spanish MICINN
(PID2021-125441OA-I00, PID2020–112625GB-I00, and CEX2019-000910- S), the
European Union (fellowship LCF/BQ/PI21/11830019 under the Marie Skłodowska-
Curie Grant Agreement No. 847648), Generalitat de Catalunya (2021 SGR 01443)
through the CERCA program, Fundació Cellex, and Fundació Mir-Puig. F.H.L.K.
acknowledges support from the ERC TOPONANOP (726001), the government of Spain
(PID2019-106875GB-I00) and Generalitat de Catalunya (CERCA, AGAUR, 2021 SGR
01443). Furthermore, the research leading to these results has received
funding from the European Union’s Horizon 2020 under grant agreement no.
881603 (Graphene flagship Core3) and 820378 (Quantum flagship).

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