# Meta-Optics Triplet for Zoom Imaging at Mid-Wave Infrared

Anna Wirth-Singh1,∗
&Arturo Martin Jimenez2 &Minho Choi3 &Johannes E. Fröch1,3
&Rose Johnson3 &Tina Le Teichmann1 &Zachary Coppens2 &Arka Majumdar1,3,∗∗
&
1Department of Physics, University of Washington, Seattle, WA, USA
2CFD Research Corporation, Huntsville, AL, USA
3Department of Electrical and Computer Engineering, University of Washington,
Seattle, WA, USA
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Lenses with dynamic focal length, also called zoom functionality, enable a
variety of applications related to imaging and sensing. The traditional
approach of stacking refractive lenses to achieve this functionality results
in an expensive, heavy optical system. Especially for applications in the mid-
infrared, light weight and compact form factor are required. In this work, we
use a meta-optic triplet to demonstrate zoom imaging at mid-wave infrared
wavelengths. By varying the axial distances between the optics, the meta-optic
triplet achieves high quality imaging over a zoom range of 5x, with 50∘ full
field of view in the widest configuration and an aperture of 8 mm. This
triplet system demonstrates the potential for meta-optics to reduce
conventional components in complex and multi-functional imaging systems to
dramatically thinner and lighter components.

##  1 Introduction

Thermal imaging has extensive applications in defense, night vision, and
meteorology [1, 2, 3, 4]. In particular, the mid-wave infrared (MWIR, 3-5
$\mu$m) is especially attractive for long-range imaging and sensing due to its
superior penetration in high-humidity atmospheric conditions [5]. Wide field
of view and high resolution are often desirable for these imaging
applications. For imaging dynamic targets, it is particularly helpful to have
both wide field of view images for context and narrow field of view for
zoomed-in detail, with rapid switching mechanism between the views. However,
achieving both qualities simultaneously requires a large, high-resolution
camera sensor, and such sensors are particularly expensive at MWIR
wavelengths. Alternatively, these applications can benefit from zoom
functionality, wherein the image magnification is changed by repositioning
optics within the lens system so that a range of image magnifications can be
captured using a single aperture and sensor.

Most conventional zoom systems are composed of a series of refractive lenses,
some of which are axially translated to change the zoom state. While
effective, stacked refractive lenses quickly become heavy, which is
undesirable for many applications where zoom functionality is required, such
as airborne scopes and cameras. Due to the long total track length of zoom
imaging systems, excess weight can cause significant torque on the mechanical
support system. In addition, to achieve high quality imaging with zoom
constraints, aspheric lenses are often used [6, 1] that are difficult to
manufacture due to non-constant curvature across the surface. Combined with
the limited availability of suitable materials for MWIR refractive lenses,
such as germanium and calcium fluoride, this contributes to the increased cost
of thermal zoom imaging systems.

Ultra-thin meta-optics have demonstrated great potential to miniaturize
imaging systems, including those at MWIR wavelengths [7, 8, 9]. However,
introducing zoom functionality into meta-optics, and other forms of
metasurfaces, is challenging. Various techniques have been employed, including
mechanical stretching [10, 11, 12], MEMS [13], and Alvarez [14, 15] lenses,
but these techniques are limited in zoom range and aperture. Similarly to the
conventional approach of stacking a series of refractive lenses, meta-optics
can also be stacked to improve imaging performance. Meta-optic doublets for
applications including aberration correction [16, 17] and large field of view
[18] have been demonstrated, but without a tuning mechanism. Recently,
researchers have demonstrated zoom imaging by axial translation in a doublet
lens system at 940 nm using two meta-optics with extended depth of focus for
extended zoom range, but for a relatively small aperture of 1 mm [6]. Further
extension into triplet meta-optic systems is a relatively unexplored area,
with existing demonstrations focused mainly on chromatic aberration correction
[19, 20].

Figure 1: The zoom lens concept. (a) Ray tracing diagrams for all five
configurations. By translating the compensator and variator optics, the
effective focal length of the system is changed from wide field of view (top)
to narrow field of view (bottom). The blue lines represent normally incident
rays, while the yellow rays represent $25^{\circ}$ and $5^{\circ}$ angle of
incidence for the wide and narrow configurations, respectively. Intermediate
angles are shown in shades of green. (b) Illustration of zoom imaging on a
car. Under 5x zoom, the details of the side mirror and windows become clear.

In this work, we demonstrate an 8 mm aperture, 50∘ full field of view (FoV)
zoom imaging meta-optic triplet at 3.4 $\mu$m wavelength. We follow the
traditional approach by varying the position of two of the three meta-optic
elements to increase the image magnification from 1x to 5x. Additionally, our
zoom imaging system is parfocal, meaning that the focal plane remains fixed
while achieving the intermediate zoom states. For optimal transmission through
the triplet lens system, we realize the meta-optics in a silicon-on-sapphire
material platform. We demonstrate high-quality imaging performance up to 50∘
FoV in the wide field of view configuration. This development enables MWIR
applications with zoom functionality and lightweight multi-element imaging
systems.

##  2 Zoom Lens Design

At minimum, most optical zoom systems consist of a mechanically adjustable
afocal frontend and imaging backend. In order to keep a fixed back focal
length, the front part of the system must be composed of two moving groups
[2], resulting in a triplet lens system. The design for the meta-optic zoom
lens follows this standard three-group zoom structure (consisting of a
compensator, a variator, and a focusing optics) and the zoom behavior is
schematically depicted in Figure 1. The system is optimized for five discrete
zoom states, denoted configurations 1 through 5, with this change accomplished
by varying the axial distances between the metasurfaces. Specifically, the
meta-optic nearest the sensor, termed the focusing optic, remains a fixed 15
mm distance from the sensor while the positions of the other two optics
(denoted as the compensator optic nearest the object and the variator optic in
the middle) are adjusted to change the zoom state. This change is primarily
accomplished by axially translating the variator optic, which is adjusted
within a range of 14.6 mm. The compensator optic has negative power to defocus
the light and its axial position is adjusted within a smaller range of 5.21 mm
to correct for aberrations. Ray tracing diagrams of all configurations are
shown in Figure 1a.

The designed F-number, effective focal length (EFL), and full field of view of
each configuration are summarized in Table 1. Configuration 1, also called the
"wide" configuration, provides the widest field of view at $50^{\circ}$ full
FoV and F-number of 3.7. Configuration 5, also called the "tele"
configuration, provides the most zoomed-in image with $10^{\circ}$ full FoV at
an F-number of 8.0. The intermediate Configurations 2 through 4 provide
equally spaced intermediate zoom states.

Table 1: System Design Specifications Configuration | 1 (Wide) | 2 | 3 (Mid) | 4 | 5 (Tele)
---|---|---|---|---|---
F Number | 3.71 | 3.94 | 4.40 | 5.33 | 7.98
EFL (mm) | 8.78 | 10.53 | 14.19 | 21.74 | 43.89
Full FoV | 50∘ | 40∘ | 30∘ | 20∘ | 10∘
$d_{1}$ (mm) | 16.60 | 13.57 | 9.64 | 5.71 | 2.00
$d_{2}$ (mm) | 2.00 | 2.82 | 4.51 | 7.88 | 16.60

The phase profiles of the meta-optics were optimized using commercial ray
tracing software (Zemax OpticStudio). Each meta-optic was modeled as a
Binary-2 phase profile according to the equation

$\Phi(\rho)=\sum_{i=1}^{6}A_{i}\left(\frac{\rho}{M}\right)^{2i}$ (1)

where $M$ is a normalization constant, $\rho$ is the radial coordinate, and
$A_{i}$ are polynomial coefficients. To optimize the zoom system, we initially
varied the phase profile coefficients $A_{i}$ to optimize performance at both
the tele and wide configurations at predetermined positions. Specifically,
wavefront error (with respect to an ideal spherical wavefront) was defined as
the merit function during optimization. The optimized coefficients of each
metasurface are provided in the Supplementary Material. Once the phase for
each metasurface was determined, the spacings between the surfaces were varied
to obtain the desired intermediate zoom states. As labeled in Figure 1a, the
distance between the compensator and variator optics is denoted $d_{1}$ and
the distance between the variator and the focusing optics is denoted $d_{2}$.
The distance between the focusing lens and the image plane, denoted $d_{3}$,
was held constant at 15 mm while $d_{1}$ and $d_{2}$ were allowed to vary.

Figure 2:  Metasurface design and fabrication. (a) Top-down SEM image of a
fabricated optic. (b) SEM image of a fabricated optic taken at a slightly
oblique (10∘) angle, illustrating fabrication quality. (c) The simulated phase
(red) and transmission (blue) of the unit cell. The widths selected for the
pillar library are denoted by red circles. (d) A photograph of the fabricated
optics. A person is holding the compensator optic in the foreground while the
variator (left) and focusing (right) optics are resting on the table in the
background. (d) A photograph of the meta-optics in the experiment setup.

##  3 Meta-Optics Design and Fabrication

The meta-optics are physically realized as crystalline silicon pillars on
sapphire substrate (commercially available from UniversityWafer). While
silicon is transparent at MWIR wavelengths, bulk silicon incurs significant
Fresnel reflections due to its high index of refraction (n = 3.5) as compared
to the air (n = 1) [21]. Therefore, to provide increased transmission
efficiency in the multi-layer system, we chose sapphire (n = 1.7) substrate.
Transmission through the sapphire substrate is expected to be 93%. The silicon
pillars are 1.5 $\mu$m tall arranged on a lattice with periodicity 1.5 $\mu$m,
and the sapphire substrate thickness is 460 $\mu$m. Using rigorous coupled
wave analysis [22], we simulated the phase and transmission of silicon pillars
as a function of pillar width as shown in Figure 2c. From these results, we
selected ten pillars with widths ranging from 500 nm to 1.0 $\mu$m to comprise
the pillar library.

The meta-optics were fabricated using electron beam lithography and
inductively coupled plasma etching. All three meta-opics were fabricated from
a single 1.5 $\mu$m thick silicon-on-sapphire wafer that was diced into chips.
Each sample was cleaned in acetone and isopropyl alcohol before an oxygen
plasma treatment to promote resist adhesion. Next, we spin-coated ZEP-520A
resist and exposed the resist by electron beam lithography. After writing the
pattern, we developed the resist in amyl acetate followed by a short oxygen
plasma treatment to remove residue. In order to achieve a large etching depth
of 1.5 $\mu$m, we deposited a hard mask for etching. Specifically, we
deposited 70 nm of alumina on top of the patterned samples via electron beam
evaporation. To form the hard mask, we lift-off the resist with
N-Methyl-2-pyrrolidone. Finally, the silicon was etched via reactive ion
etching with SF6 and C4F8 gas chemistry.

Top-down and oblique scanning electron microscope (SEM) images of the
fabricated optics are shown in Figure 2a and b, respectively, illustrating the
fabrication quality. A photograph of the fabricated optics is shown in Figure
2d. Except for the meta-optic area, the thin silicon layer has been etched
away, revealing the transparent substrate. The surface of the optic, while
appearing opaque brown under visible illumination, is transparent under
infrared illumination.

##  4 Experiment Imaging Results

The imaging quality of the meta-optical system was characterized in a simple
experimental setup consisting of an illumination source and target object, a
narrowband filter, the meta-optics, relay optics, and a camera sensor. A
photograph of the fabricated optics in the measurement setup is shown in
Figure 2e. As the source of MWIR illumination, we use a hot plate (Torrey
Pines Scientific HP50) set to 130∘C. This provides broadband infrared
radiation that we filter to a narrower band near the design wavelength using a
narrowband filter (Thorlabs FB3500-500, with 3.5 $\mu$m center wavelength and
FWHM bandwidth of 500 nm). A matte black anodized aluminum target projecting
the USAF 1951 resolution chart is placed just after the hot plate to form the
imaging object. In order to demonstrate imaging up to $50^{\circ}$ FoV, we
place the target at a distance $D$ from the first meta-optic such that the
edges of the target are $25^{\circ}$ from normal incidence. For our target
which is 12.5 cm wide, this corresponds to $D=13.4$ cm in imaging experiments.
Images were collected on a FLIR A6751 MWIR InSb camera sensor with 640
$\times$ 512 pixels at 15 $\mu$m per pixel resolution. Due to the camera
sensor being located further than the designated 15 mm back focal length
inside the protective camera housing, it was necessary to use relay optics to
re-image the meta-optics image onto the sensor. For this purpose, we used an
F/1 MWIR-coated, germanium plano-convex singlet lens (Edmund Optics #69-649,
25 mm focal length) with an F/2.5 compound refractive lens assembly (FLIR
#4218538, 25 mm focal length). To ensure that the relay optics do not restrict
measurement of the meta-optics, we ensure that the numerical aperture (NA) of
the relay optics ($NA>0.45$) exceeds the maximum NA of the meta-optic zoom
system ($NA=0.42$). The relay optics additionally aid in precise alignment of
the system; details of the alignment procedure are provided in the
Supplementary Material.

Figure 3:  Simulated and experimental imaging results. (a) Experiment imaging
results for Configurations 1 (left) through 5 (right). The imaging object, a
hot plate and USAF resolution chart, is placed such that the edges of the
target cover 50∘ FoV with respect to the meta-optics. (b) Experimental results
that have been computationally enhanced with sharpening and denoising. (c)
Simulated imaging results via ray tracing.

The simulated and experimental imaging results for all five configurations are
shown in Figure 3, with the USAF resolution target positioned such that the
horizontal edges of the imaging target correspond to 50∘ full FoV. To correct
for noise due to ambient radiation on the sensor, we performed a flat field
correction on all experimental results by collecting an image with the object
blocked and subtracting that background image from the signal; the resulting
image captures are shown in Figure 3a. We observe good imaging quality and
close agreement with the simulated results shown in Figure 3c. However, the
somewhat broad linewidth of the illumination source (approximately 500 nm FWHM
with the narrowband filter) introduces chromatic aberrations that reduce the
sharpness of the image as compared to a single-wavelength measurement. To
mitigate this effect, simple computational postprocessing techniques can be
introduced to improve the image quality. In Figure 3b, we show the experiment
imaging results after postprocessing with a Gaussian kernel sharpening filter
and bm3d denoising algorithm [23]. While there still exists some diffuse noise
that we attribute primarily to reflections from the narrowband filter, the
USAF groups are effectively sharpened for close qualitative agreement with the
simulated single-wavelength results in Figure 3c. Moving from left to right,
the system exhibits clear zoom functionality and qualitatively good imaging
performance in Configurations 1 through 5, as well as close agreement between
the simulation and experiment.

Figure 4:  Tolerance to meta-optic misalignment. (a) Nominal MTF for
Configuration 1 for angles of incidence up to 25∘. The reported MTF is the
average over both sagittal and tangential directions. The diffraction limit is
plotted in black for comparison. The vertical red line indicates the 20 lp/mm
= 0.18 cyc/mrad resolution that was chosen as a metric for the tolerance
analysis. (b) Schematic depicting the categories of misalignments that were
considered. In the Monte Carlo analysis, all three metasurfaces were
misaligned according to the specified tolerances. (c)-(e) Tolerance analysis
for the wide configuration. For each field (up to 25∘), we plot the MTF at 20
lp/mm as a function of allowed misalignment. The misalignment is decentering
in (c), tilt in (d), and axial translation in (e). The error bars represent
one standard deviation from Monte Carlo analysis consisting of 20 trials.

##  5 Discussion

While this zoom lens system was optimized for single wavelength illumination
at 3.4 $\mu$m, we demonstrate high-quality imaging over a wavelength band of
approximately 500 nm. In meta-optics (and other forms of diffractive optics)
with set phase wraps, the focal length is inversely proportional to the
optical wavelength [24]. Therefore, light of different wavelengths does not
focus in the same plane, leading to chromatic aberrations. Despite this
effect, we demonstrate relatively good imaging quality with the addition of a
narrowband filter. While single-wavelength rather than broadband illumination
is predicted to improve imaging performance in simulation, further filtering
ambient light to a narrower range would also reduce the signal to noise ratio
(SNR) due to the reduced light entering the system, unless some active
illumination source (such as an LED or laser) is used to illuminate the
target. However, such active illumination would be impractical for many
applications of this zoom system, such as remote sensing. Therefore, we
present imaging results over a broader band, for higher SNR, coupled with
computational postprocessing to regain some quality lost due to undesirable
chromatic effects. Extension to broader bandwidth operation may be possible
using dispersion engineering [25] to match the required phases over a range of
wavelengths or applying inverse design techniques [26] to extend the operating
wavelength range.

In multi-element optical systems such as this one, misalignment can have
deleterious effects on lens performance. To quantify the effects of
misalignment on the performance of the system, we conducted a tolerance
analysis of Configuration 1 in Zemax at the design wavelength of 3.4 $\mu$m,
shown in Figure 4. The nominal MTF is shown in Figure 4a. In the absence of
misalignment, the imaging performance is nearly diffraction limited at normal
incidence and decreases slightly at larger angles of incidence. As a metric of
lens system performance, we report the average (sagittal and tangential) MTF
at 20 lp/mm resolution as a function of alignment tolerances. As illustrated
in Figure 4b, we specifically consider the effects of translation in Z (axial
displacement), translation in X/Y (decentering), and rotation about the X axis
(tilt). We varied each category of misalignment separately in this analysis.
In detail, we conducted a Monte Carlo analysis comprising 20 trials wherein
each element was misaligned by a random amount within the specified tolerance
along the horizontal axis. We considered reasonable experimental alignment
tolerances of up to 1 mm of translation and up to $5^{\circ}$ of tilt in each
element. In Figure 4c through 4e, we report the average MTF value as a
function of worsening misalignment for angles of incidence up to 25∘. As
expected, the average MTF decreases with increasing misalignment, however this
decrease is approximately linear. This indicates robustness in the system that
we attribute to the relatively small phase gradient of each metasurface.

Our meta-optics-based zoom lens system achieves a zoom range of 5x, which is
sufficient for many applications. To further increase the zoom range, it would
generally require decreasing the NA of the system, which results in reduced
imaging quality and field of view [6]. Similarly to refractive optics,
additional layers of optics provide degrees of freedom that can be used to
improve performance and achieve additional functionalities, such as increased
zoom range. This successful demonstration of a meta-optic triplet system
highlights the potential for meta-optics to replace refractive elements in
increasingly complex multi-layer and multi-functional imaging systems. In
particular, we envision meta-optical systems like this one finding utility in
weight-constrained systems requiring zoom functionality and wide field of
view, such as airborne remote sensing applications.

## Funding

Funding for this work was supported by the federal STTR program.

## Acknowledgements

Part of this work was conducted at the Washington Nanofabrication Facility /
Molecular Analysis Facility, a National Nanotechnology Coordinated
Infrastructure (NNCI) site at the University of Washington with partial
support from the National Science Foundation via awards NNCI-1542101 and
NNCI-2025489.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Author Contributions

Z.C. and A.M. conceived and supervised the project. A.W.-S. designed the meta-
optic unit cells, led the experimental measurements, and analyzed the data.
A.M.J. designed the zoom lens system in Zemax. M.C. fabricated the meta-
optics. J.F. and M.C. advised the experimental setup and edited the
manuscript. R.J. and T.L.T. contributed to the experimental measurements.
A.W.-S. wrote the manuscript with input from all coauthors.

## Disclosures

A.M. is a co-founder of Tunoptix, which is commercializing similar meta-optics
in the visible. Z.C. and A.M.J. work at CFDRC, which is commercializing
infrared imaging systems.

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# Intrinsic Orbital Origin for the Chirality-Dependent Nonlinear Planar Hall
Effect of Topological Nodal Fermions in Chiral Crystals

Mingxiang Pan  School of Physics, Peking University, Beijing 100871, China
Hui Zeng  State Key Laboratory of Low-Dimensional Quantum Physics, Department
of Physics, Tsinghua University, Beijing 100084, China  Erqing Wang  School of
Physics, Peking University, Beijing 100871, China  Huaqing Huang
<EMAIL_ADDRESS>School of Physics, Peking University, Beijing 100871,
China  Collaborative Innovation Center of Quantum Matter, Beijing 100871,
China  Center for High Energy Physics, Peking University, Beijing 100871,
China

###### Abstract

Topological semimetals in chiral crystals, which possess both structural
handedness and band crossings (or nodes) with topological chiral charge,
exhibit many exotic physical properties. Here we demonstrate that the
structural and electronic chirality of these systems can endow them with
another fascinating phenomenon—the intrinsic nonlinear planar Hall effect
(INPHE), which is prominent around the nodes and reverses sign upon chirality
reversal in opposite enantiomers. Taking chiral tellurium as an example, we
reveal an intrinsic orbital mechanism, which manifests diverging orbital
magnetic moments with hedgehog-like textures around nodes and, therefore,
generates a dominant contribution to the INPHE that is proportional to the
topological charge. Furthermore, we show that multifold fermions in
topological chiral semimetals with B20 structures (e.g., CoSi and PtAl) induce
a giant INPHE conductivity reaching the order of $1\sim
10\;\mathrm{A}\cdot\mathrm{V}^{-2}\cdot\mathrm{T}^{-1}$, which is detectable
in experiments. Our study not only relates nonlinear transport to band
topology and enantiomer recognition but also offers a new way to explore the
exotic physical properties associated with unconventional chiral fermions.

Chiral crystals which possess a well-defined handedness due to the lack of
inversion and mirror symmetries, exhibit numerous fascinating non-reciprocal
phenomena [1, 2, 3, 4], such as electric magnetochiral anisotropy [5, 6, 7, 8]
and chirality-induced spin selectivity [9, 10, 11, 12]. In topological chiral
semimetals, such as tellurium (Te) [13, 14, 15, 16] and B20 compounds (e.g.,
CoSi and PtGa) [17, 18], in addition to structural chirality, a definite
electronic chirality can be assigned to electron wavefunctions at point-like
two- or multi-band crossings (nodes) of the quasiparticle dispersions, which
are known as Weyl or multifold fermions. These nodal fermions are
characterized by an integer topological chiral charge $\mathcal{C}$, which is
the Chern number for any surface enclosing the nodal point in momentum space
[19, 20]. The electronic chirality, which reverses signs in chiral variants
with opposite handednesses (enantiomers), also results in a plethora of novel
phenomena, such as long Fermi-arc surface states [21, 22, 23], unconventional
radial spin textures [13, 15, 14, 16, 24, 25], and most notably the quantized
circular photogalvanic effect whose magnitude associated with optical
transitions near a nodal point is proportional to its topological charge [26,
27, 28, 29, 30]. In this Letter, we demonstrate another novel chirality-
dependent phenomenon—the nonlinear planar Hall effect (NPHE), which strongly
enhances around nodal points and reverses sign in opposite-handed enantiomers.
We further reveal a previously overlooked intrinsic orbital origin of this
effect, which is endowed by the electronic chirality.

The NPHE generates a nonlinear Hall response current ($j_{\rm{H}}\propto
E^{2}B$) by a driving $E$ field under an in-plane magnetic $B$ field, and
therefore, can be regarded as the magnetic perturbated response of the
recently-discovered nonlinear anomalous Hall effect [31, 32, 33, 34, 35, 36,
37, 38, 39]. The NPHE was experimentally observed at the two-dimensional (2D)
surface of topological insulators [40] and recently were reported in chiral
semimetals Te and CoSi with a clear dependence on structural chirality [41,
42]. Previous theoretical studies on this effect mostly focus on the extrinsic
mechanisms that depend on scattering and scale as $\tau^{2}$ in electron
relaxation time $\tau$ [43, 44, 45, 46]. More recently, a $\tau$-independent
intrinsic contribution was proposed by considering the Zeeman-type coupling
between the in-plane $B$ field and electron spin in 2D noncentrosymmetric
materials [47]. However, the orbital degree of freedom which also couples to
the $B$ field and may result in NPHE, has been relatively unexplored. Amongst,
the orbital effect in chiral crystals that exhibit unique orbital textures has
remained elusive [48]. Moreover, despite the success of the experiments, the
underlying microscopic mechanism of the chirality-dependence of NPHE in chiral
semimetals has not been fully recognized.

In this Letter, we demonstrate a direct connection between the NPHE and nodal
fermions in chiral crystals and reveal an intrinsic orbital mechanism that
dominates the effect. Taking Te as a prototypical example of topological
chiral semimetals, we show that the intrinsic NPHE (INPHE) is strongly
enhanced around the Weyl point (WP) and reverses sign in opposite-handed
enantiomers. Based on a generic $k\cdot p$ model, we analytically demonstrate
that the orbital magnetic moment around WPs exhibits a hedgehog-like texture
that can be characterized by electronic chirality. Consequently, the
pronounced INPHE is mainly attributed to the enhanced orbital magnetic moment
surrounding WPs and proportional to the topological chiral charge. In
addition, we show that multifold fermions in topological chiral semimetals,
including CoSi, RhSi, RhSn, PtGa, and PtAl, can also result in chirality-
dependent INPHE with even larger response amplitudes associated with their
high Chern numbers.

Weyl points in chiral Te.—The chiral crystal structure of Te consists of
weakly interacting atomic spiral chains, as shown in Fig. 1(a). The left- and
right-handed Te belongs to space groups $P3_{2}21$ (No. 154, $D_{3}^{6}$) and
$P3_{1}21$ (No. 152, $D_{3}^{4}$) depending on the handedness of the screw
axis along the z-axis. Our first-principles calculations 111See Supplemental
Material at http://link.aps.org/supplemental/xxx, for more details about the
analytical derivation of the INPHE based on generic $k\cdot p$ models, details
of first-principles calculation methods, and numerical results of INPHE for
chiral Te and B20 compounds Cosi, RhSi, RhSn, PtGa, and PtAl, which include
Refs. [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. [50, 51, 52, 53, 54, 55,
56, 57, 58, 59, 60] show that both the conduction band minimum (CBM) and
valence band maximum (VBM) are close to but slightly off the BZ corners (H and
H’) point with a band gap of about 0.3 eV (see Fig. 1(b). As shown in Fig.
1(d), spin-orbit coupling lifts the spin degeneracy at the bottom of the
conduction band and forms a WP at H and H’ due to the lack of inversion and
mirror symmetry [13, 15, 14, 16, 24, 61, 62]. More importantly, the WPs in the
right-handed Te behave as sources of the Berry curvature field in
${\bm{k}}$-space and the radial spin textures are outward [see Figs. 1(c,d)],
indicating their electronic chirality of $\mathcal{C}=+1$. In contrast, the
WPs in the left-handed Te have the opposite chirality $\mathcal{C}=-1$ and
inward radial spin texture. In both crystals, the chiralities are the same
between WPs at H and H’, which are linked by the time-reversal symmetry.
However, it does not violate the Nielsen–Ninomiya no-go theorem [63] because
their oppositely chiral charged partners exist above in energy. This leaves
the WPs near CBM an ideal playground to study the chirality-dependent INPHE.
Given that the WPs are about 2.5 meV above the CBM, hereafter we set the Fermi
energy $E_{F}=0$ at the WP for n-doped Te, which has been experimentally
realized by an external gating of exfoliated flakes or the hydrothermal growth
with dielectric doping technique [14, 64, 65, 66].

Figure 1: (a) Crystal structure of left-handed and right-handed Te from the
top view. (b) The band structure of right-handed Te. The insert shows the
Brillouin zone. (c) The enlarged conduction bands near the WP along $k_{z}$.
Red and blue represent positive and negative z-components of spin,
respectively. (d), Berry curvature vector field of right-handed Te around H in
$k_{x}$-$k_{y}$ plane. (e) A sketch of the experimental setup to measure
$\chi_{xzzx}$. (f,g) $\chi_{xzzx}$ versus the chemical potential $\mu$ for
left- and right-handed Te.

Intrinsic nonlinear planar Hall effect.—The INPHE conductivity $\chi^{\rm
int}$ defined by $j_{a}^{\rm int}=\chi_{abcd}^{\rm int}E_{b}E_{c}B_{d}$
($a,b,c,d$ are Cartesian indices) can be expressed in terms of band quantities
as (see Supplemental Material (SM) for more details ††footnotemark: ),

$\displaystyle\chi_{abcd}^{\rm int}$ $\displaystyle=$
$\displaystyle\sum_{n}\int_{\rm
BZ}\frac{d^{3}\bm{\mathit{k}}}{(2\pi)^{3}}f^{\prime}(\epsilon_{n})[\alpha_{abcd}^{n,\textnormal{{I}}}(\bm{\mathit{k}})+\alpha_{abcd}^{n,\textnormal{{II}}}(\bm{\mathit{k}})],$
(1) $\displaystyle\alpha_{abcd}^{n,\textnormal{{I}}}$ $\displaystyle=$
$\displaystyle v_{a}^{n}\Lambda_{bcd}^{n}-v_{b}^{n}\Lambda_{acd}^{n},$ (2)
$\displaystyle\alpha_{abcd}^{n,\textnormal{{II}}}$ $\displaystyle=$
$\displaystyle(\partial_{a}G_{bc}^{n}-\partial_{b}G_{ac}^{n})\mathcal{M}_{d}^{n},$
(3)

where $f^{\prime}$ is the derivative of the equilibrium Fermi distribution
function $f(\epsilon;T,\mu)$ [$T$ and $\mu$ are omitted in Eq. (1)]. The
superscript I and II represent the contributions from the magnetic
susceptibility $\Lambda^{n}_{abc}$ of Berry connection polarizability (BCP)
$G^{n}_{ab}$ [67, 68] and from the combination of the BCP dipole and magnetic
moment $\mathcal{M}_{d}^{n}$. Explicitly,

$G_{ab}^{n}(\bm{\mathit{k}})=2\textnormal{Re}\sum_{m\neq
n}\frac{v_{a}^{nm}v_{b}^{mn}}{(\epsilon_{n}-\epsilon_{m})^{3}},$ (4)

and $\Lambda_{abc}^{n,\textnormal{S(O)}}=\partial
G_{ab}^{n}(\bm{\mathit{k}})/\partial B_{c}$ depend on the interband spin and
orbital magnetic moments
$\bm{\mathcal{M}}^{mn,\textnormal{S}}=-g\mu_{B}\bm{\mathit{s}}^{mn}$ and
$\bm{\mathcal{M}}^{mn,\textnormal{O}}=\sum_{l\neq
n}(\bm{\mathit{v}}^{ml}+\delta^{lm}\bm{\mathit{v}}^{n})\times\mathcal{A}^{ln}/2$,
with $g$ the $g$ factor and $\mu_{B}$ the Bohr magneton. Here
$\mathcal{A}^{ln}$ is the unperturbed interband Berry connection and
$\bm{\mathit{s}}^{mn}(\bm{\mathit{v}}^{ml})$ is the spin (velocity) matrix
element [47, 46]. Therefore, both parts I and II of $\chi^{\rm int}_{abcd}$
can be further divided into spin and orbital contributions.

For usual planar Hall measurements in the x-z plane of chiral Te [41] [see
Fig. 1(e)], only two components $\chi_{xzzx}$ and $\chi_{zxxz}$ are allowed by
the $C_{2x}$ symmetry. For general cases where the applied electric and
magnetic fields are not aligned with crystal axes but make polar angles
$\theta$ and $\phi$ from the z-axis, i.e., $\bm{E}=E(\sin\theta,0,\cos\theta)$
and $\bm{B}=B(\sin\phi,0,\cos\phi)$, the angle-dependent INPHE conductivity is

$\chi_{\rm H}^{\rm
int}(\theta,\phi)=\chi_{xzzx}\sin\phi\cos\theta-\chi_{zxxz}\cos\phi\sin\theta.$
(5)

Here we focus on $\chi_{xzzx}$ to describe the INPHE in chiral Te and defer
$\chi_{zxxz}$ which exhibits similar behavior in Fig. S2 of SM ††footnotemark:
. For n-type Te with opposite handednesses, $\chi_{xzzx}^{\rm int}$ rapidly
develops peaks on the order of
$10^{-1}\mathrm{A}\cdot\mathrm{V}^{-2}\cdot\mathrm{T}^{-1}$ as $\mu$
approaches CBM, and undergoes a sign change when $\mu$ crosses the WPs, as
shown in Fig. 1(f,g). Remarkably, the left- and right-handed Te exhibit
opposite $\chi_{xzzx}^{\rm int}$, implying the chirality-dependence of INPHE
in chiral Te. Remarkably, our results are consistent with the recent
experiment on chiral Te which reported the gate-tunable and chirality-
dependent NPHE with cosine angular dependences on the $B$ field [41]. In
addition, we estimate extrinsic contributions that depend on $\tau^{2}$ and
found that their effects are negligible, indicating that INPHE is the primary
mechanism (see Sec. IV in SM ††footnotemark: ). It is also worth noting that
$\chi^{\rm int}$ mainly comes from part II of the orbital contribution
$\chi_{xzzx}^{\textnormal{{II}},{\rm O}}$ which originates from BCP dipole and
orbital magnetic moment [see Eq. (3)]. This is different from the conventional
spin-dominated case [47], indicating a distinct physical mechanism, as we will
discuss later.

Figure 2: (a) The BCP dipole
$\lambda_{xzz}^{n}({\bm{k}})=\partial_{x}G^{n}_{zz}-\partial_{z}G^{n}_{xz}$,
(b,c) orbital and spin magnetic moment $\mathcal{M}_{x}^{\rm
O}({\bm{k}}),\mathcal{M}_{x}^{\rm S}({\bm{k}})$, (d,e)
$\alpha_{xzzx}^{n,\textnormal{{I}},{\rm O}}({\bm{k}})$ and
$\alpha_{xzzx}^{n,\textnormal{{I}},{\rm O}}({\bm{k}})$ for the upper
conduction band of right-handed Te. (f) The orbital part of the integrand
$\beta_{xzzx}^{\rm
O}({\bm{k}})=\sum_{n}f^{\prime}(\epsilon_{n})(\alpha^{n,\textnormal{{I}},{\rm
O}}+\alpha^{n,\textnormal{{II}},{\rm O}})$ of Eq. (1) at 4 meV above the WP
for the right-handed Te. These quantities are shown in a patch
($[-2.2,2.2]\times[-0.8,0.8]$ in units of $10^{-3}$Å-1) of the $k_{x}$-$k_{z}$
plane around H.

To gain further insight into the dominant component
$\chi_{xzzx}^{\textnormal{{II}},{\rm O}}$, we analyze the ${\bm{k}}$-space
distribution of relevant band geometric quantities, including the BCP dipole
$\lambda^{n}_{xzz}=(\partial_{x}G_{zz}^{n}-\partial_{z}G_{xz}^{n})$, magnetic
moment $\mathcal{M}_{x}^{n}$, and
$\alpha_{xzzx}^{\textnormal{{I}}(\textnormal{{II}}),{\rm O}}$ for the upper
conduction bands. As shown in Fig. 2, these quantities are pronounced around H
where the energy difference between the two bands ($\Delta\epsilon$) is small.
This is because similar to Berry curvature [Fig. 1(c)], these quantities are
endowed by the interband coherence and scale as $1/(\Delta\epsilon)^{\eta}$
with $\eta$ being some integers (see SM ††footnotemark: ). Since both
$\lambda^{n}_{xzz}$ and $\mathcal{M}_{x}^{n,{\rm O}}$ are odd functions of
$k_{x}$ [Fig. 2(a,b)], the resultant $\alpha_{xzzx}^{\textnormal{{II}},{\rm
O}}$ exhibit a single negative peak for the upper band [Fig. 2(e)]. Moreover,
we find that the lower conduction band has an opposite peak of
$\alpha_{xzzx}^{\textnormal{{II}},{\rm O}}$ (see Fig. S5 in SM ††footnotemark:
), indicating a sign reversal of $\chi_{xzzx}^{\textnormal{{II}},{\rm O}}$
when $\mu$ passes the WP. In contrast, $\alpha_{xzzx}^{\textnormal{{I}},{\rm
O}}$ display similar quadrupole distributions for both bands, resulting in
small uncompensated contributions to the conductivity and without
$\mu$-dependent sign reversal. In addition, one observes that the spin
magnetic moment $\mathcal{M}_{x}^{n,{\rm S}}$ is orders of magnitude smaller
than $\mathcal{M}_{x}^{n,{\rm O}}$ [see Fig. 2(b,c)], hence the spin
contributions $\alpha_{xzzx}^{\textnormal{{I}},{\rm S}}$, and
$\alpha_{xzzx}^{\textnormal{{II}},{\rm S}}$ are negligible.

Analytic model study around WPs.—To better understand the behavior of
$\chi_{xzzx}^{\rm int}$ around WPs and its chirality dependence, we consider
the following Weyl model:

$\begin{split}H_{W}({\bm{k}})={\bm{h}}({\bm{k}})\cdot{\bm{\sigma}},\end{split}$
(6)

where ${\bm{\sigma}}=(\sigma_{x},\sigma_{y},\sigma_{z})$ are Pauli matrices
and ${\bm{h}}({\bm{k}})=(v_{1}k_{x},v_{2}k_{y},v_{3}k_{z})$ with $v_{1}=v_{2}$
constrained by the $D_{3d}$ symmetry at H. The model describes two energy
bands $\epsilon_{\pm}({\bm{k}})=\pm
h=\pm\sqrt{v_{1}^{2}k_{x}^{2}+v_{2}^{2}k_{y}^{2}+v_{3}^{2}k_{z}^{2}}$. The
orbital and spin magnetic moments are given by

$\bm{\mathcal{\bm{M}}}^{\pm,{\rm
O}}=-\frac{v_{1}v_{2}v_{3}}{2h^{2}}{\bm{k}}=\pm\bm{\Omega}^{\pm}h,\quad\bm{\mathcal{\bm{M}}}^{\pm,{\rm
S}}=\mp g\mu_{B}\hat{{\bm{h}}}.$ (7)

One observes that the orbital magnetic moments $\bm{\mathcal{M}}^{\pm,{\rm
O}}$ are the same for the two bands ($\pm$). Given that the directional vector
${\bm{m}}=\bm{\mathcal{M}}^{\pm,{\rm O}}/|\bm{\mathcal{M}}^{\pm,{\rm O}}|$ is
along with ${\bm{k}}$ [as shown in Fig. 3(a,b)], the orbital magnetism in
momentum space forms a hedgehog-like texture [like Berry curvature
$\bm{\Omega}({\bm{k}})$ in Fig. 1(d)] with a winding number on the Fermi
surface, $\zeta=\frac{1}{4\pi}\oint_{\rm
FS}{\bm{m}}\cdot(\partial_{\theta}{\bm{m}}\times\partial_{\phi}{\bm{m}})ds=-{\rm
sgn}(v_{1}v_{2}v_{3})=\mathcal{C}$, which directly relate to the electronic
chirality. However, the spin magnetic moments $\bm{\mathcal{M}}^{\pm,{\rm S}}$
are opposite for the two bands and only depend on the directional vector
$\hat{\bm{h}}$, which are consistent with the first-principles results (see
Fig 1(c) and SM ††footnotemark: ). Both spin and orbital magnetic moments are
pseudovectors that reverse sign under $\bm{k}\rightarrow-\bm{k}$ in reciprocal
space, i.e., $\bm{\mathcal{M}}(-\bm{k})=-\bm{\mathcal{M}}(\bm{k})$ as enforced
by time-reversal symmetry. They also reverse signs in opposite chiral variants
that are related by an inversion or mirror operation.

Figure 3: (a) Band structure with orbital moment (arrows) around H from the
$k\cdot p$ model of right-handed Te. (b) Orbital magnetic texture in the
H-centered Fermi pockets formed by the two bands ($\pm$) with $\mu$ being
chosen as 2 meV above the WP. (c) $\chi_{xzzx}(\mu)$ calculated from the
$k\cdot p$ model. (d) First-principles band structure along $k_{x}$ near H for
the right-handed Te under a biaxial strain of 3%. The solid (dashed) circle
represents WP with $\mathcal{C}=+1(-1)$. (e) Local gap between two conduction
bands in the $k_{x}$-$k_{y}$ plane, where three new WPs are labeled as
$W_{1,2,3}$. (f) $\chi_{zxxz}$ versus $\mu$ for the 3%-strained Te.

Next, we derive the geometric quantities in Eq. (3),

$\alpha_{xzzx}^{\textnormal{II},{\rm
O}}=\pm\frac{|v_{1}^{3}v_{2}v_{3}^{3}|k_{x}^{2}}{4h^{7}}\mathcal{C},\quad\alpha_{xzzx}^{\textnormal{II},{\rm
S}}=g\mu_{B}\frac{v_{1}^{3}v_{3}^{2}k_{x}^{2}}{4h^{6}}.$ (8)

Generally, $\alpha^{\rm O}$ scales as $1/(\Delta\epsilon)^{5}$ and
$\alpha^{\rm S}$ scales as $1/(\Delta\epsilon)^{4}$. Therefore, the orbital
contribution can be more enhanced than the spin part with a decreased gap
around the WP. Similarly, we derive $\alpha_{xzzx}^{\textnormal{{I}}}$ in Eq.
(2) for spin and orbital (see SM ††footnotemark: ), and eventually obtain all
contributions to the INPHE conductivity, $\chi_{xzzx}^{\textnormal{{I}},{\rm
S}}=-\chi_{xzzx}^{\textnormal{{II}},{\rm
S}}=\frac{g\mu_{B}}{24\pi^{2}\mu^{2}}\frac{|v_{3}|}{v_{1}},\chi_{xzzx}^{\textnormal{{I}},{\rm
O}}=0$, and

$\chi_{xzzx}^{\textnormal{{II}},{\rm
O}}=-\frac{v_{3}^{2}}{24\pi^{2}\mu^{3}}\mathcal{C}.$ (9)

Remarkably, the two spin parts have opposite signs which are canceled out,
resulting in zero net contribution. Therefore, the INPHE conductivity
originates primarily from part II of the orbital contribution. More
importantly, $\chi_{xzzx}^{\textnormal{{II}},{\rm O}}$ depends on the
chirality $\mathcal{C}$ of the WP, implying opposite INHPE for right- and
left-handed Te ($\mathcal{C}=\pm 1$). Moreover, $\chi^{\textnormal{{II}},{\rm
O}}$ would switch signs when the chemical potential is shifted from below
($\mu<0$) to above ($\mu>0$) the WP, which is consistent with the results in
Fig. 1(f,g). Noticing that the quadratic terms of ${\bm{k}}$, which would bend
over the lower conduction band, are neglected in the above model (6). By
including the symmetry-allowed quadratic term $H_{\rm
quad}({\bm{k}})=c_{1}(k_{x}^{2}+k_{y}^{2})+c_{3}k_{z}^{2}$, we found that
$\alpha^{\textnormal{{II}},{\rm O}}$ remains unchanged but
$\alpha^{\textnormal{{I}},{\rm O}}$ has a nonzero value which is about one
order of magnitude smaller than the former (see SM ††footnotemark: ). Based on
$H_{W}+H_{\rm quad}$ with parameters fitted from first-principles band
structures, we plot the conductivity in Fig. 3(c), which agrees well with Fig.
1(g).

Tilted WPs induced by strain.—Since the Weyl fermions are fine-tunable by
applying external strain [13, 14, 16, 69], we further study the INPHE of Te
under a biaxial strain along the hexagonal plane. As shown in Fig. 3(d,e),
three titled WPs with chirality $\mathcal{C}=-1$, which are related to each
other by the $C_{3}$ rotation, emerge around H in the 3%-strained right-handed
Te. This induces significant INHE when the Fermi level is shifted around these
WPs [see Fig. 3(f)]. Remarkably, we find a considerable contribution from
$\chi^{\textnormal{{I}},{\rm O}}$ which is larger than
$\chi^{\textnormal{{II}},{\rm O}}$ and exhibit a opposite sign inversion with
increasing $\mu$. Based on a tilted Weyl model $H_{\rm
tilt}=H_{W}({\bm{k}})+w_{1}k_{x}+w_{2}k_{y}$, we found that

$\alpha_{\rm tilt}^{\textnormal{{I}},{\rm
O}}=-\frac{w_{2}k_{y}|v_{1}v_{2}v_{3}^{5}|k_{z}^{2}}{8h^{8}}\mathcal{C}$ (10)

scale as $1/(\Delta\epsilon)^{5}$, which is comparable with
$\alpha^{\textnormal{{II}},{\rm O}}$ (see Sec. III.B in SM ††footnotemark: ).
By integrating Eq. (10) on the Fermi surface, we estimate
$\chi^{\textnormal{{I}},{\rm O}}\propto\mathcal{C}/\mu^{3}$, exhibiting the
similar behavior as $\chi^{\textnormal{{II}},{\rm O}}$ in Eq. (9). Therefore,
$\chi^{\textnormal{{I}},{\rm O}}$ should be strongly enhanced for tilted Weyl
fermions with clear dependences on the chirality and the chemical potential,
consistent with the significant peaks accompanied by a sign change as a
function of $\mu$ in Fig. 3(f). Our analysis also indicates that large INPHE
should be generally expected for type-II Weyl semimetals with over-tilt cone-
shape band structures, such as WTe2 [70, 71], TaIrTe4 [72, 73], and p-doped Te
[13, 15, 14].

Figure 4: (a) The crystal structure of the B20 compounds AB (A=Co, Rh, Pt;
B=Si, Sn, Ga, Al). The crystal is a right- or left-handed configuration when
the red or blue helices represent A atoms. (b) Band structure of the right-
handed CoSi where four- and six-fold degenerate nodes appear at the $\Gamma$
and R points. (c,d) $\chi_{zxxz}$ versus $\mu$ around the multifold nodes at
(c) $\Gamma$ and (d) R.

Multifold fermions with high Chern number.—In addition to chiral Te, we also
investigate the INPHE induced by multifold fermions in topological chiral
semimetals CoSi, RhSi, RhSn, PtGa, and PtAl. Figure 4(a) shows their chiral
cubic structures in space group $P2_{1}3$ (No. 198) which exhibit helices
along the [111] direction. We take the right-handed CoSi as an example and
defer the rest to Fig. S10-S14 in SM ††footnotemark: . Figure 4(b) shows four-
and six-fold degenerate nodes appear at the $\Gamma$ and R points [see Fig.
4(b)], which carry high Chern numbers of $\mathcal{C}=-4$ and 4, respectively.
Given that the twofold node can be described as the spin-1/2 Weyl fermions,
the four- and six-fold nodes can be regarded as spin-3/2 and double-spin-1
fermions, respectively. Based on the effective model of these multifold
fermions, we analytically derived a giant INPHE that shows similar dependence
on the chirality and chemical potential (See Sec. V in SM ††footnotemark: ).
Remarkably, we found that both $\chi^{\textnormal{{I}},{\rm O}}$ and
$\chi^{\textnormal{{II}},{\rm O}}$ make significant contributions to the
INPHE, which are proportional to their Chern numbers $\mathcal{C}$ and
dominate the INPHE over spin contributions. As shown in Fig. 4(c,d), the INPHE
is indeed strongly enhanced around these multifold nodes with its amplitude
being about $1\sim 10\;\mathrm{A}\cdot\mathrm{V}^{-2}\cdot\mathrm{T}^{-1}$,
which is $1\sim 2$ orders of magnitude larger than that of Weyl nodes.

Conclusion.—We have shown that nodal fermions in chiral crystals can lead to
prominent INPHE which depends on the electronic chirality with a compelling
connection to the structural chirality of host crystals. More importantly, we
reveal that the large INPHE mainly originates from the enhanced orbital
magnetism with node-centered hedgehog-like textures characterized by the
topological charge. Our study reveals an exotic chirality-dependent nonlinear
transport phenomenon, potentially leading to novel applications in chirality-
induced nonreciprocal devices and enantioselective processes. Our work also
paves a new way for studying other types of nodal fermions in more topological
semimetals, which opens the possibility of exploring unprecedented physical
properties of unconventional nodal fermions, such as magnetic Weyl [74, 75],
Dirac [76, 77], hourglass [78, 79], triple [80, 81], and other high-fold
chiral fermions [82].

###### Acknowledgements.

We thank Prof. Cong Xiao for the valuable discussion. This work is supported
by the National Key R&D Program of China (Grant No. 2021YFA1401600) and the
National Natural Science Foundation of China (Grant No. 12074006). The
computational resources are supported by the high-performance computing
platform of Peking University.

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