[1]Christopher Dirdal

[1]Smart Sensors and Microsystems, SINTEF Digital, Forskningsveien 1, Oslo,
0373, Norway

# Metasurface lens that is converging or diverging depending on transmission
direction enables ultra-compact MEMS tunable reflective lens.

Firehun Dullo<EMAIL_ADDRESS>Jesil Jose<EMAIL_ADDRESS>Gregory Bouquet<EMAIL_ADDRESS>Zeljko Skokic
<EMAIL_ADDRESS><EMAIL_ADDRESS>*

###### Abstract

A conventional refractive lens surface can act as a positive (converging) or
negative (diverging) lens, but the same surface cannot act as both. We show
that a geometric phase metasurface lens can have the unique property of acting
both as a positive or negative lens upon transmission through its front or
rear side, respectively. This offers certain freedom in compound lens design,
where one combines focusing and defocusing operations. We utilize this
property to make an ultra-compact, varifocal reflective lens, where a
metasurface is placed in front of a novel long-stroke piezoelectric MEMS-
micromirror. A large theoretical diopter tunability of 6330 m-1 is enabled due
to innovative thin-film piezoelectric MEMS design, offering 62 µm displacement
at 40V and low power, along with rapid actuation in the kHz region. The
achieved MEMS-displacement is an order of magnitude larger than previously
reported out-of-plane mechanical metasurface actuation. Since both metasurface
and micromirror are flat, the presented reflective lens can be assembled
without need for a spacer. It is therefore well suited for wafer-level silicon
fabrication at high volumes and low cost. A proof-of-concept implementation
using a 1550nm NIR metalens is demonstrated, attaining on the order of 1121
m-1 diopter change for a focal length shift of 270 µm caused by a 53 µm
micromirror displacement.

###### keywords:

Tunable metasurface, tunable lens, MEMS, piezoelectric MEMS

##  1 Introduction

The earliest known use of lenses dates back several thousands of years to
ancient Egypt [1]. A fairly modern concept of lenses arose already in 10th
century AD when they were understood by the Persian mathematician Ibn Sahl in
terms of refraction [2], and in 1609 Galileo presented an improved telescope
which used convergent and divergent lenses that are easily recognizable as
lenses today [3]. Throughout this long history, lenses have been either
focusing or defocusing, meaning that if e.g. a lens focuses light passing
through its front side, it will also focus light passing through its rear side
(and vice versa). From common experience, turning a magnifying glass around
does not change its optical function. This is because the property of focusing
or defocusing corresponds to the convex or concave geometry of a lens,
respectively (illustrated in Fig. 1a), and this property is kept for both
directions of light propagation. In this paper we demonstrate a notable
possibility of breaking this rule. One can make static surfaces that are both
focusing and defocusing depending on whether light propagates through their
front or back side, unlike previous lenses. We show that this can be achieved
using the recently developed geometric phase (also known as Pancharatnam-Berry
phase) metasurface lenses [4] under the condition that its sub-units are
reflection symmetric and that they are placed according to the circular
symmetry of the lens phase function. We show that particular design freedom is
thereby offered for compound lens design, when one wishes to combine focusing
and defocusing operations.

Until recently it has also been challenging to make compact lenses that can
tune their focusing ability. This is a highly desirable property in a wide
variety of optical systems. Changing the focal point and optical power
(measured in diopters) of a lens enables functionalities like projecting at
different distances, imaging at different depths or modulating the power
incident on an area. Traditional tunable optical lenses typically rely on
motors to displace lenses, leading to constraints in terms of size, cost,
power consumption and response time. However, recent developments in micro-
optical tunable lenses have paved a way for overcoming these limitations [5].

Micro-optics offer fast, small, lightweight, and cost-effective tunability of
relevance to miniaturized optical tunable lenses. These are properties
necessary for a wide range of applications from biological imaging to
applications on drones or robots [5, 6]. There exist micro-optical techniques
by which a single lens can be tuned (e.g. by deformation of polymer lenses [7,
8] or modulating the refractive index in e.g. liquid crystal lenses [9, 10]).
However, for faster actuation rates and stronger modulations, MEMS-displaced
lens doublets have emerged as a promising platform [11]: These include both
modulating the gaps between the lenses [12, 13] and sideways translation of
lenses of the Alvarez lens type [14, 15].

While traditional refractive lenses rely on curvature to achieve a specific
focal length, flat lenses utilizing subwavelength nanostructures have recently
emerged, known as metasurface lenses (metalenses). Whereas refractive lenses
are typically bulky and add weight, metalenses allow for a compact form
factor. These can furthermore be fabricated in silicon MEMS and CMOS
processing lines [16]. In addition to the benefits in form factor and
manufacturability, metasurface elements have been extensively explored for
their added degrees of freedom in manipulating light [5], also being
integrated in MEMS [17, 11] or other tunability schemes [18, 19]. For MEMS
based devices, the optical tunability is often constrained by the available
MEMS displacement range. On that note, thin-film piezoelectric MEMS
(piezoMEMS) has been proposed due to its ability to offer long-stroke quasi-
static displacement at low voltages. Thin-film piezoMEMS was first applied to
metasurfaces in our earlier work [20]. In another study, we presented a
varifocal doublet lens utilizing piezoMEMS, showcasing a maximum metalens
displacement of 7.2 µm [13]. Although this demonstrated a significant
advancement, there is still room for improving both the displacement range and
compactness of the system.

Having a lens which can both focus and defocus depending on transmission
direction gives certain design freedoms not available from conventional
lenses. For instance, this enables making a reflective equivalent to Galileo’s
telescope by placing such a lens in front of a mirror. Rather than make a
telescope, we combine this arrangement with long stroke piezoMEMS actuation to
enable an ultra-compact reflective lens doublet. Our resulting device offers
strong optical modulations and a diopter change on the order of 6330 m-1 by
displacing a piezoMEMS micromirror relative to a geometric phase metasurface
lens by 62 $\mu$m with 40V application. The concept and involved components
are shown in Fig. 1. The achieved MEMS-displacement range is more than an
order of magnitude larger than state of the art electrostatic actuation of a
metasurface doublet at half the voltage [12], and an order of magnitude larger
than in our previous work [13]. The lens can be operated up to kHz range and
is inherently low power due to utilizing thin-film piezoelectric PZT
actuators. An ultra-compact form factor is enabled by a unique feature of
using a geometric phase metalens: Notably, the metalens acts as a positive
(converging) lens in one direction and a negative (diverging) lens in the
other. Furthermore, the reflective arrangement of the device additionally
doubles the effective mirror displacement compared to transmissive varifocal
lenses. In combination these properties enable an ultra-compact lens which
offers strong modulations.

Figure 1: a) Conventional lenses either focus light or defocus light, whether
the incoming light is rightward or leftward through them. Contrary to this, a
geometric phase metasurface lens can enable focusing in one direction and
defocusing in the other. (b) A NIR geometric phase metasurface lens (300µm x
300µm) and (c) a SEM image of a portion of the metasurface lens, where the
scalebar represents 2µm. (d) A novel piezoelectric MEMS micromirror (die size
9 mm x 9 mm) capable of lifting the central mirror aperture up or down in a
piston motion. (e) An illustration of the compact reflective metalens (CRM)
concept where the dual converging/diverging metasurface on a silicon substrate
is mounted on top of the MEMS micromirror. As the central mirror is drawn away
from the metasurface, the focal length of the CRM is modulated. (f) An image
of the assembled metasurface and MEMS mirror (the metasurface is faced down
towards the mirror).

##  2 Theory

###  2.1 Reflective doublets

Figure 2(a) shows the sketch of a reflective tunable lens consisting of a
static lens with focal length $f$ placed in front of a mirror. The lens-mirror
separation is given by $L$. The mirrored (virtual) system is also drawn within
the gray box, revealing that this reflective arrangement is equivalent to a
transmissive doublet of twice the lens-mirror separation $2L$. By modulating
the separation distance, the incoming collimated light leaves the reflected
lens as either focused, defocused or collimated. In micro-optic
implementations of such a device, a MEMS mirror may be used for modulating the
distance $L$, where available modulation ranges are typically restricted to
some microns for electrostatic actuation [12] up to hundreds of microns for
thermal actuation [21]. The twice effective gap $2\Delta L$ of a reflective
arrangement relative to a transmissive lens doublet is therefore particularly
suited for MEMS-actuation where the lens-mirror displacement range is
constrained. In Sec. 3.2 we present a thin-film piezoelectric MEMS
architecture which offers lens-mirror displacements an order of magnitude
larger than the state-of-the art electrostatic and piezoelectric displacement
ranges.

In Fig. 2(a) collimation occurs when $L=f$, i.e. when the incoming collimated
light is focused to and reflected from the focal point of the lens. This is
the lens-mirror separation at which the largest changes in focal length occur:
Moving the focal point from infinite to finite distances. Therefore, a
separation on the order of $L\sim f$ is required for a tunable lens which can
offer large focal length changes. This is however a considerable gap for MEMS-
mirror actuation. If focal lengths are desired in the $f\sim 300$ µm range and
upward the necessary gap $L\sim 300\mu$m is already on the length scale of a
silicon wafer, and considerably supersedes typical MEMS-piston motion ranges.
Fabrication of a MEMS-implementation of such a lens could thus require the
addition of a spacer element between the MEMS and lens elements, which thus
may complicate wafer-level fabrication.

(a) Reflective lens

(b) Compact reflective metalens (CRM)

Figure 2: Two implementations of a reflective lens. (a) A reflective
arrangement using a conventional lens combined with a mirror. The mirrored
system is also drawn (within the gray box) to show that this reflective
arrangement is equivalent to a transmissive arrangement of twice the lens-
mirror spacing $2L$. A reflective arrangement is therefore useful when the
mirror actuation distance is limited. When using a conventional refractive
lens the lens-mirror gap $L$ must correspond to roughly the focal length
$L\sim f$ in order to attain large focal length change upon displacing the
mirror. With limited mirror displacement range, such an implementation may
necessitate placing a spacer between the mirror and the lens, which could
complicate wafer-level fabrication. (b) Using a geometric phase metasurface
lens (metalens) gives design freedom which allows the lens-mirror spacing to
be set to zero $L\to 0$ while maintaining large focal length changes. This is
because the metalens acts as a positive (converging) lens to the incoming path
and a negative (diverging) lens to the reflected path, enabling a reflective
doublet effectively consisting of a negative and positive lens.

###  2.2 Geometric phase enables compact reflective metalens (CRM)

We shall now present an alternative design which makes use of unique
properties of a geometric phase metasurface lens (GP-metalens). Fig. 2(b)
shows a similar arrangement for a tunable reflective lens, but this time a GP-
metalens is placed in front of the mirror. It will be shown that the GP-
metalens acts as a positive (converging) lens upon first transmission, and a
negative (diverging) lens upon second transmission, after being reflected by
the mirror. While the previous arrangement (Fig. 2(a)) is equivalent to a
transmissive doublet consisting of two equal positive lenses, the metasurface
variant in Fig. 2(b) is therefore equivalent with a transmissive doublet of a
positive and negative lens of equal focal length magnitude, for which the
largest changes in focal length occur near to zero lens-mirror gap $L\to 0$.
This removes the need for a spacer between lens and mirror, allowing for the
reflective lens to be ultra-compact and easily fabricated in wafer-level
processes.

The direction dependent focusing/defocusing of the GP-metalens shall now be
explained. Geometric phase (also known as Pancharatnam-Berry phase) allows for
pointwise implementation of a lens phase function by rotated structures,
working on circular polarization states of light [22, 4]. The operation
$\hat{T}$ of an array of identical metasurface elements on right $|R\rangle$
and left $|L\rangle$ circular polarized light can be shown to place the
transmitted light in a superposition of circular polarization states according
to

$\displaystyle\hat{T}|R\rangle$ $\displaystyle=$ $\displaystyle
B\exp(i2\alpha)|L\rangle+A|R\rangle,$ (1) $\displaystyle\hat{T}|L\rangle$
$\displaystyle=$ $\displaystyle A|L\rangle+B\exp(-i2\alpha)|R\rangle.$ (2)

As is observed, the resulting cross-polarized field has attained a phase $\pm
2\alpha$, where $\alpha$ can be shown to be equal the rotation angle of the
metasurface inclusions: For our implementation the inclusions are rectangular
pillars (Fig. 2). Full cross-polarization ($|A|^{2}\to 0$ and $|B|^{2}\to 1$)
can be achieved through suitable design of the metalens elements [16]. To
implement a positive (converging) lens phase function $\theta(r)$ which varies
circular-symmetrically with radius $r$ over the metalens, the rotations of the
rectangular pillars are varied pointwise according to $\alpha(r)=\theta(r)/2$
in the layout. Note that the resulting metasurface is thus not an array of
identical elements, and Eqs. (1) and (2) are therefore considered
approximations. Figures 3(a) and 3(b) illustrate the front and rear view of a
GP-metalens. It may be observed that flipping between front and rear face of
the metasurface is equivalent to a sign change in the rotation angles of the
rectangular pillars $\alpha(r)\to-\alpha(r)$, as indicated in the two zoom-in
portions of the metalens displayed (red solid and blue dashed squares). With a
sign change on all angles, the reflected metalens thus implements a lens
function of opposite sign $\theta(r)\to-\theta(r)$. Therefore, assuming the
front side of the metalens implements a positive (converging) lens, the
opposite side therefore implements a negative (diverging) lens, or vice versa.

Given the observation above, it is natural to ask whether flipping a geometric
phase metasurface always is equivalent changing the sign of the corresponding
rotation angles. It turns out that this is only the case when the sub-unit
structures are (i) individually reflection symmetric, and (ii) the rotations
of each equidistant pair of sub-units from the reflection axis are identical.
This is explained in further detail in Sec. 5. As a final remark on the
metalens design, note that the above discussion presumes that the incidence
light has the same circular polarization state traveling through the front and
rear of the metalens. Recalling from Eqs. (1)-(2) that the light is cross-
polarized after transmission, our compact-reflective lens (CRM) concept relies
on the mirror to ensure that the light has the same circular polarization
state upon transmission through the GP-metalens in both directions: After
first transmission through the GP-metalens, the mirror reverts the
polarization state from the cross-polarized light (e.g. $|L\rangle$) back to
the incident polarization state (i.e. $|R\rangle$), before the light re-enters
the GP-metalens the second time.

(a) Front view.

(b) Rear view (reflected metasurface).

(c) Cross-section

(d) Top view

Figure 3: Geometry of the metasurface.(a)-(b) Illustration of a geometric
phase (GP) metalens seen from front (a) and from rear side (b). The front and
rear view are interchanged by flipping the metalens around the y-axis (i.e.
the rotation axis). Doing so is seen to correspond to a sign change in the
rotation angle $\alpha\to-\alpha$ of the rectangles. Hence, if the rotation
angles $\alpha(r)$ corresponding with various radii $r$ are chosen so as to
implement the phase function of a lens $\theta(r)=2\alpha(r)$, then the one
side will correspond to a positive (converging) lens whereas the other will
correspond to a negative (divergent) lens for a given circular polarization
state. (c)-(d) SEM images of fabricated Silicon metasurface by UV-Nanoimprint
lithography and Bosch DRIE etching. The details of the fabrication process are
discussed in Sec. 3.1 and more details are given in [16].

###  2.3 Thin lens model of the Compact Reflective Lens

This section presents a thin-lens model of the CRM. Despite the simplicity of
the model it coincides well with both experimental measurements and Zemax
simulations (see Sec. 4 and Supplementary Information 2.1). In Fig. 4 relevant
parameters are defined for deriving the effective focal point distance $S_{2}$
of the reflective lens in terms of the lens-mirror spacing $L$, assuming
collimated incoming light and a GP-metalens designed to be positive upon first
transmission for right circular polarization $|R\rangle$ and negative for
second transmission of $R\rangle$ (after reflection from the mirror). The
dashed lines identify the refracted rays directed towards the lens focal point
at a distance $|f|$ from the GP-metalens. A triangle with base length $\delta$
is drawn to help express the distance to the object point $S_{1}$ for second
(reflected) transmission through the lens. Upon second transmission the GP-
metalen will act as a negative lens with focal length $-|f|$, for which
application of the thin lens equation gives the final focal point distance
from the GP-metalens:

$\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle-\frac{|f|(|f|-2\delta)}{2|f|-2\delta},$ (3) $\displaystyle=$
$\displaystyle-\frac{|f|(2L+|f|)}{2L}.$

The lens-mirror spacing is given by $L=\delta-|f|$ having set the origin at
the lens intersection with the optical axis (i.e. assuming all lens-mirror
spacings are negative $L=-|L|$). The solid curve in Fig. 4(b) plots the
normalized effective focal point $S_{2}/|f|$ vs the normalized lens-mirror
spacing $L/|f|$. As expected one observes rapid change in the focal length
$S_{2}$ close to zero gap $L=0$, and the effective focal point approaches
$S_{2}\to-|f|$ at large gaps (as expected, corresponding with the image point
for a negative lens when an object is placed at infinity). At $L=-0.25|f|$ the
effective focal length becomes equal to the focal length of the metalens
$S_{2}=|f|$. A MEMS-displacement range of 25% of the focal length of the lens
$|f|$ therefore allows to modulate the focal point between infinity and a
distance $|f|$ from the metalens.

Note that it is possible to combine the metalens with a refractive surface
beneath the metasurface structures or on the back side of the silicon
substrate. This allows also for shifting the curve of Fig. 4(b) so as to
achieve a finite focal length at zero lens-mirror gap. This design freedom is
further discussed in the SI 1.1.

The preceding discussion assumes a metalens designed to act as a positive lens
upon first transmission for right circular incoming light $|R\rangle$. In the
SI 1.2 it is shown that applying cross-polarized light (i.e. $|L\rangle$) to
CRM, the device becomes equivalent with a doublet consisting of consecutive
negative and positive lenses with equal focal length magnitudes.

(a)

(b)

Figure 4: (a) Same compact reflective lens (CRM) as illustrated in Fig. 2(b)
with defined quantities and ray-tracing for deriving the thin lens model in
Eqs. (3). (b) A normalized plot of effective focal lengths from GP-metalens of
the CRM upon normalized lens-mirror displacement by the thin-lens model when
assuming $|R\rangle$.

##  3 Implementation

###  3.1 Fabrication of metasurface lens

The metasurface design employed in this publications targets an operation
wavelength of 1550nm and is fabricated in silicon by use of UV-nanoimprint
lithography (UV-NIL) patterning of resist and subsequent Bosch Deep Reactive
Ion Etching (DRIE). The resutling structures are shown in Fig. 3(c) and 3(d).
Two metalenses were utilized, one with focal length $|f|=10$ mm and size 1.5
mm x 1.5 mm, and one with $|f|\approx 215$ $\mu$m with size 300µm x 300µm. The
latter lens does not have a well-defined focal length because its phase
function was found through Zemax optimization of the lens placed in the
doublet configuration utilized in this paper. As mentioned, the phase function
is implemented in terms of pointwise rotations of the silicon pillars at the
center of each unit cell. The targeted rectangular pillar dimensions of the
UV-NIL template are: Widths $w=237$nm, lengths $l=361$nm, heights $h=1200$nm,
and the pitches of the square unit cells are $p=835$nm.

As can be seen from Fig. 3(c) and 3(d), smooth sidewalls are attained despite
using the three steps passivation, de-passivation and isotropic etch of the
Bosch process: The Bosch process was tuned to aim for a lateral etch depth
(scallop depth) on the order of 10nm. A detailed account of this process
offering structures with good pattern fidelity and high focusing efficiency
have been earlier published [16].

###  3.2 Novel micromirror with long stroke piezoelectric MEMS actuation

We have previously demonstrated a piezo-electric actuated micromirror which
can achieve tilting motion of ±0.5° degrees and translational ”piston” motion
of 7.2µm [23, 13]. A suspended central circular gold mirror of diameter 3mm is
actuated by applying voltages to a thin-film PZT piezoelectric membrane
encircling it, deposited on typically 700µm wide and 10µm thin actuator arms.
A similar concept is realized here, using a cascade of concentric actuator-
rings to increase the stroke-length. Here, the downward/upward push/pull of
diameter 3 mm, 400 µm thick, planar gold-plated central mirror is generated by
actuating four independent segments of thin film PZT actuator-pairs. As shown
in Fig. 5, biasing the inner actuators relative to the mirror, push the mirror
down, while biasing the outer rings pull it up. It is foudn that 10 µm thick
membrane sections of 700 µm width maximize the piston stroke-length while
maintaining mirror-planarity and stability, given the mechanical boundary-
conditions for the present design.

A vertical displacement of up to ± 31 µm (a 62 µm total stroke) at 40V of
applied voltage has been measured using white light interferometry, as shown
in Fig. 5. This is an order of magnitude larger than that of our previously
reported micromirror [23, 13]. Versions in which the electrode segments have
been further divided into quadrants allow for enabling tip/tilt motion, e.g.
for deflecting a laser beam. Observing a reflected laser beam under tip/tilt
actuation from 0.5 kHz to 6 kHz reveal resonances at around 1.86 kHz, 2.8 kHz
and 4.5-5.5kHz.

Thin-film piezoelectric MEMS has the advantage of requiring ultra low power.
From similar architectures previously characterized we have measured power
consumption of a few 100nW at 23V. Thin-film piezoelectric PZT MEMS can offer
long-term durability and reliability making them relevant for wide range of
applications[24, 25].

Figure 5: Long-stroke MEMS-micromirror. (a) Cross-sectional height profile of
the MEMS micromirror along the line indicated in the inset (b), as measured by
white-light interferometry. The mirror height corresponding to three distinct
actuation states is shown. At a 40V application to the innermost electrodes,
the PZT membrane pulls the ring downward, placing the mirror at its lowest
position. Conversely, a 40V application to the outermost electrodes causes the
PZT membrane to draw the ring upward, resulting in the highest position. In
the absence of voltage, an intermediate position is achieved. Notably, a
displacement range of 62 µm is observed, and a high level of planarity is
maintained throughout movement. (c) Picture of a MEMS mirror. (d) Presents 3D
images of the MEMS mirror at the three specified actuation states.

##  4 Results

###  4.1 Verification of lens concept

To verify the reflective metalens concept, a mirror was displaced from the GP-
metalens using the setup illustrated in Fig. 6(a). First manual displacement
of the mirror was used with a metalens of focal length $|f|=10$mm and the
results are compared with Eq. (3) of the thin lens model, and thereafter the
MEMS micromirror presented in Sec. 3.2 was used together with a metalens of
focal length $|f|\approx 215\mu$m to verify the compact refletive metalens
(CRM) concept using the setup illustrated in Fig. 7(a).

As shown in Fig. 6(a), the lens-mirror doublet is illuminated by a collimated
laser beam from a 1.55µm fiber laser, which undergoes dual reflection off two
mirrors (for alignment). It then passes through a right-handed circular
polariser (CPR) to place it in right circular $|R\rangle$ state. A 50/50 non-
polarizing beam splitter (BS) is used for illuminating and collecting along
normal incidence angle. The light then travels through the GP-metalens, is
cross polarized to $|L\rangle$ and focused. The light then reflects off the
mirror positioned behind the GP-metalens, reverting the polarization state
from $|L\rangle$ to $|R\rangle$ again. The reflected light re-enters the GP-
metalens a second time, is cross-polarized to $|L\rangle$ and then diverged.
The light then passes through the beam splitter. The light for imaging is
directed through a x20 infinity-corrected microscope objective, a tube lens,
and an IR-camera. In reality, due to fabrication imperfections, not all light
is cross-polarized during the transmissions through the metalens. Improved
image quality is achieved by selecting only the $|L\rangle$ light, by use of a
left-handed circular polarizer (CPL) placed between the microscope objective
and the planoconvex lens. Both the metalens and mirror are mounted to an XYZ
translation stage for alignment and distance adjustments. Additionally, the
mirror is attached to a kinematic mount, allowing for a few degrees of tip and
tilt adjustment. The objective lens is mounted on a single-axis motorized
translation stage equipped with a controller for precise movements (with step
accuracy $1\mu$m).

Figures 6(c) and 6(d) show focusing and defocusing of the reflective lens by
changing the lens-mirror displacement. When the metalens and mirror are well-
aligned and parallel a strong back reflection (from the air-silicon interface)
occurs which obstructs the focused spot. However, by carefully adding a slight
mirror tilt, we successfully separated the focus spot and selectively bringing
it into the camera’s field of view and thus significantly reduce interference
from strong back reflections. Addition of an anti-reflectance coating would be
helpful in reducing such back reflections from the silicon substrate. To
compare the the performance of the reflective lens with the thin lens model of
Eq. (3), an experiment was performed where the focal distance was found for
various lens-mirror displacements and plotted in Fig. 6(b): For each position
of the objective lens, the focal point of the reflective metalens is brought
into view by modulating the lens-mirror distance. The focal length $S_{2}$ is
calculated by first finding the focal point with the objective lens, and then
subtracting the working distance of the objective lens (20mm) from the
measured distance between the silicon backside of the metalens to the
objective lens. The objective lens is moved using the motorized translation
stage. The lens-mirror separation $L$ is found from first bringing the
metalens and mirror into contact $L\to 0$ and then noting the rotation of the
micrometer screw which translates the mirror, as the lens-mirror gap is
increased. The minimum unit of the micrometer screw is 10µm. After finding the
focal spot with the objective lens for each measurement, slight adjustments to
the mirror position which leads to visible defocus on either side of the focus
are used to define the measurement uncertainty, and the average value is used
to calculate the actual lens-mirror spacing. The resulting measurements are
represented as blue dots (with error bars) in Fig. 6(b). For the metalens with
focal length $|f|=10$ mm a focal tunability of 4 cm was measured by changing
the separation between the mirror and metalens from 400 µm to 700 µm. Notably,
the measurement results closely align with the predictions of the thin lens
model given by Eq. (3). The thin lens model is furthermore verified with
results obtained from simulating the system in Zemax (red crosses) - more
information on this is given in SI 2.1.

(a) Experimental setup

(b) Prediction based on thin lens model and Zemax simulations, and
experimental results

(c) Focus

(d) Defocus

Figure 6: Tuning the effective focal length using reflective lens. (a)
Schematic of the optical setup. (b) Outcome from the thin lens model, Zemax
simulations and experimental results. Focus (c) and defocus (d) of the
reflective lens by changing the lens-mirror displacement.

To verify a compact reflective metalens (CRM) we used the long stroke
displacement MEMS mirror of Sec. 3.2 which offers 62 µm for an applied voltage
of 40V, as shown in Fig. 5. To ensure large diopter changes, a focal length is
chosen that corresponds with the available MEMS range (as discussed in Sec.
2.3): We combined the micromirror with the metalens of focal length
$|f|\approx 215\mu$m (Sec. 3.1). According to the normalized, thin-lens model
plot of Fig. 4(b), this MEMS-range should in theory allow for a considerable
diopter change of around 6330 m-1 by shifting the focal point from infinity to
157$\mu$m away from the metalens. Figures 7(c) and 7(d) show focusing and
defocusing of the CRM by changing the lens-MEMS mirror displacement. In Fig.
7(b) we present the change in focal length vs applied voltage to the MEMS-
actuated CRM. For the $|f|\approx 215\mu$m lens, the uncertainty of
measurement of both focal length and lens-mirror gap are considerable compared
to the involved length scales. At the starting point, where a voltage
$V_{\text{Outer}}=30$V is applied to the outer electrodes of the micromirror,
neither lens-mirror gap or focal distance are therefore known with certainty.
However, given the precise movement ($1\mu$m) of the objective lens due to
movement by the motorized stage, accurate measurements of the focal length
_change_ corresponding to voltage application to the outer and inner
electrodes of the micromirror are obtained and presented in Fig. 7(b).
Furthermore, micromirror displacements corresponding to voltage application
are known from separate white-light interferometric measurements (Fig. 5), and
are added to the right y-axis of Fig. 7(b). Combined, the experimental results
demonstrate a focal shift of $\Delta f_{\text{meas}}=270\mu$m for a
micromirror displacement of $\Delta L=53\mu$m from a actuation voltage of 30V
between outer and inner electrodes.

An estimate of the corresponding diopter change to the focal length change can
be obtained by fitting the measured focal length change $\Delta
f_{\text{meas}}$ and micromirror shift $\Delta L$ to the thin-lens model
curve, when accounting for the silicon substrate. Under paraxial assumptions
the correction factor $t(1-n_{\text{air}}/n_{\text{Si}})$ should be added to
the thin lens model focal length $S_{2}$, where $t=500\mu$m and
$n_{\text{Si}}=3.5$ are the thickness and refractive index of the silicon
substrate, respectively, and $n_{\text{air}}=1$ is the refractive index of
air. This fitting leads to estimated start and end focal lengths of
$f_{1}=650\mu$m and $f_{2}=376\mu$m, respectively, and corresponding lens-
mirror gaps of $L_{1}=-45.5\mu$m and $L_{2}=-98.5\mu$m (see SI 2.2 for more
details). From these values the diopter change is estimated to be 1121 m-1.

Given the substrate thickness $t=500\mu$m, the estimate $f_{2}=376\mu$m
indicates that the focal point of the CRM has entered into the silicon
substrate where both the focal point and the working distance of the objective
lens become shifted relative to their position in air due to refraction. Under
paraxial assumptions it is easy to show that the effects of the focal point
and working distance shifts compensate, so that the measured focal distance
under the setup described above is equivalent to measuring the focal distance
outside of the silicon (see SI 2.2 where also Zemax simulations are shown to
confirm this).

(a) Experimental setup

(b) MEMS actuation

(c) Focus

(d) Defocus

Figure 7: Tuning the effective focal length using CRM. (a) Schematic of the
optical setup. (b) Left axis: Experimentally measured focal length shift vs
voltage actuation. Right axis: Experimentally measured mirror displacement vs
voltage application. Focus (c) and defocus (d) of the CRM by changing the
lens- MEMS mirror displacement.

##  5 Discussion

An interesting property of the geometric phase metalens is that flipping it
around its central axis is equivalent to changing the signs on the rectangle
rotations, as discussed in Sec. 2.2. As mentioned there, this is not a general
result applicable to all geometric phase metasurfaces. The specific conditions
shall now be discussed. Mathematically, flipping the metalens around the
central axis (about the y axis in Fig. 3(b)) corresponds to a reflection
transform of the metalens. For a shape to equal its reflection transform apart
from differences in rotations, it must itself be reflection symmetric. This
can be shown by demanding that the sets of points of a shape and its
reflection transform coincide when the shapes are suitably rotated and
translated to the origin (SI 3). As a contrary example, one may for instance
draw right angled triangles between three corners of the shown rectangles
(i.e. non-reflection-symmetric sub-units) displayed in Fig. 3(a) and observe
how the reflection transform no longer corresponds to a sign change in
rotations. In addition to having reflection symmetric structures, it matters
how they are collectively placed: There must be identical placement of
structures on either side of the rotation axis. In Fig. 3(b) one can observe
pairwise identical rectangles placed on either side of the central column of
rectangles in every row. This ensures that the reflection transformed sub-
units are placed at the coordinates that correspond to their oppositely
rotated counterparts in the non-reflected metasurface. This requirement is
automatically fulfilled when implementing the phase function of a lens
$\theta(r)$ because of its circular symmetry. As a contrary example, if one
considers the reflection transform of the zoom in portion of Fig. 3(a)
(enclosed by a red solid square) placed over itself, it may be observed in
some of the rectangles that the transformation is not equivalent to a sign
change in rotation. This occurs due to the lack of pairwise identical
placement of rectangles about the central column of the zoom-in portion. As a
side note, observe that due to the inversion symmetry of the rectangular
structures used in this work (in addition to their reflection symmetry), the
metasurface in Fig. 3(a) attains inversion symmetry.

##  6 Conclusion

Throughout the history of optics, lenses have been either converging or
diverging. In this work we present a notable exception to this rule, using a
geometric phase metasurface lens (metalens) which can both act as a positive
(converging) or negative (diverging) lens, depending on transmission through
the front or rear of the lens, respectively. This is enabled by the geometric
phase principle and certain symmetry requirements outlined in this work. The
unique property of the metalens allows for certain freedom in design of
compound lenses. In this work we present a varifocal reflective doublet
concept that is equivalent to a transmissive doublet with a consecutive
positive and negative lens. This enables an ultra-compact tunable lens of
large diopter changes, which is only possible with the given metalens. The
lens works without the need for a spacer between the metalens and mirror,
making the device well-suited for wafer-level silicon fabrication at high
volumes and low cost.

A proof-of-concept implementation using a 1550nm NIR metalens is demonstrated,
coupled with a novel long stroke (62 µm at 40V) and high resonance frequency
($\sim 1.86$ kHz) piezoelectric MEMS-micromirror. The metasurface displacement
range achieved by this MEMS-mirror surpasses that of state-of-the-art
electrostatic actuation by more than an order of magnitude, and theoretically
enables a considerable diopter change on the order of 6330 m-1. Our proof-of-
concept implementation demonstrated a diopter change of around 1121 m-1. The
manuscript also presents a thin lens model of the compact reflective lens
(CRM) and explains the geometric phase metalens working principle.
Additionally, the manuscript provides a detailed account of the design and
working principle of the novel long-stroke MEMS-micromirror.

Experimental results validate the CRM concept, demonstrating focal tunability
through manual adjustments of the lens-mirror separation in the case of a 10mm
focal length metalens, and voltage-controlled displacement of a MEMS mirror
configuration in the case of a 215µm metalens. The experimental outcomes
closely align with simulations and the derived thin lens model.

Overall, the proposed reflective lens system offers a promising solution for
achieving fast and strong modulations in a ultra-compact form factor. The
large focal length shifts and diopter changes it enables is of relevance to a
wide range of micro-optical applications within sensing and imaging.

Supplementary information Please refer to the Supplementary Information for
further details.

Acknowledgments

This research was funded by EEA Financial Mechanism 2014–2021, under project
no. EEA-RO-NO2018-0438—ElastoMETA, and the ATTRACT programme that has received
funding from the European Union’s Horizon 2020 Research and Innovation
Programme under Grant Agreement no. 101004462. We thank our colleagues Anand
Summanwar and Karolina Milenko for assistance with fabricating the metalenses,
Aina Herbjørnrød for assistance with fabricating the micromirrors.

Author Contributions CD conceived of the presented idea and developed the
theory. GB performed the ray tracing simulations with input from CD. The
design and fabrication of metasurface and MEMS micromirror were lead by CD and
ZS, respectively. FD and JJ carried out the experiment with input from CD, GB
and ZS. JJ performed the micromirror characterization. CD and FD wrote the
manuscript with inputs from the other authors. All authors discussed the
results and contributed to the final manuscript.

## Declarations

The authors declare no conflicts of interest.

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$\pi$. Assume that $\Pi_{\infty}$ is relevant [Liu-FJcycles, Definition 1.2].
Then the following statements are equivalent.

  1. (1)

The twisted arithmetic functional $\mathrm{FJ}_{\epsilon}\not=0$.

  2. (2)

We have $L^{\prime}(1/2,\Pi,As_{E/F}\otimes\mu)\not=0$ and
$\mathrm{Hom}_{\mathbb{C}[\mathrm{U}(V)(\mathbb{A}_{f})]}(\pi_{f}\otimes\Omega(\mu,\epsilon),\mathbb{C})\not=0.$

There is a relative trace formula approach towards refinements of above
arithmetic twisted Gan-Gross-Prasad conjecture. Following [Liu-FJcycles], we
focus on the geometric side.

For test functions $f$, $f^{\vee}$ for $\pi,\pi^{\vee}$ respectively, and
$\phi\in\mathrm{Hom}_{E}(A_{K},A_{\mu},\epsilon)$ and
$\phi_{c}\in\mathrm{Hom}_{E}(A_{K},A_{\mu},-\epsilon)$. Consider the
Beilinson–Bloch–Poincaré (BBP) height pairing ([Liu-FJcycles, Conjecture 1.5])

$\mathrm{Int}(f,\phi,f^{\vee},\phi^{\vee})=\mathrm{vol}(K)^{2}\left<\mathrm{FJ}(f,\phi)_{K},\mathrm{FJ}(f^{\vee},\phi_{c})_{K}\right>_{Y_{K},A_{\mu}}^{\mathrm{BBP}}$

The BBP height pairing is defined via Beilinson–Bloch height pairings on
$Y_{K}\times X_{K}\times X_{K}$ using the Poincare line bundle (a Cartier
divisor) $\mathcal{P}_{\mu}\in\mathrm{CH}^{1}(A_{\mu}\times A_{\mu}^{\vee})$.

We may enlarge the field $E^{\mathrm{RSZ}}$ to a number field $E^{\prime}$
such that $D_{K}^{n-1}$ is a sum of $E$-rational points of $X_{K}$. Use
$(-)^{\prime}$ to denote the base change to $E^{\prime}$, for instance
$X_{K}^{\prime}=(X_{K})_{E^{\prime}}\to Y_{K}^{\prime}=(Y_{K})_{E^{\prime}}$
and $A_{\mu}^{\prime}=(A_{\mu})_{E^{\prime}}$. For a point $P\in
D_{K}^{n-1}(E^{\prime})$, we consider the Albanese morphisms
$\alpha_{P}:X_{K}^{\prime}\to A_{\mu}$.

We do a first reduction of the height pairing to Belinson-Bloch height pairing
on $Y_{K}^{\prime}\times X_{K}^{\prime}\times X_{K}^{\prime}$. For $P,Q\in
X_{K}(E^{\prime})$, let

$\Delta^{\phi,P}X:=(\mathrm{id}_{Y_{K}^{\prime}}\times(\phi^{\prime}\otimes\alpha_{P})_{*})(\Delta^{E_{0}\times
F_{0}}X_{K})\in\mathrm{CH}^{n-1+[M_{\mu}:\mathbb{Q}]/2}(Y_{K}^{\prime}\times
A_{\mu}^{\prime}),$
$\Delta^{\phi_{c},Q}X:=(\mathrm{id}_{Y_{K}^{\prime}}\times(\phi^{\prime}_{c}\otimes\alpha_{Q})_{*})(\Delta^{E_{0}\times
F_{0}}X_{K})\in\mathrm{CH}^{n-1+[M_{\mu}:\mathbb{Q}]/2}(Y_{K}^{\prime}\times
A_{\mu}^{\prime\vee}).$

We modify these cycles to cohomological trivial cycles by replacing
$\Delta^{E_{0}\times F_{0}}X_{K}$ by $\Delta^{E_{0}\times
F_{0}}_{z}X_{K}=z^{*}\Delta^{E_{0}\times F_{0}}X_{K}$ by taking a Hecke system
of projectors $z=(z_{K})_{K}:\mathrm{CH}^{*}(Y_{K}\times
X_{K})\to\mathrm{CH}^{*}(Y_{K}\times X_{K})^{0}$ [Liu-FJcycles, Definition
4.35]. We always use the notation $\Delta_{z}$ to mean taking the Hecke
projector $z$.

As [Liu-FJcycles, Lemma 5.1.], above BBP height pairing
$\mathrm{Int}(f,\phi,f^{\vee},\phi^{\vee})$ is a sum of similar height
pairings

$\left<(T^{f}_{K}\otimes
T_{\mu}^{\mathrm{can}})^{*}\Delta^{\phi,P}_{z}X,(T^{f^{\vee}}_{K}\otimes
T_{\mu^{c}}^{\mathrm{can}})^{*}\Delta^{\phi_{c},Q}_{z}X\right>_{Y_{K}^{\prime},A_{\mu}}^{\mathrm{BBP}}.$

indexed by $P,Q\in D_{K}^{n-1}(E^{\prime})$.

######  Definition 9.0.4.

Let the _weak magic divisor_ for $(\phi,\phi_{c})$ and $P,Q\in
X_{K}(E^{\prime})$ be the divisor

$\mathcal{Q}^{\phi,\phi^{c},P,Q}_{\mu,K}=(\phi^{\prime}\circ\alpha_{P})\times(\phi^{\prime}_{c}\circ\alpha_{Q})^{*}(T_{\mu}^{\mathrm{can},t}\otimes
T_{\mu}^{\mathrm{can}})^{*}\mathcal{P}_{\mu}\in\mathrm{CH}^{1}(X_{K}^{\prime}\times
X_{K}^{\prime}).$

Consider $(y,x_{1},x_{2})\in Y_{K}^{\prime}\times X_{K}^{\prime}\times
X_{K}^{\prime}$. We have the divisor
$Y_{K}^{\prime}\times\mathcal{Q}^{\phi,\phi^{c},P,Q}_{\mu,K}$ on
$Y_{K}^{\prime}\times X_{K}^{\prime}\times X_{K}^{\prime}$. Let
$p_{i}:Y_{K}^{\prime}\times X_{K}^{\prime}\times X_{K}^{\prime}\to
Y_{K}^{\prime}\times X_{K}^{\prime}$ be the projection map sending
$(y,x_{1},x_{2})$ to $(y,x_{i})$. Consider the cycle

$\Delta^{E_{0}\times F_{0}}_{f,f^{\vee}}X_{K}:=(T^{f^{t}*f^{\vee}}_{K}\times
id_{X_{K}^{\prime}})^{*}\Delta^{E_{0}\times
F_{0}}X_{K}\in\mathrm{CH}^{2(n-1)}(Y_{K}\times X_{K}).$

######  Proposition 9.0.5.

For $P,Q\in X_{K}(E^{\prime})$, we have

$\left<(T^{f}_{K}\otimes
T_{\mu}^{\mathrm{can}})^{*}\Delta^{\phi,P}_{z}X,(T^{f^{\vee}}_{K}\otimes
T_{\mu^{c}}^{\mathrm{can}})^{*}\Delta^{\phi_{c},Q}_{z}X\right>_{Y_{K}^{\prime},A_{\mu}}^{\mathrm{BBP}}$
$=(p_{1}^{*}\Delta_{z}^{E_{0}\times
F_{0}}X_{K},(Y_{K}^{\prime}\times\mathcal{Q}^{\phi,\phi^{c},P,Q}_{\mu,K}).p_{2}^{*}\Delta^{E_{0}\times
F_{0}}_{z,f,f^{\vee}}X_{K})^{\mathrm{BB}}_{Y_{K}^{\prime}\times
X_{K}^{\prime}\times X_{K}^{\prime}}.$

###### Proof.

The proof follows from the argument of [Liu-FJcycles, Lemma 5.2]. ∎

Now we need to understand the three cycles

$p_{1}^{*}\Delta^{E_{0}\times F_{0}}X_{K}=\\{(y,x_{1},x_{2})|y=x_{1}\\},\quad
p_{2}^{*}\Delta^{E_{0}\times
F_{0}}X_{K}=\\{(y,x_{1},x_{2})|y=x_{2}\\}\in\mathrm{CH}^{2(n-1)}(Y_{K}^{\prime}\times
X_{K}^{\prime}\times X_{K}^{\prime})$

and the divisor
$Y_{K}^{\prime}\times\mathcal{Q}^{\phi,\phi^{c},P,Q}_{\mu,K}\in\mathrm{CH}^{1}(Y_{K}^{\prime}\times
X_{K}^{\prime}\times X_{K}^{\prime})$. Weak magic divisors motivate the
definition of doubling divisors [Liu-FJcycles, Definition 5.8] and is related
to Kudla’s generating series of divisors (see also Proposition 6.3.2)

$Z(\phi)_{K}:=Z(h=1,\phi)\in\mathrm{CH}^{1}(X_{K})$

on $X_{K}$ for $\phi\in\mathcal{S}(V(\mathbb{A}_{f}))^{K}$.

Let $\mathrm{cl}_{B}:\mathrm{CH}^{1}(X_{K}^{\prime}\times X_{K}^{\prime})\to
H^{2}_{B}(X_{K}\times X_{K},\mathbb{C})$ by the cycle class map to Betti
cohomology group of $(X_{K}\times X_{K})(\mathbb{C})$. We have the following
”partial Fourier transform” provided by Kudla-Millson theory [Kudla-Millson].

######  Proposition 9.0.6.

([Liu-FJcycles, Lemma 5.5, Lemma 5.6]) There is a natural isomorphism of
$\mathbb{C}[\mathrm{U}(V)(\mathbb{A}_{f})]$-modules

(62)
$\mathfrak{d}:\Omega(\mu,\epsilon)\otimes_{M_{\mu}}\Omega(\mu_{c},-\epsilon)\otimes_{M_{\mu}}\mathbb{C}\cong\mathcal{S}(V(\mathbb{A}_{f})).$

Moreover, there is a unique $\mathbb{C}$-linear map

$c\mathcal{Q}_{\mu,K}:\mathcal{S}(V(\mathbb{A}_{f}))^{K}\to
H^{2}_{B}(X_{K}\times X_{K},\mathbb{C})$

such that
$c\mathcal{Q}_{\mu,K}(\mathfrak{\phi\otimes\phi_{c}})=\mathrm{cl}_{B}(\mathcal{Q}^{\phi,\phi^{c},P,Q}_{\mu,K})$,
and under the diagonal $\Delta:X_{K}\to X_{K}\times X_{K}$ we have

$\Delta^{*}c\mathcal{Q}_{\mu,K}(\phi)=\mathrm{cl}_{B}(Z(\phi)_{K}).$

######  Definition 9.0.7.

A codimension $1$ cycle $Z^{D}_{K}(\phi)$ on $X_{K}\times X_{K}$ is called a
doubling divisor (or magic divisor) of level $K$ for an element
$\phi\in\mathcal{S}(V(\mathbb{A}_{f}))^{K}$ if
$\mathrm{cl}_{B}(Z_{K})=c\mathcal{Q}_{\mu,K}(\phi)$, and $Z_{K}$ has proper
intersection with the diagonal $\Delta:X_{K}\to X_{K}\times X_{K}$.

Now consider the Shimura variety $Y_{K}\times Y_{K}\times X_{K}\times X_{K}$
and in which we denote a point by $(y_{1},y_{2},x_{1},x_{2})\in Y_{K}\times
Y_{K}\times X_{K}\times X_{K}$. Choose a doubling divisor $Z_{K}^{D}(\phi)$ on
$M\times M$ for $\phi$. We obtain the cycle

$T^{f}_{K}\times Z_{K}\in\mathrm{CH}^{2(n-1)+1}(Y_{K}\times Y_{K}\times
X_{K}\times X_{K}).$

We have another two cycles

$P_{y_{1},x_{1}}:=\\{y_{1}=x_{1}\\}\in\mathrm{CH}^{2(n-1)}(Y_{K}\times
Y_{K}\times X_{K}\times X_{K}).$
$P_{y_{2},x_{2}}:=\\{y_{2}=x_{2}\\}\in\mathrm{CH}^{2(n-1)}(Y_{K}\times
Y_{K}\times X_{K}\times X_{K}).$

We denote by $P_{y_{1},x_{1}}^{z}$ (resp. $P_{y_{2},x_{2}}^{z}$) their
modifications under the Hecke systems $z$ of projectors for
$\mathrm{CH}^{*}(Y_{K}\times X_{K})$.

######  Definition 9.0.8 (Global arithmetic invariant functional).

$\mathcal{I}_{K}^{z}(f,\phi):=\left<P_{y_{1},x_{1}}^{z},(T^{f}_{K}\times
Z^{D}_{K}(\phi)).P_{y_{2},x_{2}}^{z}\right>^{\mathrm{BB}}_{Y_{K}\times
Y_{K}\times X_{K}\times X_{K}}.$

Note in the untwisted case, [Liu-FJcycles, Definition 5.13] uses a section
$T_{K}\mapsto(\mathrm{id}\times T_{K})$ from Hecke correspondences on $X_{K}$
to Hecke correspondences on $X_{K}\times X_{K}$, which is not available in our
twisted set up. Similar to [Liu-FJcycles, Proposition 5.10], the height
pairing

$(p_{1}^{*}\Delta_{z}^{E_{0}\times
F_{0}}X_{K},(Y_{K}^{\prime}\times\mathcal{Q}^{\phi,\phi^{c},P,Q}_{\mu,K}).p_{2}^{*}\Delta^{E_{0}\times
F_{0}}_{z,f,f^{\vee}}X_{K})^{\mathrm{BB}}_{Y_{K}^{\prime}\times
X_{K}^{\prime}\times X_{K}^{\prime}}$

is reduced to above global arithmetic invariant functionals
$\mathcal{I}^{z}_{K}(-,-)$. To proceed, we consider the local part of the
global arithmetic invariant functional at a good place $v_{0}$ of $F_{0}$.

######  Definition 9.0.9.

We say a finite place $v_{0}$ of $F_{0}$ is a good place (with respect to $F$
and $\phi$), if

  1. (1)

$v_{0}$ is inert in $F$ and unramified in $E_{0}$.

  2. (2)

the underlying rational prime $p$ is odd and unramified in $E^{\mathrm{RSZ}}$.

  3. (3)

Let $v$ be any place of $F_{0}$ above $p$. Then there exists a self-dual
lattice $L_{v}\subseteq V_{v}$ such that
$K_{v}=\mathrm{U}(L_{v}),K_{v}^{E_{0}}=\mathrm{U}(L_{v,O_{E_{0}}})$. And
$F_{v}=1_{K_{v}^{E_{0}}}$, $\phi_{v}=1_{L_{v}}$.

Fix a good place $v_{0}$ of $F_{0}$, and denote by $q_{v_{0}}$ the size of its
residual field $k_{v_{0}}$. We assume that $v_{0}$ is inert in $E_{0}$
otherwise the reduction is done in [Liu-FJcycles, Theorem 5.25]. As Section
6.2, choose large enough $\Delta$ such that no $p$-adic places are in
$\Delta$. Consider the embedding of punctured integral models over the semi-
local ring $O_{E_{v_{0}}^{\mathrm{RSZ}}}$

$\Delta^{E_{0}}:\mathcal{X}_{K}:=\mathcal{M}_{O_{E_{v_{0}}}^{\mathrm{RSZ}}}\to\mathcal{Y}_{K}:=\mathcal{M}^{E_{0}}_{O_{E_{v_{0}}}^{\mathrm{RSZ}}}$

which are proper smooth schemes. Let
$\mathcal{P}_{y_{i},x_{i}}=\\{y_{i}=x_{i}\\}$ ($i=1,2$) in
$\mathcal{Y}_{K}\times\mathcal{Y}_{K}\times\mathcal{X}_{K}\times\mathcal{X}_{K}$.

######  Definition 9.0.10 (Local arithmetic invariant functional at a good
place $v_{0}$).

Assume that $F\otimes\phi$ has regular supports at a place of $F_{0}$. The
local arithmetic invariant functional at $v_{0}$ is the Euler–Poincaré
characteristic

$\mathcal{I}_{v_{0}}(f,\phi):=(2\log
q_{v_{0}})\chi(\mathcal{Y}_{K}\times\mathcal{Y}_{K}\times\mathcal{X}_{K}\times\mathcal{X}_{K},\mathcal{O}(\mathcal{P}_{y_{1},x_{1}})\otimes^{\mathbb{L}}\mathcal{O}((T^{f}_{K}\times\mathcal{Z}^{D}_{K}(\phi)).\mathcal{P}_{y_{2},x_{2}}))$

We expect the local arithmetic invariant functional is the main term of the
global arithmetic invariant functional at $v_{0}$. Let $\mathcal{Z}(\phi)_{K}$
be the Zariski closure of the geometric theta series $Z(\phi)_{K}$ on
$\mathcal{X}_{K}$ which is related to the arithmetic theta series of Kudla-
Rapoport divisors in Section 6.3.

######  Proposition 9.0.11.

Assume the support of

$\mathcal{P}_{y_{1},x_{1}}\cap(T^{f}_{K}\times\mathcal{Z}^{D}_{K}(\phi))\cap\mathcal{P}_{y_{2},x_{2}}$

is in $\mathcal{Y}_{K}\times\mathcal{Y}_{K}\times\Delta X_{K}$. Then we have

$\mathcal{I}_{v_{0}}(f,\phi)=(2\log
q_{v_{0}})\chi(\mathcal{Y}_{K}\times\mathcal{Y}_{K},\mathcal{O}(\mathcal{Z}(\phi)_{K}\times\mathcal{X}_{K})\otimes^{\mathbb{L}}\mathcal{O}(T^{f}_{K})\otimes^{\mathbb{L}}\mathcal{O}(\mathcal{X}_{K}\times\mathcal{X}_{K})).$

###### Proof.

This follows from the same argument of [Liu-FJcycles, Proposition 5.19]. ∎

Note that the support assumption is automatic in the untwisted case [Liu-
FJcycles, Proposition 5.19]. We expect the support condition to hold for good
choices of $F$ and $Z_{K}^{D}(\phi)$. Then by the argument of [Liu-FJcycles,
Theorem 5.25], using Definition 7.1.1 and Proposition 7.4.2, we see that
$\xi$-the Fourier coefficients of $\mathcal{I}_{v_{0}}(f,\phi)$ at a good
place $v_{0}$ of $F_{0}$ is related to our intersection pairing (34)

$\mathrm{Int}_{v_{0}}(\xi,\Phi=f\otimes\phi),$

hence to the twisted arithmetic fundamental lemma (Theorem 4.0.8) by
Proposition 7.4.2. See also Corollary 7.4.6.

## References


# Inhomogeneous Superconducting State Probed by 125Te NMR on UTe2

Genki Nakamine1<EMAIL_ADDRESS>Katsuki Kinjo1 Shunsaku
Kitagawa1<EMAIL_ADDRESS>Kenji Ishida1
<EMAIL_ADDRESS>Yo Tokunaga2 Hironori Sakai2 Shinsaku Kambe2 Ai
Nakamura3 Yusei Shimizu3 Yoshiya Homma 3 Dexin Li3 Fuminori Honda3 and Dai
Aoki3,4 1Depertment of Phisics1Depertment of Phisics Kyoto University Kyoto
University Kyoto Kyoto 606-8502 606-8502 Japan
2ASRC Japan
2ASRC Japan Atomic Energy Agency Japan Atomic Energy Agency Tokai-mura Tokai-
mura Ibaraki 319-1195 Ibaraki 319-1195 Japan
3IMR Japan
3IMR Tohoku University Tohoku University Oarai Oarai Ibaraki Ibaraki 311-1313
311-1313 Japan
4University Grenoble Japan
4University Grenoble CEA CEA IRIG-PHERIQS IRIG-PHERIQS F-38000 Grenoble
F-38000 Grenoble France France

###### Abstract

UTe2 is a recently discovered promising candidate for a spin-triplet
superconductor. In contrast to conventional spin-singlet superconductivity,
spin-triplet superconductivity possesses spin and angular momentum degrees of
freedom. To detect these degrees of freedom and obtain the solid evidence of
spin-triplet superconductivity in UTe2, we performed 125Te-NMR measurement,
which is sensitive to the local spin susceptibility at a nuclear site. We
previously reported that the shoulder signal appears in NMR spectra below the
superconducting (SC) transition temperature $T_{\rm c}$ in $H\parallel b$, and
that a slight decrease in the Knight shift along the $b$ and $c$ axes ($K_{b}$
and $K_{c}$, respectively) below $T_{\rm c}$ at a low magnetic field $H$.
Although the decrease in $K_{c}$ vanished above 5.5 T, the decrease in $K_{b}$
was independent of $H$ up to 6.5 T. To clarify the origin of the shoulder
signal and the trace of the decrease in $K_{b}$, we compared the 125Te-NMR
spectra obtained when $H~{}\parallel~{}b$ and $H~{}\parallel~{}c$ and measured
the 125Te-NMR spectra for $H~{}\parallel~{}b$ up to 14.5 T. The intensity of
the shoulder signal observed for $H~{}\parallel~{}b$ has a maximum at $\sim 6$
T and vanishes above 10 T, although the superconductivity is confirmed by the
$\chi_{\rm AC}$ measurements, which can survive up to 14.5 T (maximum $H$ in
the present measurement). Moreover, the decrease in $K_{b}$ in the SC state
starts to be small around 7 T and almost zero at 12.5 T. This indicates that
the SC spin state gradually changes with the application of $H$. Meanwhile, in
$H~{}\parallel~{}c$, unexpected broadening without the shoulder signals was
observed below $T_{\rm c}$ at 1 T, and this broadening was quickly suppressed
with increasing $H$. We construct the $H$–$T$ phase diagram for
$H~{}\parallel~{}b$ and $H~{}\parallel~{}c$ based on the NMR measurements and
discuss possible SC states with the theoretical consideration. We suggest that
the inhomogeneous SC state characterized by the broadening of the NMR spectrum
originates from the spin degrees of freedom.

##  1 Introduction

Superconductors exhibit two types of spin states in general: spin-singlet ($S$
= 0) and spin-triplet ($S$ = 1). For most superconductors, an $s$-wave pairing
with a spin-singlet state and zero momentum results in a superconducting (SC)
state with no degrees of freedom. In contrast, the spin degrees of freedom
remain in the spin-triplet state, and they possibly cause unusual behaviors.
This includes the magnetic field ($H$)-boosted superconductivity, multiple SC
phases, and rotation of the $\bm{d}$ vector, which is an SC order parameter
for the spin-triplet pairing[5, 1]. However, there are few materials in which
the spin-triplet state is considered to be realized. In addition, the
experimental observations for these phenomena are quite limited. Some of the
most promising candidates for spin-triplet superconductors are uranium
(U)-based ferromagnetic (FM) superconductors UGe2, URhGe, and UCoGe[2, 3, 4],
in which the ferromagnetism coexists with the superconductivity. For these
superconductors, many experimental and theoretical studies have suggested that
spin-triplet superconductivity is induced by the Ising-type FM spin
fluctuations[5]. One of the most convincing results is the $H$-boosted
superconductivity caused by the development of the FM spin fluctuations, which
has been observed in the vicinity of the $H$-induced quantum critical point
for the U-based FM superconductors URhGe[6] and UCoGe[7]. Therefore, research
on the spin-triplet pairing state has been conducted for these
superconductors.

Recently, the superconductivity of UTe2 was discovered[8]; since then, UTe2
has attracted significant attention because of its unique SC properties. UTe2
has the orthorhombic crystal structure with the space group $Immm$ (#71,
$D_{2h}^{25}$). Its SC upper critical fields ($H_{\rm c2}$) for all three
crystalline axes greatly exceed the Pauli limiting field[8, 9]; particularly
for $H\parallel b$, the SC transition temperature $T_{\rm c}$ has an anomalous
upturn at $\mu_{0}H~{}=~{}15$ T, which is similar to the $H$-boosted
behavior[10]. This $H_{\rm c2}$ behavior cannot be understood by the scenario
in the spin-singlet superconductors, and it is reminiscent of the FM
superconductors. In addition to $H_{\rm c2}$, the Ising-type anisotropy of the
static magnetic susceptibility[8, 9], and the presence of the moderate Ising-
type fluctuations[11] are similar to those observed in FM superconductors[12,
4, 13, 14, 15, 16, 17]. However, UTe2 does not show any magnetic order at the
ambient pressure[8]. Therefore, it is suggested that UTe2 is located in the
vicinity of the FM quantum critical point and that the spin-triplet
superconductivity mediated by the Ising-type FM spin fluctuations is realized
in UTe2. Furthermore, multiple SC phases have been reported in the pressure
($P$)–temperature ($T$) phase diagram for UTe2[18, 19, 20]. Another SC phase
is induced by applying pressure, and the evolution to more complex multiple SC
phases in $H~{}\parallel~{}a$ under pressure was reported[21]. These
experimental results strongly suggest that UTe2 is a spin-triplet
superconductor with degenerate multiple SC order parameters.

To investigate the SC properties of UTe2 from a microscopic point of view, we
previously performed 125 Te-NMR measurements at the ambient pressure[22, 23].
This is because the NMR measurement is sensitive to the changes in the local
spin susceptibility at a nuclear site even in the SC state. When the $\bm{d}$
vector is pinned to one of the crystalline axis, Knight shift proportional to
the spin susceptibility decreases in the SC state in $\bm{H}\parallel\bm{d}$,
but remains unchanged in $\bm{H}\perp\bm{d}$. Thus, the magnetic properties of
the spin-triplet SC order parameters can be investigated. During our previous
measurements[23], we focused on the reduction in the spin susceptibility in
the SC state and measured the NMR Knight shift along the $b$ and $c$ axes
($K_{b}$ and $K_{c}$, respectively). $K_{b}$ and $K_{c}$ decrease slightly at
1 T, which indicates that the $\bm{d}$ vector has the $b$ and $c$ components.
This result limits the SC pairing symmetry in UTe2 to $A_{u}$ or $B_{3u}$ in
odd-parity states within the irreducible representations of $D_{2h}$. We also
reported the anisotropic response of the Knight shift reduction against $H$.
The reduction in $K_{b}$ is nearly constant of $H$ up to 6.5 T; however, the
reduction in $K_{c}$ decreases with increasing $H$. In addition, $K_{c}$ does
not change even below $T_{\rm c}$ at 5.5 T. This anisotropic response of the
spin susceptibility is considered to be related to the SC pairing symmetry.

Although we have performed intensive research thus far, the following issues
remain unresolved. First, the origin of additional NMR signals in the SC state
for $H~{}\parallel~{}b$ is unclear. As previously reported[23], in addition to
the shift of the main peak, additional shoulder signals appear in the SC-state
NMR spectrum in $H~{}\parallel~{}b$. Although the presence of additional
signals suggests the appearance of another ordered state inside the SC phase,
we cannot clarify the origin and characteristics from the previous
experiments. Second, it must be clarified whether $\bm{d}$-vector rotation
occurs with a high $H$ along the $b$ axis. The reduction in $K_{b}$ in the SC
state implies that the SC state is unstable with a high $H$, which is due to
the competition between the SC-condensation and the Zeeman energies. In such a
case, $\bm{d}$-vector rotation that is related to a change in the SC pairing
symmetry is expected at a critical magnetic field $H_{\rm pin}$. We estimated
the $\mu_{0}H_{\rm pin}\sim$ 13 T from the decrease in $K_{b}$, as reported in
our previous paper[23].

To solve these issues, in the present study, we performed precise NMR spectrum
measurements up to 6.5 T in $H~{}\parallel~{}b$ and $H~{}\parallel~{}c$, and a
high-$H$ NMR measurement up to 14.5 T in $H~{}\parallel~{}b$. We found that
the additional NMR signals appear well below $T_{\rm c}$ and the area fraction
for those at 0.1 K against the main peak reaches a maximum at $\sim$ 6 T, and
it becomes zero above 10 T. Meanwhile, an unexpected large broadening of the
NMR spectrum was observed in the SC state for $H~{}\parallel~{}c$, and it is
quickly suppressed by $H$. Furthermore, the reduction in $K_{b}$ starts to
decrease at $\sim$ 7 T and it becomes zero at $\sim$ 12.5 T. This indicates
that the $b$ components of the $\bm{d}$ vector disappear above 12.5 T.
Considering all these results, we discuss the possible SC states based on the
phase diagram.

##  2 Experimentals

The single-crystal UTe2, which is the same sample used in our previous
study[23], was grown using the chemical transport method with iodine as a
transport agent[24]. Natural uranium and 99.9% 125Te-enriched metals were used
as the starting materials for the present sample. There are two
crystallographically inequivalent Te sites, 4$j$ and 4$h$, with point
symmetries $mm$2 and $m$2$m$, respectively. We denote these sites as Te(1) and
Te(2) according to the previous reports[23]. The measurements were performed
using the single crystal with a size of $3.5\times 0.7\times 1.4$ mm3. $T_{\rm
c}$ was determined by the AC susceptibility that was measured by recording the
resonance frequency of the NMR-tank circuit during cooling. The frequency-
swept 125Te ($I$ = 1/2, gyromagnetic ratio ${}^{125}\gamma_{n}/2\pi$ = 13.454
MHz/T)-NMR spectrum was obtained by the integration of the Fourier transform
(FT) for a spin-echo signal that was observed after a radio frequency (RF)
pulse sequence with a 3 kHz step for a fixed $H$. The typical RF pulse length
is 12 $\mu$s. The NMR spectra are plotted against
$K~{}\equiv~{}(f-f_{0})/f_{0}$. Here, $f$ is the NMR frequency, and $f_{0}$ is
the reference frequency determined as $f_{0}~{}=~{}(\gamma_{n}/2\pi)\mu_{0}H$.
The magnetic field was calibrated using 63Cu (${}^{63}\gamma_{n}/2\pi$ =11.285
MHz/T) and 65Cu (${}^{65}\gamma_{n}/2\pi$ =12.089 MHz/T) NMR signals from the
NMR coil. In the present measurement, the errors of the NMR intensity and the
linewidth are determined from the background noise amplitude and the
resolution of FT signals, respectively. All the NMR spectra in the SC state
were recorded with the field-cooling process. Low-temperature NMR measurements
down to 80 mK were performed using a 3He-4He dilution refrigerator, and the
single-crystalline sample was immersed into the mixture to reduce the heating
effect. In the low-$T$ NMR measurements, we reduced the power of the RF-pulses
for the observation of the spin-echo signal as small as possible. Due to this
effort, we detected the broadening of the NMR spectra below $T_{\rm c}$
originating from SC diamagnetism in all measurements. We performed the
measurements using two setups. In the first setup (Setup A), we used the SC
split magnet, which generates a horizontal field up to 6.5 T with a single-
axis rotator to apply $H$ exactly parallel to the $b$ axis or $c$ axis with
the $a$ axis as the rotation axis. This is the same setup as in the previous
measurements[23]. The sample alignment and angle dependence of the NMR
spectrum are reported in the supplemental materials in the previous
report[23]. In the second setup (Setup B), we used the SC solenoid magnet,
which generates a vertical field up to 14.5 T to apply a higher $H$ along the
$b$ axis. In Setup B, because the sample cannot be rotated in $H$, a
misalignment with $H$ occurred, which is evaluated to be $3\sim 13$ degrees
tilted from the $b$ axis. The evaluation of the misalignment is discussed in a
later section.

##  3 Experimental results

###  3.1 Setup A: measurements with a rotator

Figure 1:  (Color online) (a) $T$ variation of 125Te-NMR spectrum against
$K~{}\equiv~{}(f-f_{0})/f_{0}$ measured at 5.5 T along the $b$ axis. The
arrows indicate the peak of the “main peak” signal. The dashed and dotted
lines denote the Knight shift in the normal state and $K~{}=~{}5.25$%,
respectively. (b) Temperature dependence of the NMR intensity at $K=5.25$%
normalized by the main-peak intensity at $\mu_{0}H=1,2,3,$ and 5.5 T.
Background (BG) noise level is shown by dotted lines. To avoid overlapping
data, the offset value (0.5) is added. The SC transition temperatures $T_{\rm
c}$ obtained by the AC susceptibility measurement are shown by the dotted
arrows. The solid arrows indicate the characteristic temperature $T_{x}$ at
which the intensity starts to increase upon cooling. (c) 125Te-NMR spectrum at
the lowest temperature at $\mu_{0}~{}H~{}=~{}1,3,$ and 5.5 T along the $b$
axis. For comparison purposes, the normal-state NMR spectrum at 3 T is also
shown.

First, we summarized the experimental results in Setup A. Figure 1(a) shows
the 125Te-NMR spectrum at several temperatures measured at 5.5 T in
$H~{}\parallel~{}b$. The peak shown by the arrows shifts to a lower $K$
direction below $T_{\rm c}$. This indicates a decrease in the spin
susceptibility by forming Cooper pairs. In addition, we found that additional
signals appear at low temperatures. Hereafter, we refer to the peak of the NMR
spectrum and the additional signal that is observed at low temperatures as the
“main peak” and “shoulder signal”, respectively. To investigate the origin of
the shoulder signal appearing in the SC state upon cooling, the temperature
dependence of the NMR intensity at $K~{}=~{}$5.25% [shown by the dotted line
in Fig. 1(a)], which is normalized by the main peak intensity, is plotted in
Fig. 1(b). We can define the characteristic temperature $T_{x}$, at which the
intensity at $K~{}=~{}$5.25% starts to increase. $T_{x}$ shown by the solid
arrows in Fig. 1(b) is $\sim 1$ K and is almost independent of $H$ in the
measured $H$ region, which is different from $T_{\rm c}$. These results
indicate the presence of another phase below $T_{x}$. The difference between
$T_{\rm c}$ and $T_{x}$ is $\sim$ 0.6 K at 1 T, which is significantly larger
than the separation in the double transition that is observed by a specific
heat at zero fields ($\sim$ 0.2 K)[20, 25]. It is noted that the shoulder
signal has a characteristic structure. There are two peaks at the lower $K$
side and a broad peak that tails to a higher $K$ than that of the main peak,
as shown in Fig. 1(c). This indicates that the spin susceptibility is
inhomogeneously distributed in the low-$T$ phase. The area fraction of the
shoulder signal at 0.1 K against the main peak increases with increasing $H$
up to 5.5 T, although the main peak still remains. Since we require higher-$H$
NMR measurements than 6.5 T to clarify how the area fraction of the shoulder
signal changes against $H$, the measurement was performed using Setup B as
described in section 3.2.

We also performed 125Te-NMR measurements in $H~{}\parallel~{}c$ by Setup A.
Figure 2(a) shows the 125Te-NMR spectrum against $K$ at several temperatures
measured at 1 T in $H~{}\parallel~{}c$. Although a single symmetric peak was
observed in the normal state, this peak consists of two NMR signals arising
from the Te(1) and Te(2) sites because of the closeness of the Knight shift at
the two sites when $H$ is parallel to the $c$ axis[11, 23]. The symmetric NMR
spectrum becomes broader in the SC state without showing any shoulder peak
structure observed in $H~{}\parallel~{}b$, and it slightly shifts to a lower
$K$ direction, as we previously reported [23]. The temperature dependence of
the full width at the half maximum (FWHM) of the NMR spectrum, from which the
normal-state value is subtracted ($\Delta$FWHM), in the frequency units at 1,
3, and 5.5 T are shown in Fig. 2(b). Although the $\Delta$FWHM for all the $H$
values increases below $T_{\rm c}$, the $\Delta$FWHM at the lowest $T$ becomes
smaller with increasing $H$, as shown in the inset of Fig. 2(b). It is noted
that the broadening at the lowest $T$ is one order of magnitude larger than
the broadening expected from the conventional SC diamagnetic effect. The
broadening of the NMR spectrum by the conventional SC diamagnetic effect is of
the order of the SC lower critical field $\mu_{0}H_{\rm c1}\sim 1.2$ mT[23].
This gives the upper limit of the 125Te-NMR spectrum broadening $\sim$16 kHz.
Therefore, it is concluded that the broadening of the 125Te-NMR in
$H~{}\parallel~{}c$ is not simply due to the conventional SC diamagnetic
effect. Instead, it originates from the unconventional SC properties in UTe2.
The unexpected large broadening of the NMR spectrum in the SC state suggests
that the spin susceptibility is inhomogeneously distributed also in
$H\parallel c$. It is also noteworthy that the magnetic field dependence of
the broadening of the NMR spectrum is different between $H~{}\parallel~{}b$
and $H~{}\parallel~{}c$. The NMR spectrum at the lowest temperature, which is
shown against $K$, quickly becomes narrower with an increase in $H$ when
$H~{}\parallel~{}c$, as shown in the inset of Fig. 2(b) and Fig. 2(c).
Although the magnetic field at which the broadening behavior disappears is
ambiguous in $H\parallel c$, it is roughly estimated as $\sim$ 3 T above which
the $\Delta K$ is almost independent of $H$ as shown in the inset of Fig.
2(b). Here, $\Delta K$ is the $\Delta$FWHM at the lowest temperature divided
by the measured frequency. In contrast, the broadening behavior becomes more
obvious with an increase in $H$ when $H~{}\parallel~{}b$, as shown in Fig.
1(c). This indicates that the spin-susceptibility distributed signal, observed
in the low fields, is suppressed in $H\parallel c$, but it seems to be
enhanced in $H\parallel b$.

Figure 2:  (Color online) (a) $T$ variation of 125Te-NMR spectrum against $K$
measured at 1 T along the $c$ axis. The dashed line denotes the Knight shift
in the normal state. (b) Temperature dependence of the FWHM for the NMR
spectrum from which the normal-state value is subtracted ($\Delta$FWHM) at
$\mu_{0}H=1,3,$ and 5.5 T. The arrows show $T_{\rm c}$s. The inset shows the
magnetic field dependence of the $\Delta$FWHM at the lowest temperature
divided by the measured frequency ($\Delta K$). (c) 125Te-NMR spectrum at the
lowest temperature at $\mu_{0}H=1,3,$ and 5.5 T along the $c$ axis. For
comparison purposes, the normal-state NMR spectrum at 5.5 T is also shown.

###  3.2 Setup B: high-$H$ measurements in $H\parallel\sim b$

To investigate the origin of the shoulder signal and the SC properties for a
high $H$, we performed the NMR measurements using Setup B. As described in
Section 2, there is a small sample misalignment in Setup B because the field
direction cannot be controlled. Figure 3(a) shows the 125Te-NMR spectrum in
Setup B along with those in Setup A at several field angles from the $b$ to
$c$ axis at 1 T and 1.65 K. The NMR spectrum of Setup B almost overlaps with
the NMR spectrum at 12∘ in Setup A. From the comparison between the Knight
shift in Setup B and the angular dependence of the Knight shift in Setup A, we
estimated that the tilted angle is 12.5∘ from the $b$ axis to the $c$ axis
when we assume that the sample misalignment originates from only the $b$–$c$
rotation as shown in the inset of Fig.3 (a). We can also estimate that the
tilted angle is $\sim$ 3∘ from the $b$ axis to the $a$ axis because there is a
possibility of sample misalignment with the $b$–$a$ rotation. Here, we used
the Knight shift along the $a$ axis $K_{a}$ $\sim$ 40% that was obtained by
extrapolating the temperature dependence of $K_{a}$ from 20 K to 1.6 K. It has
been reported that the NMR signal for $H~{}\parallel~{}a$ disappears below 20
K due to the divergence of $1/T_{2}$[11]; thus, $K_{a}$ at low temperatures
could not be measured. To check the effect of the misalignment on the
superconductivity, we measured the magnetic field dependence of $T_{\rm c}$ in
Setup B from the $T$-scan of $\chi_{\rm AC}$, as shown in the inset of Fig.
3(b). $T_{\rm c}$ was determined by the intersection point of the slope of the
SC diamagnetism and the normal state. The obtained magnetic field dependence
of $T_{\rm c}$ is plotted in Fig. 3(b) with that measured in Setup A, where
$H$ is exactly parallel to the $b$ axis. Although $T_{\rm c}(H)$ is slightly
reduced from that in $H~{}\parallel~{}b$, the shape of the phase diagram is
almost the same up to 14.5 T. This indicates that the effect of the
misalignment is small for the superconductivity.

Figure 3:  (Color online) (a)125Te-NMR spectrum in Setup B together with those
in Setup A at several field angles from the $b$ to $c$ axis at 1 T and 1.65 K.
(Inset)Angular dependence of the Knight shift in Setup A (squares) together
with the Knight shift in Setup B (dashed line). (b)The $H$–$T$ phase diagram
for $H\parallel b$ in Setup A and for $H\parallel\sim b$ in Setup B. The
broken curves represent $H_{\rm c2}$ from the reference[10].
(Inset)Temperature dependence of $\chi_{\rm AC}$ at several $H$ in Setup B.
$T_{\rm c}$s, which are represented by the arrows, were determined by the
intersection point of the slopes of the SC diamagnetism and the normal state.

Next, we show the magnetic field dependence of the magnetic properties in the
normal state. Figure 4(a) shows the 125Te-NMR spectrum at several $H$ values
in the normal state. A peak in the NMR spectrum gradually shifts to a higher
$K$ direction with increasing $H$. The Knight shift in the normal state,
$K_{\rm normal}$ is almost independent of $T$ below 2 K, as shown in the inset
of Fig. 4(a). Because $K_{\rm normal}$ is related to the Pauli paramagnetism,
the increase in $K_{\rm normal}$ against $H$ can be interpreted as an increase
in the density of the states, which is consistent with the previous bulk
measurements[26, 27]. The magnetic field dependence of $K_{\rm normal}$ and
the Sommerfeld coefficient $\gamma$[26] is presented in Fig. 4(b). The
observed increase in $K_{\rm normal}$ was scaled with $\gamma$.

Figure 4:  (Color online) (a)125Te-NMR spectrum at several $H$ in the normal
state. The dashed line is a guide for the eye. (Inset)Temperature dependence
of $K_{\rm normal}$ at 14 T. (b) Magnetic field dependence of $K_{\rm normal}$
and the Sommerfeld coefficient $\gamma$[26].

The shoulder signal in the SC state displays peculiar behavior. Figure 5(a)
shows the 125Te-NMR spectrum measured at $\sim 0.1$ K up to 5.5 T in Setup B.
As previously mentioned and illustrated later, the Knight shift in the normal
and SC states depend on $H$. Thus, to simply compare the NMR line shape, in
which the NMR spectra are normalized with the maximum peak, it is plotted
against $K-K_{\rm peak}$, where $K_{\rm peak}$ is the Knight shift at the
maximum peak of the NMR spectrum. Although the peaks were located at slightly
different positions due to the misalignment from the $b$ axis, the shoulder
signal observed in Setup A was also observed in Setup B. Therefore, we could
investigate the $H$ evolution of the shoulder signal in the high $H$ region.

Figure 5:  (Color online) 125Te-NMR spectrum at $\sim$0.1 K in (a)
$\mu_{0}H\leq$ 5.5 T and (b) $\mu_{0}H\geq 5.5$ T in Setup B. The dashed line
denotes the position of $K-K_{\rm peak}~{}=~{}-0.2$%. (Inset) Magnetic field
dependence of the NMR intensity at $K-K_{\rm peak}~{}=~{}-0.2$% measured at
0.1 K.

Figure 5(b) shows the 125Te-NMR spectrum that was measured at $\sim 0.1$ K
above 5.5 T. This is plotted in the same manner as Fig. 5(a). As illustrated,
the intensity of the shoulder signal decreases with increasing $H$, and it
almost vanishes at approximately 10 T. We plotted the magnetic field
dependence of the intensity at $K-K_{\rm peak}~{}=~{}-0.2$%, as shown by the
dotted line in Fig. 5(b) at $\sim 0.1$ K in the inset of Fig. 5(b). From the
figure, we can determine the critical field $\mu_{0}H_{x}$ where the shoulder
signal disappears to be 10 T.

Figure 6:  (Color online) Magnetic field dependence of the change in the spin
component of the Knight shift in the SC state, $\Delta K_{\rm spin}$ in Setup
A and B. The dotted line indicates $\Delta K_{\rm spin}$ = 0. The dashed lines
are a guide for the eye. (Inset) 125Te-NMR spectrum at $\sim 0.1$ K (SC state)
measured at 8, 10.5, and 12.5 T in Setup B. For comparison purposes, the
normal-state NMR spectrum is also shown.

In addition to $H_{x}$, we also investigated the critical field at which the
decrease in $K_{b}$ becomes zero. In the inset of Fig. 6, the NMR spectrum at
$\sim 0.1$ K (SC state) is compared with the normal-state NMR spectrum at 8,
10.5, and 12.5 T. While the NMR spectrum at 8 T and 10.5 T shifts to a lower
$K$ direction in the SC state, no spectrum shift was observed between the
normal and SC states at 12.5 T. For the quantitative discussion, we evaluated
the change in the spin component of the Knight shift in the SC state, $\Delta
K_{\rm spin}$, which is described as

$\Delta K_{\rm spin}=K(T\sim 0.1\,{\rm K})-K_{\rm normal}-\Delta K_{\rm dia},$

and we plotted the magnetic field dependence of $\Delta K_{\rm spin}$ in Fig.
6. Here, $K(T)$ is the Knight shift at $T$, and $\Delta K_{\rm dia}$ is the
contribution from the conventional SC diamagnetism. $\Delta K_{\rm dia}$ was
estimated from the formula suggested by de Gennes[28], which is expressed as

$\Delta K_{{\rm dia}}=-\frac{H_{\rm c1}}{H}\frac{\ln\left(\frac{\beta
d}{\sqrt{e}\xi}\right)}{\ln{\kappa}}.$

As shown in Fig. 6, the obtained $\Delta K_{\rm spin}$ is almost independent
of $H$ up to 7 T, which is consistent with the previous results[23].
Meanwhile, the absolute value of $\Delta K_{\rm spin}$ decreases gradually
above 7 T and becomes almost zero above $\sim$12.5 T. We comment that the
intensity of the shoulder signal starts to decrease at around 7 T, indicating
that the SC state might be gradually changed from this field. It is noted that
the superconductivity survives at least up to 14.5 T because the SC transition
was confirmed by the $\chi_{\rm AC}$ measurement and the broadening of the NMR
spectrum. The zero value of $\Delta K_{\rm spin}$ implies that the $\bm{d}$
vector has no $b$ component above 12.5 T.

##  4 Discussion

Here, we discuss how the $\bm{d}$ vector changes against $H$ along the $b$
axis. In a previous report[23], we pointed out that the low-$H$ SC state is
unstable; moreover, it should change in the high-$H$ region in
$H~{}\parallel~{}b$ because $K_{b}$ decreases in the SC state, and the
competition between the SC-condensation and Zeeman energies is expected. We
estimated the critical magnetic field $\mu_{0}H_{\rm pin}$ to be $\sim$ 13 T
from the decrease in $K_{b}$, which is consistent with the observed critical
field in which the decrease in $K_{b}$ becomes zero. The observed magnetic
field dependence of $\Delta K_{\rm spin}$ is gradual, not drastic. One
possibility is that the SC pairing symmetry below and above 12.5 T is the same
representation in $H~{}\parallel b$. Therefore, the change in the SC state is
a crossover rather than a first-order phase transition, which gives strong
constraints for the SC pairing symmetry.

Next, we discuss the low-$T$ SC state, in which the sharp signal arising from
the high-$T$ SC phase and the broad spin-susceptibility-distributed signal
(the shoulder signal in $H\parallel b$ and the unexpected broad signal in
$H\parallel c$) are observed. One of the characteristic features of the
low-$T$ SC state is that the broad signal exists only below $T_{c}$ ($H$),
indicating that this signal is closely related to the superconductivity. The
other is the anisotropic field response of the spectrum broadening. The NMR
spectrum at the lowest temperature quickly becomes narrower with an increase
in $H$ when $H~{}\parallel~{}c$, although the broadening behavior becomes more
obvious with an increase in $H$ and disappears at 10 T when
$H~{}\parallel~{}b$. These results exclude the possibility of the magnetic
origin for the spectrum broadening. Therefore, one possible scenario to
interpret the origin of the broad signal is that other SC states with
different SC pairing symmetries from that in the high-$T$ SC phase are
realized in the specific regions of the sample. ($i.e.$ the broad signal
arises from the low-$T$ SC phase.) Since the NMR spectrum is extremely broad,
a spatially distributed order parameter or multi-component order parameter
generating the internal field are expected for the low-$T$ SC phase. As only
the high-$T$ SC phase exists between $T_{c}$ and $T_{x}$, the low-$T$ SC phase
is expected to grow to replace a part of high-$T$ SC phase. In general, such
coexistence is induced by disorders or impurities. However, the spectrum in
the normal state is very sharp as shown in Figs. 1(a) and (c), and there are
no indications of the presence of impurities or disorders. Therefore, it is
considered that the high-$T$ and low-$T$ SC phases are highly degenerated, and
the ground state can change with a slight difference in the local situation.

Figure 7:  (Color online) Summarized $H$–$T$ phase diagram for (a)
$H~{}\parallel~{}b$ and (b) $H~{}\parallel~{}c$. The diamonds, triangles and
circles represent $H_{\rm c2}$ determined by $\chi_{\rm AC}$ in this study.
The broken curves represent $H_{\rm c2}$ from reference[10]. The stars
represent $T_{x}$ and $H_{x}$ as described in the text. The SC phase can be
distinguished into four regions based on the NMR results. Region boundaries
are indicated by dotted lines. Each region in the phase diagrams is numbered
as I$\sim$IV.

We now summarize the $H$–$T$ phase diagram for $H~{}\parallel~{}b$ and
$H~{}\parallel~{}c$ in UTe2 in Figs. 7 (a) and (b), respectively. Based on the
present results, we can distinguish the SC phase into four regions, and number
each region in the phase diagrams as I$\sim$ IV. In region I, only the main
peak was observed, and $K_{b}$ and $K_{c}$ decrease in the SC state. This
corresponds to the high-$T$ SC phase and indicates the single-phase SC state
with the nonzero $b$ and $c$ components in the $\bm{d}$ vector ($\bm{d_{b}}$
and $\bm{d_{c}}$, respectively). Considering the $D_{2h}$ point group
symmetry, the $A_{u}$ or $B_{3u}$ state is a candidate for SC pairing
symmetry, which is consistent with the theoretical suggestion[29, 30, 31]. In
region II, the shoulder signal appears in addition to the main peak in
$H~{}\parallel~{}b$. The region II corresponds to the low-$T$ SC state where
the high-$T$ SC phase and the low-$T$ SC phase coexist as discussed above. The
determination of the boundary of the region II for $H\parallel c$ is
ambiguous. We consider that region II in $H\parallel c$ appears also below 1 K
from the constraint that the transition between region I and region II occurs
at the same temperature at the zero field. In region III, $K_{b}$ does not
decrease in the SC state. This indicates that the SC state with
$\bm{d}\perp\bm{b}$ is realized. Meanwhile, $K_{c}$ does not decrease in the
SC state in region IV, indicating that the SC state with $\bm{d}\perp\bm{c}$.
Because the change in $\Delta K_{\rm spin}$ against $H$ is gradual in
$H\parallel b$ and $H\parallel c$, it seems that the transitions between
region I and III and between region I and IV are crossovers. Since no phase
transitions were detected between the different regions with other
experimental techniques, a further experimental confirmation is desired.

Finally, we discuss a possible SC pairing symmetry in each region that is
based on the present experimental results and the theoretical considerations.
The Table 1 represents the character tables for point groups (a) $D_{2h}$, (b)
$C^{b}_{2h}$, and (c) $C^{c}_{2h}$[29]. Here, the $D_{2h}$ symmetry is reduced
to $C_{2h}^{b}$ in $H~{}\parallel~{}b$ or $C_{2h}^{c}$ in $H~{}\parallel~{}c$.
The SC pairing symmetry is classified into the one of the irreducible
representations listed on the tables, and the $\bm{d}$ vector can be
represented by the corresponding basis function.

Table 1: Character tables for point groups (a) $D_{2h}$, (b) $C^{b}_{2h}$, and
(c) $C^{c}_{2h}$[29]. Only odd-parity irreducible representations (IR) are
listed. The notation $x,y,z$ in the reference[29] correspond to $a,b,c$ in the
table.

(a) $D_{2h}$ (zero magnetic field)
---
IR | $E$ | $C_{2c}$ | $C_{2b}$ | $C_{2a}$ | $I$ | $\sigma_{c}$ | $\sigma_{b}$ | $\sigma_{a}$ | Basis functions
$A_{u}$ | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 |  $k_{a}\hat{a}$, $k_{b}\hat{b}$, $k_{c}\hat{c}$
$B_{1u}$ | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 |  $k_{b}\hat{a}$, $k_{a}\hat{b}$
$B_{2u}$ | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 |  $k_{a}\hat{c}$, $k_{c}\hat{a}$
$B_{3u}$ | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 |  $k_{c}\hat{b}$, $k_{b}\hat{c}$

(b) $C_{2h}^{b}$ ($H\parallel b$)
---
IR | (IR)$\uparrow D_{2h}$ | $E$ | $C_{2b}$ | $I$ | $\sigma_{b}$ | Basis functions
$A_{u}$ | $A_{u}+B_{2u}$ | 1 | 1 | -1 | -1 |  $k_{a}\hat{a}$, $k_{b}\hat{b}$, $k_{c}\hat{c}$, $k_{a}\hat{c}$, $k_{c}\hat{a}$
$B_{u}$ | $B_{1u}+B_{3u}$ | 1 | -1 | -1 | 1 |  $k_{b}\hat{a}$, $k_{a}\hat{b}$, $k_{c}\hat{b}$, $k_{b}\hat{c}$

(c) $C_{2h}^{c}$ ($H\parallel c$)
---
IR | (IR)$\uparrow D_{2h}$ | $E$ | $C_{2c}$ | $I$ | $\sigma_{c}$ | Basis functions
$A_{u}$ | $A_{u}+B_{1u}$ | 1 | 1 | -1 | -1 |  $k_{a}\hat{a}$, $k_{b}\hat{b}$, $k_{c}\hat{c}$, $k_{b}\hat{a}$, $k_{a}\hat{b}$
$B_{u}$ | $B_{2u}+B_{3u}$ | 1 | -1 | -1 | 1 |  $k_{a}\hat{c}$, $k_{c}\hat{a}$, $k_{c}\hat{b}$, $k_{b}\hat{c}$

As mentioned above, the plausible SC pairing symmetry in region I is $A_{u}$
or $B_{3u}$. In region III(IV), the $K_{b}$($K_{c}$) does not decrease and
thus, the $B_{2u}(B_{1u})$ state without $d_{b}$($d_{c}$) component is
expected. From the crossover behavior of the boundary between region I and
region III(IV), the SC pairing symmetry of region I should be the same
irreducible representation as that of region III(IV) under $H\parallel b$
($H\parallel c$). As shown in Table 1, the $A_{u}$ state which is in the same
representation as $B_{2u}$ state in $C_{2h}^{b}$ and as $B_{1u}$ state in
$C_{2h}^{c}$ is more suitable than the $B_{3u}$ state for the SC pairing
symmetry of region I. To determine the SC pairing symmetry at region I, the
Knight shift measurement in $H\parallel a$ is crucial.

A recent theoretical calculation that is based on a periodic Anderson model
reveals that the $A_{u}$ and $B_{3u}$ states are almost degenerate and the
dominant component of the $\bm{d}$-vector for the $A_{u}$($B_{3u}$) state is
$\bm{d_{b}}$($\bm{d_{c}}$)[30]. This is consistent with our experimental
results. According to the calculation, it seems that the degeneracy of the
$A_{u}$ and $B_{3u}$ states results in the coexistence of the high-$T$ and
low-$T$ SC phases expected in region II. In region II, the broad signal is
quickly suppressed against $H$ along the $c$-axis, indicating that the low-$T$
SC phase includes $B_{3u}$ component with a large $\bm{d_{c}}$. We suggest
that the $\alpha A_{u}+\beta B_{3u}$ state, whose complex coefficients
$\alpha,\beta$ are changed depending on the local situations, can be
consistent with the characteristic NMR spectrum observed in region II.

For multi-component SC state, the time-reversal symmetry-breaking SC state
originating from a chiral or a nonunitary state is expected. Although our
measurements were performed under fields, our results are possibly related to
the recent results with the polar Kerr effect[25] and the scanning tunneling
microscopy measurements[32]. We also point out that the spin-susceptibility-
distributed signal in region II can be also explained by the properties of the
superconductivity with the spin degree of freedom. It might be possible that
the texture of $\bm{d}$-vector induces such a spatially modulated SC state in
spin-triplet superconductors.

##  5 Conclusion

We performed 125Te-NMR measurements on a single-crystal UTe2 to investigate
the SC properties from a microscopic point of view. A shoulder signal appeared
inside the SC phase when $H\parallel b$. The area fraction of the shoulder
signal shows a maximum at $\sim 6$ T and vanishes above 10 T, although the
superconductivity confirmed by the $\chi_{\rm AC}$ measurements survives to at
least 14.5 T. In contrast, an unexpected broadening without the shoulder
signal was observed when $H~{}\parallel~{}c$ at a temperature below $T_{\rm
c}$. This broadening is quickly suppressed by increasing $H$. In addition, the
decrease in $K_{b}$ in the SC state starts to be small at around 7 T and
almost zero at 12.5 T. This indicates that the SC spin state gradually changes
with the application of $H$. Based on the experimental results, we constructed
phase diagrams for both field directions. The SC phase can be distinguished
into four regions based on the character of the SC spin state. In particular,
an inhomogeneous low-$T$ SC state, where the high-$T$ and low-$T$ SC phases
coexist, is expected to be realized in the low-$T$ and low-$H$ regions. The
low-$T$ SC phase originates from the presence of the spin degrees of freedom.
This finding places strong constraints on the SC pairing symmetry of UTe2, and
it also opens up new possibilities for the spin-triplet SC states.

###### Acknowledgements.

The authors would like to thank M. Manago, J. Ishizuka, Y. Yanase, S.
Fujimoto, Y. Maeno, and S. Yonezawa for their valuable discussions. This work
was supported by the Kyoto University LTM Center, Grants-in-Aid for Scientific
Research (Grant Nos. JP15H05745, JP17K14339, JP19K03726, JP16KK0106,
JP19K14657, JP19H04696, JP20H00130, and JP20KK0061, and a Grant-in-Aid for
JSPS Research Fellow (Grant No. JP20J11939) from JSPS.

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