Topologically reconstructing Pancharatnam-Berry phase via encircling exceptional point for chiral spin-orbit interaction steering

Domain: physics
Published: 2026-03-15
Editorial outcome: minor_revision

Abstract
Photonic spin-orbit interaction (SOI) is related to Pancharatnam-Berry (PB) phase, which is
usually imposed on the cross-polarized components of light incident on a metasurface in
broadband manner. However, metasurfaces exhibiting exceptional points (EPs) can block the
cross-polarization conversion channel, hinting at the possibility of EPs influencing SOI. Here, we
demonstrate that the accumulation of PB phase is equivalent to the encircling of an EP, leading to
topologically reconstructed PB phase. Through spin-Hall effect and spin-to-vortex conversion, we
experimentally realize chiral SOI steering within chosen spectral ranges and polarization, showing
suppression, inversion and doubling of the PB phase accumulated via rotation. Our work
introduces the possibility to control SOIs via a topologically reconstructed PB phase, offering
promising possibilities for applications such as information encryption.

Introduction
Photonic systems exhibit spin orbit interaction (SOI)1, which describes the coupling between the
spin and orbital angular momentum of light as it propagates through anisotropic media 2,3. This
interaction provides a paradigm for regulating light-matter interaction and has led to various
intriguing phenomena, such as transverse spin 4,5, spin-momentum locking 6,7, and optical force
effects 8. Photonic SOI fundamentally stems from geometric phases: the spin-redirection phase
originating from wavevector evolution in momentum space, and the Pancharatnam-Berry (PB)
phase arising from polarization-state evolution on the Poincaré sphere 9,10. PB phase are widely
used in metasurfaces, a platform composed of periodically arranged sub-wavelength
nanostructures 11-13, to exhibit SOI phenomena, such as the photonic spin Hall effect 14,15, spin-
orbit conversion 16-18, spin-orbit mapping 19, and support applications including optical holography
20-22 and lenses 23. PB phase in metasurfaces are caused by the rotation of nanostructures, and is
imposed on the orthogonal polarization states 24,25. The integration of PB phase with nonlinear
optics 26,27, disorder 28, and phase-change materials (PCMs) 29-31,49 provides a means to steer
photonic spin-orbit interactions and achieve diverse functionalities. Among these methods,
additional tunability is introduced by PCMs, yet with increased fabrication or preservation
demands. Thus, discovering paradigms that use stable materials with simple fabrication to steer
photonic SOI is imperative. Metasurfaces, in addition to being platforms for SOI, are inherently
open systems where each nanostructure continuously exchanges energy with the incident light 32.
Such non-Hermitian systems naturally contain exceptional points (EPs) 32-35. EPs, non-Hermitian
degeneracies where both eigenvalues and eigenvectors degenerate, introduce various phenomena,
including enhanced sensitivity to perturbations, asymmetric mode switching, etc. 36-39. The
significance of EPs varies across different systems, such as micro-ring resonators and acoustic
systems 40-43. In metasurface systems, non-Hermitian Jones matrices serve as effective
Hamiltonians 33-34,44, leading to degenerate eigenstates that correspond to either left-circularly
polarized (LCP) or right-circularly polarized (RCP) states, reflecting the chirality of the EP 34. EPs
in this regime indicate that circular polarization conversion channel can be completely blocked
while leaving the opposite conversion channel unaffected 32. Thus, EPs provide a unique physical
mechanism that governs polarization conversion processes with chirality, and have inspired a
broad range of studies.45-48. As PB phase-mediated SOI processes operate via polarization
conversion, understanding and harnessing the influence of EPs on these processes becomes a
promising task.

Here, we demonstrated EP’s role in PB phase-mediated photonic SOI, and hence revealed three
chiral-sensitive mechanisms that can steer photonic SOI across visible to near-infrared spectra. In
detail, PB phase accumulation via rotation is topologically equivalent to encircling a polarization
conversion singularity, such as an EP, which reveals that PB phase-based SOI processes are
topologically protected. By introducing static elliptical perturbations, a rotational anisotropy is
induced, as their coupling to the main structure changes with rotation. This ensures that the
encircling trajectory avoids full EP collapse, encountering EPs only at discrete points. Such
encounters disrupt topological protection, change the winding number, and trigger three novel
steering mechanisms: chiral SOI suppression, inversion, and doubling (Fig. 1). These correspond
to suppressed, inverted, or doubled PB phase, respectively, at target wavelength range and
polarization. The chirality is governed by the main structure of the meta-atom. To show the
winding number changing mechanisms can steer photonic SOI on the geometric phase level,
photonic spin-Hall effect and spin-to-vortex conversion are both examined across visible and near-
infrared spectra. These effects are experimentally manifested as wavelength-sensitive spin-
dependent splitting in positional space and orbital-angular-momentum (OAM) generation.
Moreover, photonic SOI steering with different chirality and wavelength range is also verified
through experiments. The observed results match well with the simulations and showcase novel
mechanisms to steer SOI via PB phase with different winding numbers, which we term
topologically reconstructed PB phase (TR-PB phase). Different from nonlinear geometric phase
or active metasurfaces, our work fundamentally enriches PB phase accumulation rule with
fabrication simplicity, and adopts stable metal to steer SOI on the fundamental wave via
topological control with controlled chirality. Comparisons are made in Supplementary Table 1.
Our findings open up promising possibilities for the topological control in systems with
singularities 40 and advanced applications such as information encryption 49.

Results
Three phenomena mediated by TR-PB phase are shown in Fig. 1a-c, i. For suppression in Fig. 1a,
wavelength tuning under RCP transforms PB phase from -2θ to a θ independent constant,
decoupling SOI from rotation. This manifests as OAM transition ℓ = -2 to ℓ = 0 in spin-to-vortex
conversion shown in Fig. 1a, ii. The doubling phenomenon results in a PB phase change from −2θ
to −4θ under RCP incidence, as shown in Fig. 1b. For inversion in Fig. 1c, PB phase flips from -
2θ to +2θ by tuning wavelength under RCP incidence, yielding OAM inversion from ℓ = -2 to ℓ =
+2 shown in Fig. 1c, ii. However, LCP incidence maintains +2θ PB phase response throughout all
configurations in Fig. 1a-c, demonstrating the chirality of the phenomena. Meta-atoms, shown in
Fig. 1d-e, integrate rotating dual-L main structures with static perturbation ellipses on a platform
that employs a 160-nm indium-tin-oxide (ITO) film and 40-nm gold layer on glass, without the
use of PCMs. The meta-atoms for Fig. 1Ea, b, ii and Fig. 1c, ii (Marked by black dashed lines) are
schematized in Fig. 1d and Fig. 1e, respectively. These discovered phenomena cannot be
thoroughly interpreted in conventional Poincaré sphere formalism. To resolve this problem, we
adopted a complex plane perspective, reinterpreting PB phase accumulation.

We first reexamine the mathematics behind the PB accumulation process, which can be described
by acting the rotation operation on the Jones matrix. It is described by the following expression.
= − = 

[ − ] [ ] [
−
]    
  
   
J ( ) R ( ) J R ( )
c
s
o
in
s
c
s
o
in
s
T
T
x x
y x
T
T
x y
y y
c o
s
s
in
s
c
in
o s
(1)
Suppose the transmission Jones matrix of the meta-atom is described by J.
(T T ) (t ei xx t ei xy ) (a+ib c+id)
J = xx xy = xx xy = ( )
( T yx T yy ) ( t yx ei yx t yy ei yy ) ( c+id e+if ) (2)
Where Tij is the transmission from j-polarized input light to i-polarized output light. The meta-atom
can impart amplitude and phase modulation, represented byt and .
ij ij
Due to the reciprocity and
the anisotropy, T =T and T T , respectively. In addition, a-f are real values obtained from
xy yx xx yy
the complex amplitude modulation. After been rewritten in the basis of circular polarization, it is
easy to derive equation (3).
[ ] ( ) [ ] ( )
{ReT = a−e+2d 2 {ReT = a−e−2d 2
+− −+
[ ] ( ) [ ] ( )
ImT = b− f −2c 2 ImT = b− f +2c 2
+− −+
(3)
Where “+” represents LCP, while “-” represents RCP. By taking the non-Hermitian form Jones
matrix into equation (1), the Jones matrix J() can be obtained after rotation, which is described
by a-f. By thinking
( )  J as a new Jones matrix, based on equation (3), equation (4) can be derived.
{
[
+
+
−
−
]
=
=
+
+
−
−
+
−
+
+
−
−
  
  
R
I m
e T
T
(
(
)
)
R
I m
e (
(
T
T
)
)
c
c
o
o
s
s
2
2
I
R
m
e
(
(
T
T
)
)
s
s
in
in
2
2
{
[
−
−
+
+
]
=
=
−
−
+
+
−
+
−
−
+
+
  
  
R
I m
e T
T
(
(
)
)
R
I m
e (
(
T
T
)
)
c
c
o
o
s
s
2
2
I
R
m
e
(
(
T
T
)
)
s
s
in
in
2
2
(4)
From these expressions, the rotation operation is shown to drive an encirclement around the origin
of the complex plane, which corresponds to an EP of the non-Hermitian Jones matrix, as
established in supplementary note 1. This shift in perspective reveals that the accumulation of the
PB phase is equivalent to dynamically encircling an EP, with the direction of encirclement
determined by the chirality of the input circular polarization. As shown in Fig. 1(f,g), when under
LCP incidence, T () encircles the LEP in counterclockwise (CCW) manner, whereas under
−+
RCP incidence, ( )  T
+ −
encircles the REP in clockwise (CW) manner, while  continuously
increases from 0 to π. Building upon this point of view, we first dissect chiral SOI suppression and
its evolution in the complex plane.

Topological Reconstruction Controlled by a Single EP under Fixed Wavelength
When there is only one EP induced during rotation at a fixed wavelength, chiral SOI suppression
occurs, decoupling the photonic SOI from structural rotation. Here, we designed two metasurfaces
to demonstrate chiral SOI suppression under different chirality and response wavelength using
finite-difference time-domain (FDTD) parameter sweep simulations.

For these simulations, metasurfaces are set to the size of 10 μm * 10 μm, while perfectly matched
layers are applied at the x, y and z directions. We adopted the material parameters from “Au (gold)
Johnson and Christy” for simulating the structures and the refractive index is set to 1.6 for ITO.
Such configuration was applied throughout our work. The chiral SOI suppression was
demonstrated under RCP incidence within the wavelength range of 700-750 nm as shown in Fig.
2a. The meta-atom that we adopted for simulation is shown in Fig. 2a, i. The meta-atom has a
period of 650 nm with the following geometric dimensions: dx = 80 nm, dy = 45 nm, w = 75 nm,
L1 = 263 nm, and L2 = 145 nm. The perturbation ellipse has axes of 75 nm and 37.5 nm. The
centers of the two reference points (red dots) are separated by a displacement of (245 nm, 245 nm).
See supplementary note 2 for specific discussion. The cross-polarization conversion rate and PB
phase response are presented in Fig. 2a, ii-iii, where the x-axis represents rotation angle of the
main structure, and the y-axis represents input wavelength (See the line plots at the wavelengths
where the EPs occur in supplementary note 7, and the opposite chiral incidence results in
supplementary note 8). Only one EP emerge at 43° under 750 nm, with similar EP identified at 2°
under 700 nm incidence, as shown in Fig. 2a, ii. Corresponding PB phase response within the
range in Fig. 2a, iii became nearly rotation independent, which was in stark contrast with PB phase
response outside the EP influenced wavelength. We manifested such chiral SOI suppression in
photonic spin-Hall effect, using metasurface whose meta-atom rotates following equation (5)
shown below. As shown in Fig. 2a, iⅴ-ⅵi, under LCP excitation, output RCP remains leftward
deflected regardless of the excitation wavelength, while under RCP excitation, output LCP remains
rightward deflected until experiencing SOI suppression and becomes non-deflected in the EP
influenced region. Here we take response at 790 nm and 710 nm as an example, as shown in Fig.
2a, iv-ⅵi.

In Fig. 2b, the chiral SOI suppression was demonstrated under LCP incidence within the
wavelength range of 706 - 823 nm, and we manifested it in spin-to-vortex conversion, using
metasurface whose meta-atom rotates following equation (6), with topological charge set to 3 (ℓ =
3). Equations (5-6) are shown below.

ij
= ( j − 1 )  1 5 o  (5)
ij
=
1
2
 a r c t a n
( y
x
i
j
)
 (6)
In which i and j represent the label of meta-atom located at the ith row and jth column of the
metasurface, respectively, with coordinates referenced to the metasurface center. The main
structure of the meta-atom is a mirror-symmetric dual-L configuration to induce SOI suppression
with opposite chirality. As shown in Fig. 2b, i, the meta-atom has a period of 663 nm with the
following geometric dimensions: dx=80 nm, dy=47 nm, w=75 nm, L1=285 nm, L2=160 nm,
L3=185 nm. The perturbation ellipse has axes of 100 nm and 65 nm. The centers of the two
reference points (red dots) are separated by a displacement of (236 nm, 268 nm). See specific
discussion in supplementary note 2. As shown in Fig. 2b, ii-iii, only one EP emerges at 130° under
823 nm, with a similar EP identified at 68° under 706 nm incidence. (See the opposite chiral
incidence results in supplementary note 8). It is evident that when wavelength decreases across
823 nm from higher wavelength, PB phase accumulation is suppressed to a trivial case, with nearly
rotation-independent response. As shown in Fig. 2b, iⅴ-ⅵi, the metasurface generates OAM with
ℓ = +3 under LCP excitation above 823 nm (here we take the phase distribution at 850 nm as an
example), and is transformed to a near plane wave phase distribution below 823 nm (here we take
the phase distribution at 800 nm as an example). However, under RCP excitation, ℓ = -3 regardless
of the wavelength shown in Fig. 2b, ⅴi-ⅵi. Fig. 2c, d i-iii details the suppression evolution in the
cross-polarization conversion complex plane and corresponding TR-PB phase response under
wavelengths marked by purple in Fig. 2b, i. In Fig. 2c, ii, the topological protection was disrupted,
and the EP encircling trajectory altered its winding number near 823 nm and stopped encircling
EP (origin) to adopt the trajectory shown in Fig. 2c, iii, which corresponds to a trivial PB phase
response in Fig. 2d, iii. The supplementary movie 1 demonstrates the suppression evolution.
When EPs emerge consecutively while changing wavelength, it exists the possibility of chiral SOI
doubling. As shown in Fig. 2b, ii, multiple EP responses emerged below 730 nm under LCP
incidence, resulting in the doubled PB phase accumulation (+4θ) within the spectral window from
682 to 692 nm, as shown in Fig. 2b, iii. The multiple EP emergence signifies the topological
protection is further disrupted, leading to a winding number of 2, as shown in Fig. 2c, iv, under
wavelength marked by yellow dashed lines in Fig. 2b, ii, for demonstration. The supplementary
movie 8 reveals its evolution. Such a status results in a 4π geometric phase accumulation over 180°
rotation (Fig. 2d, iv) and is verified in experiments shown in Fig. 3e-f, iii. See more discussion of
chiral in supplementary note 9.

It is worth pointing out that EPs despite sharing the same handedness, they may exhibit opposite
phase-increment in the wavelength-angle space, which reflects the opposing topological
manipulations. For instance, in Fig. 2b, ii-iii, the SOI suppression is caused by the black dashed
EP at 823 nm, and is closed by the red dashed EP at 705 nm. The SOI doubling is caused by the
red dashed EP at 691 nm, and is closed by the black dashed EP at 681 nm. Since the winding
number only changes upon traversing an EP, the corresponding low PCR around such transition
has no impact on the TR-PB phase response. See supplementary note 6 for a complete discussion.

Experiments shown in Fig.3 validate chiral SOI suppression via optical setups in Supplementary
Fig. 1 in the supplementary information. The scanning electron micrographs (SEMs) of designed
meta-atoms are displayed in Fig. 3a-b, the scale bars are set to 2 μm, and it is evident that the two
metasurfaces possess main structure with opposite chirality. Spin dependent splitting in Fig. 3c-d,
i-ii, corresponds well to Fig. 2a, iii-ⅵ, demonstrating the consistency with simulations and the
existence of chiral SOI suppression. Similarly, OAM interference pattern in Fig. 3e-f, i-ii, confirms
chiral SOI suppression in spin-to-vortex conversion (with opposite chirality), corresponding to
results simulated in Fig. 2b, iii-ⅵ. Above the wavelength of 820 nm, LCP excitation yields output
with ℓ = +3 and is transitioned to ℓ = 0 under 820 nm. Here we take responses at 850 nm and 800
nm for demonstration, while RCP maintains ℓ = −3 output regardless of wavelength. Experiment
movies are provided in the supplementary movie 2-3 to demonstrate the suppression process for
the spin-dependent splitting and topological charge transitions under RCP or LCP incidence,
respectively, while changing the wavelength. The interference pattern at 730 nm shows that
topological charge has changed from +3 to +6, while the topological charge under RCP excitation
remains ℓ= −3 (Fig. 3 c-d, iii). The corresponding intensity distribution are provided in
Supplementary Fig. 16. The spectral shift of chiral SOI doubling phenomenon from 692 nm
(simulated in Fig. 2d, iv) to 730 nm in experiment arises from the interplay of two factors. First,
disrupting the topological protection of PB phase accumulation with one EP (leading to a winding
number of 0) destabilizes EP encircling evolution on the polarization-conversion complex plane.
It means the actual evolution trajectory in experiment is not the same as the simulation. Second,
fabrication errors and inhomogeneities of meta-atoms also amplify such deviations from simulated
trajectories in Fig. 2c, iv.

Topological Reconstruction Controlled by dual EPs under Fixed Wavelength
The introduction of two exceptional points (EPs) during rotation at a fixed wavelength enables
chiral SOI inversion, reversing the PB phase accumulation principle relative to rotation angle
under target spin. The corresponding numerical simulations and evolution analyses of chiral SOI
inversion are shown in Fig. 4. The meta-atom adopts the configuration in Fig. 4a. It has a period
of 700 nm with the following geometric dimensions: dx =80 nm, dy=45 nm, w=75 nm, L1=263
nm, L2=145 nm. The perturbation ellipses all possess axes of 75 nm and 37.5 nm. The centers of
the four perturbation ellipses and the main structure are separated by an absolute displacement of
(220 nm, 220 nm). The four perturbation ellipses elevate the main structure’s rotational symmetry
from twofold to fourfold, so that two exceptional points can be introduced during a 180° rotation
cycle of the main structure. In Fig. 4b, cross-polarization conversion rate of the meta-atom under
RCP excitation was mapped as a function of the main structure’s rotation angle and input
wavelength. Two EPs are observed at 51° and 141° under the wavelength of 748.12 nm, with
similar EPs identified at 60° and 150° under 691.73 nm incidence. See supplementary note 3 for
EP verification, and also see supplementary note 10 for the near-field analysis of the EP origin.
Correspondingly, the PB phase response in Fig. 4c reveals the opposite PB phase accumulation for
RCP excitation within the spectral window formed by 750nm and 690nm. It is worth pointing out
that the abrupt changes near 700 nm and 742nm in Fig. 4a-b are caused by Rayleigh anomalies,
but it is irrelevant to the inversion process. (See the opposite chiral incidence results in
supplementary note 8). The evolution of EP encirclement and the resulting PB phase accumulation
at wavelengths marked by gray dashed lines are shown in Fig. 4d, ⅰ to ⅳ and Fig. 4e, ⅰ to ⅳ. It can
be clearly seen from Fig. 4d, ⅰ to ⅳ, that the encirclement is clockwise oriented at first, and the
corresponding PB phase declines monotonically with rotation angle -2θ. Further evolving to Fig.
4d, ii, the corresponding PB phase began to decline non-monotonically. In Fig. 4d, iii, two points
simultaneously reach the EP symmetrically, while all other trajectory points remain outside the
singularity.

Such evolution on the complex plane produces the stepwise PB phase distribution observed in Fig.
4e, iii. The progression from Fig. 4d, iii to iv reveals even more intriguing evolution. After
traversing two EPs, the encircling trajectories maintain their original motion, crossing one another,
forming counterclockwise EP encircling path. This evolution manifests as inverted PB phase
accumulation +2θ under RCP incidence that increases monotonically shown in Fig. 3e, iv. The
movie is provided in the supplementary movie 4 to demonstrate such process. Such unconventional
phase behavior arises from the deliberate introduction of EPs during rotation, which disrupts the
topological protection and flips the winding number to +1. Without the EP induced evolution in
the complex plane, neither resonance nor phase-change materials could perturb the conventional
PB phase–rotation correspondence.

TR-PB phase fundamentally enriches metasurface design toolbox. To validate this claim, we
designed three distinct metasurfaces shown in Fig. 4f–h. These metasurfaces demonstrated the
chiral SOI inversion phenomenon either through spin-to-vortex conversion (Fig. 4f-g) or photonic
spin Hall effects (Fig. 4h), satisfying Equation (5) or Equation (6), respectively. For the first
(second) metasurface, the topological charge ℓ is set to 1 (2). The simulation results are shown in
Fig. 4f-g. As wavelength decreases from 800 nm to 710 nm under RCP excitation, topological
charge inverts from ℓ =-1 to +1 in Fig. 4f, i-iv and from ℓ = -2 to +2 in Fig. 4g, i-iv. In contrast,
LCP excitation maintains invariant topological charge across the entire spectrum for both
metasurfaces. For the third metasurface, photonic spin Hall effect is demonstrated under RCP
excitation shown in Fig. 4h, i to iv. The beam deflection shifts from rightward to leftward as the
wavelength decreases from 800 nm to 710 nm. LCP maintains a persistent leftward deflection
without inversion.

Figure 5 experimentally validates chiral SOI inversion via optical setups detailed in Supplementary
Fig. 1 in the supplementary information. Corresponding to simulations in Fig. 4e, topological
charge inversion from ℓ = −1 to ℓ = +1 under RCP excitation as the wavelength shifts from 795
nm to 715 nm are demonstrated in Fig. 5a, while the LCP excitation preserves the topological
charge throughout the spectral range. Similarly, the chiral SOI inversion for ℓ =2 via spin-vortex
conversion is shown in Fig. 5b. Wavelength-driven beam deflection inversion in the spin-Hall
effect is demonstrated in Fig. 5c, i‑iv, matching the simulations in Fig. 4g, i‑iv, with LCP
maintaining persistent leftward deflection. Supplementary movie 5-7 captures experimental
inversion process for both topological charge transitions and spin-dependent splitting under RCP
incidence. While the corresponding intensity distribution are provided in Supplementary Fig. 16.
Control experiments confirm these effects originate solely from TR-PB phase modulation, which
is achieved through anisotropic coupling between main structure and perturbation ellipses, rather
than perturbation ellipse, rotation of the dual-L configuration seen from Supplementary Fig. 9 in
the supplementary materials. The SEM images of the metasurfaces are presented in Fig. 5d. It can
be clearly seen that only the main structures rotate according to Equations (5) and (6), while
perturbation ellipses remain fixed. Scale bars are set to 1 μm.

TR-PB phase used for information encryption
In this work, six possible combinations of meta-atoms are designed, as shown in Fig. 6(a). Their
PB phase responses are catalogued by incident polarization and wavelength in a TR-PB phase
library, shown in Fig. 6 (b), from which the PB phase acquisition rule of any structure under
specific polarization and input wavelength can be directly extracted.

The encryption design workflow is presented in Fig. 6(c). A modified GS algorithm first computes
the near-field phase distributions of the decoy image and the hidden image. The two phase-profiles
are linked through the rules of reversed, doubled, and conventional PB phase accumulation. By
comparing the two distributions, a ternary mask consisting only of values -1, +1, and +2 is obtained.
By consulting the TR-PB library, every specific meta-atom type of the metasurface and the rotation
angle of each main structure are determined , together with the correct decryption key K
4
(σ, λ) that
encodes the required polarization and wavelength. As shown in Fig. 6(d), the hidden image is
retrieved only with the correct key. For instance, under LCP illumination, the PB phase remains
conventional (+2θ) across the entire spectrum, and the reconstructed hologram consistently
displays the cat image (Fig. 6(d), i-ii). In contrast, when the illumination switches to the
polarization corresponding to the REP, the PB phase exhibits inversion and doubling at 730 nm in
our design. As the wavelength changes from 850 nm to 730 nm, the reconstructed hologram
accordingly switches from an inverted cat image (Fig. 6(d), iii) to the dog image (Fig. 6(d), iv). This
spin-decoupling effect, arising from the chiral nature of the proposed TR-PB phase, enables
the observed wavelength and polarization dependent image switching, thereby demonstrating the
information encryption capability.

Methods
Device fabrication
The substrate used here is 160 nm ITO-coated silica glass. The substrates were cleaned by
sequential ultrasonication for 5min in acetone, methanol, and ultrapure water. The fabrication
involved electron beam lithography, sequential deposition of a 6nm titanium (Ti) layer and a 40nm
gold (Au) film, and liftoff in butanone.

Optical setup
In order to fully characterize the OAM beams, we used a Mach-Zehnder interferometer as shown
in Supplementary Fig. 1(a). In this configuration, a Ti: Sapphire ultrafast femtosecond laser (pulse
length: ≈115 fs, repetition rate: 76 MHz) centered at 800 nm is used as a light source. A
horizontally-polarized beam is produced by using a linear polarizer (LP1). Then it is split in two
beams by means of a 50/50 beam splitter (BS1). the upper arm of the interferometer passes through
a quarter waveplate (QWP1) to produce a circularly polarized beam. A weakly focusing lens (L1,
f=150 mm) is used to reduce the beam size to be comparable with the diameter of the patterned
metasurface (≈500 μm). A 20× objective lens, a lens (L2, f=150 mm) and a camera behind the
metasurface is used for recording the OAM beam created by the metasurface. The OAM beam
passes through a polarization filter made of a quarter waveplate (QWP2), which has its fast axis
oriented at 90° with respect to QWP1 and a linear polarizer (LP2) in cross-polarization with respect
to LP1. This polarization filter is used to eliminate non-converted light passing through the
metasurface. In the reference arm, the beam passes through a (HWP) that rotates the polarization
of the laser beam from horizontal to vertical and it then is sent to a 5x beam expander (BE). The
reference beam and the OAM beam are then recombined via a second 50:50 beam-splitter (BS2)
and imaged on the camera. The interference of the two beams results in the measured characteristic
spiral interference pattern (as seen in Fig. 3 and Fig. 5). In order to ensure the time overlap between
the two beams, the optical path in one arm of the interferometer was adjusted using a delay line.

In Supplementary Fig. 1(b), the optical setup used to examine the photonic spin Hall effect is
demonstrated. Same femto-second laser is adopted as light source. A ND filter is also used to
attenuate the background. A horizontally-polarized beam is subsequently produced by a linear
polarizer (LP1) and it passes through a quarter waveplate (QWP1) to produce a circularly polarized
beam. After passes through a 20× objective lens, light is effectively focused on the metasurface,
and the transmitted beam is collected by another 20× objective lens. Then, the beam passes through
a quarter waveplate (QWP2), which has its fast axis oriented at 90° with respect to QWP1, and a
linear polarizer (LP2) to eliminate the undesired polarization. Eventually, spin dependent splitting
is captured by the camera after passing through a focusing lens (L1, f=150 mm). It is worth
mentioning that the generally low polarization conversion efficiency of our sample results in a
relatively strong co-polarized transmission component, which remains detectable after passing
through the linear polarizer, and form the 0th order diffraction captured in photonic spin-Hall effect.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this work, the authors demonstrate that PB phase in metasurfaces can be topologically reconstructed by encircling EPs in the complex plane of certain cross-polarization channel in the circular polarization basis. Three SOI mechanisms are numerically and experimentally verified, including chiral SOI of suppression, inversion, and doubling, manifesting by metasurface functionalities of wavelength- and polarization-dependent OAM control and beam steering. The idea is very interesting, which reveals a new perspective in understanding and designing the chiral phase responses for broader applications. However, the presentation requires further improvement in clarity, precision, and structural organization. Before the final decision, the following comments should be addressed.\n1. The PB phase acquisition is strongly affected by the EP. However, the cross-polarization amplitude is also affected, as EP corresponds to the zero cross-polarization conversion point. This would bring influence to the desired functionalities. The authors should clarify and provide discussion on this point.\n2. Only the amplitude and phase distributions of one cross-polarization channel in the wavelength-angle space are presented in each steering mechanism. It is necessary to present the corresponding results in the other cross-polarization channel for clear comparison.\n3. One of the core ideas and key contributions of this work is the complex plane perspective for reinterpreting the PB phase accumulation. However, the authors place this discussion in the Supplementary Materials, which significantly weakens the readability and impact of the manuscript. I strongly recommend presenting this key concept and content in the main text, with concise formulas and clear descriptions, while leaving the detailed derivations in the Supplementary Materials.\n4. It will be helpful if the authors provide the resonance analysis at the EP points and offer a clear explanation of the physical origin of the EP phenomena.\n5. Can the authors talk about the underline mechanism of why the emergence of such an EP can steer the PB phase response? Besides, it seems that the SOI suppression likely emerges between the wavelength range of two EPs, and the two EPs have opposite “handedness” in the wavelength-angle space. Could the authors also check this point, since it determines whether the SOI suppression effect is induced by single or dual EPs?\n6. Are the laser spots in the middle parts of Fig. 5c caused by the 0t-order diffraction of the co-polarized transmission? If yes, why its influence on the vortex beam generations is hard to observe?\n7. In Fig. 3Eiii and 3Fiii, the experimental results for chiral SOI doubling seems not clear without the aid of the white dashed guidelines. It may be more straightforward to be demonstrated using photonic spin-Hall effect, where the deflection angle will be enlarged under LCP incidence.\n8. In the conclusion, the authors state their method can be used in information-encryption. However, related information in the main text and Supplementary Materials is quite not enough, which reads confusing and weakens the narrative. It is meaningful if the authors give an metasurface design example to show the encrypt ability, either in simulation or experiment.\n9. It will be helpful if the authors present the corresponding meta-atom schematics in Fig. 2 and Fig. 4, and give out their dimensions in the main text. This can make a clearer guidance to the readers. It is quite easy to be confused that Fig. 2Ai and Fig. 2Bi are the two cross-polarized transmission results of the same rotation meta-atom.\n10. Some other small suggestions:\na) The definitions of the signs of “+ and –”, the abbreviations of CCW and CW, should be defined in the main text.\nb) The authors state that “PB phase are widely used in metasurfaces…to exhibit SOI phenomena, such as the spin Hall effect…” However, the term of “spin Hall effect” should be used more precisely as “photonic spin Hall effect”.\nc) The thickness of the ITO film mentioned in the main text (160 nm) and the methods (150 nm) are different. The material parameters of the metal structures and the ITO film should also be presented.\nd) The tick marks of the axes in Fig. 2A(i,ii), Fig. 2B(i,ii), and Fig. 4A,4B are not clear, and the label of the horizontal axes are also not presented. Besides, the axes of all the simulated and measured results, as well as corresponding scalebars should be presented.\ne) In Fig. 2Ai and 2Bi, the mapping of t−+ and t+− are not presented in an intuitive manner. I suggest that the authors provide, in the Supplementary Materials, all line plots versus wavelength at the angles where the EP points occur.\nf) In Fig. 3 and Fig. 5, the authors present the interference pattern to characterize the vortex beam features. It will be good if authors also include the corresponding intensity distributions without interference from the reference beam.", "reviewer_2": "The author innovatively proposed a design scheme for achieving nonlinear PB phase by introducing perturbations to construct EP points in chiral metasurfaces, and elaborated on the underlying principles. The selection of the relative position between the EP point and the incident wavelength was demonstrated to suppress, reverse, and double the SOI effect, and experimental verification was carried out. The theoretical foundation is robust and the approach demonstrates significant innovation. However, the following questions require solution prior to publication:\n1. Page 7, line 143: Please explain why SOI suppression in Figure 2A occurs between 700 and 750 nm. However, SOI suppression occurs from 790 to 710 nm under RCP incidence. Does SOI suppression occur at wavelengths below 700 nm? Could you provide the wavelength ranges for which SOI suppression does and does not occur?\n2. Could you add |t-+| and arg[t-+] for metasurface 2A, and |t+-| and arg[t+-] for metasurface 2B in the supplementary material.\n3. For metasurface 2A, the phase distribution in Figure 2Aii shows that the phase variation with rotation angle within a 2π range at an incident wavelength of 790 nm has little correlation with the EP point at 750 nm, and the region surrounding the EP point does not appear to be the EP point at 750 nm. Is it possible to obtain a 2π phase modulation range at approximately 740 nm?\n4. According to Figure 2Bi, the EP generation point does not seem to be at 819.55 nm, but at least exceeds 825 nm. Please correct the wavelength of the EP point or the drawing method.\n5. The two light spots shown in Figures 3C and 3D are rather odd. Is one of them the center point?\n6. Figure 2(C-D)iv shows the generation of multiple EP points and should not be placed in this section, and this section does not discuss this figure.\n7. Why are Au and TIO materials chosen for the metasurface in the paper? Can the emergence of EP and the introduction of interference be achieved with all-dielectric materials such as Si or Ge?\n8. For the metasurface in Figure 2B, at an incident wavelength of 689.22 nm, two circles around the EP point exhibit a phase variation of 4π. However, in Figure 3Eiii, the wavelength at which the doubled PB phase appears is 730 nm. The meta-atom corresponding to the every results in Figures 3 should be clarified in the figure legend.\n9. Is the doubled PB phase in Figure 3E induced by the presence of two EPs in Figure 2B? If so, would this structure be better addressed in the \"Dual EPs induced topological reconstruction\" section? On the other hand, are the opposite PB phases due to the presence of two EPs at different frequencies? If so, could you please explain in the manuscript the specific EPs induce the PB phase doubling in each of these three phenomena?\n10. Why can elliptical metal columns introduce interference behavior, and why can the elliptical particles maintain their original nonlinear characteristics when the main structure rotates while the main structure does not rotate? Please briefly explain this principle in the paper.\n11. The adopted meta-atoms in the manuscript involves many parameters. How to determine the structural size parameters during the design process? Are there any theoretical reference values?\n12. The descriptions of the horizontal and vertical axes in Figure 2 are missing. The vertical axis in Figure 2D ranges from -3 to 3. Are the units in radians? The horizontal axis is in degrees. It is recommended that the units used be clearly indicated in the descriptions of the horizontal and vertical axes.\n13. Figure S7 and legend are separated in the supporting materials."}, "author_response": "Dear reviewers:\nThank you for taking the time to review our manuscript entitled “Chiral\nSpin-Orbit-Interactions Steering via EP induced topological reconstruction”\n(Manuscript ID: NCOMMS- 25-67039-T).\nWe greatly appreciate your valuable comments and suggestions, which have been\ncarefully considered in our revisions. Below, we have provided a point-by-point\nresponse, along with the corresponding changes made in the revised manuscript and\nsupplementary materials for your reference. We believe that all the comments have\nbeen addressed, the revised manuscript meets the standard of Nature Communicaitons\nfor publication.\nReply to Reviewer 1:\nGeneral Comment: In this work, the authors demonstrate that PB phase in\nmetasurfaces can be topologically reconstructed by encircling EPs in the complex\nplane of certain cross-polarization channel in the circular polarization basis.\nThree SOI mechanisms are numerically and experimentally verified, including\nchiral SOI of suppression, inversion, and doubling, manifesting by metasurface\nfunctionalities of wavelength- and polarization-dependent OAM control and\nbeam steering. The idea is very interesting, which reveals a new perspective in\nunderstanding and designing the chiral phase responses for broader applications.\nHowever, the presentation requires further improvement in clarity, precision,\nand structural organization. Before the final decision, the following comments\nshould be addressed.\nResponses:\nWe greatly appreciate your positive comments and detailed summary of our work.\nMany of the suggestions you have given us are enlightening our research. We have\ncarefully revised and proofread the manuscript according to your suggestions.\nBelow is a more detailed reply and revisions. The revised sentences are\nhighlighted in red and added to the revised manuscript, while the sentences in\nblack remain unchanged. Deleted sentences are marked in black with a\nstrikethrough.\nComments1: The PB phase acquisition is strongly affected by the EP. However,\nthe cross-polarization amplitude is also affected, as EP corresponds to the zero\ncross-polarization conversion point. This would bring influence to the desired\nfunctionalities. The authors should clarify and provide discussion on this point.\nResponses:\nWe greatly appreciate your comments. The cross-polarization amplitude is\naffected by the EP, as the EP corresponds to the zero cross-polarization conversion\npoint. We assume you are worried that this might influence the phenomena discussed\nin our manuscript. Below, we will provide a detailed response to address this issue,\nand we have made corresponding modifications to both the main text and\nsupplementary materials to include a more thorough discussion of this point.\nIn response to this concern, we would like to clarify that while it is true that\npassing through the EP involves encountering states with low cross-polarization\namplitude, this does not affect the desired functionalities according to our theoretical\nanalysis. Specifically, the winding number of the PB phase acquisition process is only\ninfluenced by the zero cross-polarization conversion point (the EP), not by the low\ncross-polarization conversion rate (PCR) at intermediate states. Therefore, the low\nPCR observed during the rotation process cannot steer the PB phase-mediated\nspin-orbit interaction (SOI) processes. It merely indicates low cross-polarization\nconversion efficiency at the corresponding rotation angle.\nThis conclusion can be verified by examining the encircling trajectory in the\ncross-polarization conversion complex plane, along with the PB phase response and\nPCR at the relevant wavelength. Fig. R1(A, B) details the PCR and PB phase across\nthe wavelength-angle space, with three specific wavelengths examined: 756.89 nm\n(red line), 749.37 nm (yellow line), and 738.10 nm (blue line), as shown in Fig.\nR1(C-E). In Fig. R1(C), the cross-polarization amplitude in panel (i) is low but does\nnot reach the EP (marked by the blue circle). The corresponding trajectory in the\ncomplex plane (panel (ii)) encircles the EP, yielding a winding number of -1.\nConsequently, the resulting PB phase accumulation (panel (iii)) follows a\nconventional profile. Fig. R1(D) shows the case where the EP is reached. Here, the\nencircling trajectory (panel (ii)) passes directly through the EP. This condition\nprevents the PB phase from achieving a full 2π modulation range, as shown in panel\n(iii). Despite the similarly low cross-polarization amplitude in panel (i) of Fig. R1(E)\nand (C), their trajectories in panel (ii) differ fundamentally. The trajectory in Fig.\nR1(E) does not enclose the EP, yielding a winding number of 0. Therefore, as panel\n(iii) demonstrates, the PB phase accumulation is suppressed.\nFig. R1. The influence of low PCR on TR-PB phase. (A and B) The PCR and PB\nphase response at wavelength-angle space. We fixed the wavelength at the positions\nof the red (756.89 nm), yellow (749.37) and blue (738.10 nm) dotted lines to\ninvestigate the influence of PCR on the desired functionalities. (C-E) The\ncross-polarization conversion rate, encircling trajectory in the cross-polarization\nconversion complex plane and corresponding PB phase response at different\nwavelength.\nIn summary, the standard PB phase acquisition is preserved as long as the\nwinding number is preserved, even when the PCR is low. Only after passing through\nthe EP can the winding number be altered, leading to the desired steering\nfunctionalities. The low PCR after passing through the EP is a collateral effect of\ninducing the EP and does not cause any changes in the winding number. As a result,\nlow PCR has no impact on the topologically reconstructed PB (TR-PB) phase\nresponse or the desired functionalities. This behavior is consistent across the entire\nstudy.\nChanges in the manuscript:\nWe add one sentence in the main text before Fig.3 to provide the corresponding\ndiscussion.\n“Since the winding number only changes upon traversing an EP, the corresponding\nlow PCR around such transition has no impact on the TR-PB phase response. See\nsupplementary note 6 for a complete discussion.”\nWe also add supplementary note in the supplementary materials with the subtitle\n“6. The influence of low PCR on TR-PB phase.” to provide a more detailed\ndiscussion.\n“6. The influence of low PCR on TR-PB phase\nDuring the rotation process, reaching the EP inevitably involves passing through\nstates with a low cross-polarization amplitude. However, based on our analysis, such\nlow amplitude does not affect the desired functionalities. Specifically, the winding\nnumber of the PB phase acquisition process is influenced only by the zero\ncross-polarization conversion point, which corresponds to the EP. Therefore, a low\ncross-polarization conversion rate (PCR) does not steer the PB phase-mediated SOI\nprocesses. It simply indicates a low cross-polarization conversion efficiency at the\ncorresponding rotation angle.\nThis can be verified by examining the encircling trajectory in the cross-polarization\nconversion complex plane, alongside the PB phase response and the\ncross-polarization conversion rate at the relevant wavelength. Fig. S11(A,B) details\nthe PCR and PB phase across the wavelength-angle space, with three specific\nwavelengths examined: 756.89 nm (red line), 749.37 nm (yellow line), and 738.10 nm\n(blue line), as shown in Fig. S11(C-E).\nIn Fig. S11(C), the cross-polarization amplitude in panel (i) is low but does not reach\nthe EP (marked by the blue circle). The corresponding trajectory in the complex plane\n(panel (ii)) encircles the EP, yielding a winding number of -1. Consequently, the\nresulting PB phase accumulation (panel (iii)) follows a conventional profile. Fig.\nS11(D) shows the case where the EP is reached. Here, the encircling trajectory (panel\n(ii)) passes directly through the EP. This condition prevents the PB phase from\nachieving a full 2π modulation range, as shown in panel (iii). Despite the similarly\nlow cross-polarization amplitude in panel (i) of Fig. R1(E) and (C), their trajectories\nin panel (ii) differ fundamentally. The trajectory in Fig. S11(E) does not enclose the\nEP, yielding a winding number of 0. Therefore, as panel (iii) demonstrates, the PB\nphase accumulation is suppressed.\nIn summary, the standard PB phase acquisition is preserved as long as the winding\nnumber is preserved, even when the PCR is low. Only after passing through the EP\ncan the winding number be altered, leading to the desired steering functionalities. The\nlow PCR after passing through the EP is a collateral effect of inducing the EP and\ndoes not cause any changes in the winding number. As a result, low PCR has no\nimpact on the topologically reconstructed PB (TR-PB) phase response or the desired\nfunctionalities. This behavior is consistent across the entire study.\nFig. S11. The influence of low PCR on TR-PB phase. (A and B) The PCR and PB phase\nresponse at wavelength-angle space. We fixed the wavelength at the positions of the red\n(756.89 nm), yellow (749.37) and blue (738.10 nm) dotted lines to investigate the influence of\nPCR on the desired functionalities. (C-E) The cross-polarization conversion rate, encircling\ntrajectory in the cross-polarization conversion complex plane and corresponding PB phase\nresponse at different wavelength.”\nComments2: Only the amplitude and phase distributions of one\ncross-polarization channel in the wavelength-angle space are presented in each\nsteering mechanism. It is necessary to present the corresponding results in the\nother cross-polarization channel for clear comparison.\nResponses:\nThank you very much for your comment. The other cross-polarization channel is\nessential for a clear comparison in the wavelength-angle space, as it allows the\nchirality of our phenomenon to be clearly demonstrated. The corresponding amplitude\nand phase distributions in the wavelength-angle space under opposite chirality\nincidence are illustrated in Fig. R2.\nFig. R2(A) and (D) illustrate the PCR and PB phase response of opposite\nchirality incidence corresponding to the right-handed SOI suppression process in Fig.\n2(A) ii, iii. Fig. R2(B) and (E) show the PCR and PB phase response of opposite\nchirality incidence corresponding to the left-handed SOI suppression and doubling\nprocesses in Fig. 2(B) ii, iii. Fig. R2(C) and (F) demonstrate the PCR and PB phase\nresponse of opposite chirality incidence corresponding to the right-handed SOI\ninversion process in Fig. 4(B-C). It is evident that the other cross-polarization channel\nshows a trivial PB phase response in the wavelength-angle space, with no EP\nobserved.\nDespite the presence of Rayleigh anomalies in Fig. R2, the conventional PB\nphase accumulation remains clearly visible. This is manifested as a linear dependence\non the rotation angle. Under LCP incidence, the PB phase increases by 2θ, while\nunder RCP incidence, it decreases by −2θ, respectively. The θ stands for rotation\nangle.\nFig. R2. The result of opposite chirality incidence in the wavelength-angle space.\n(A, D) PCR and PB phase response of opposite chirality incidence corresponding to\nthe right-handed SOI suppression process in Fig. 2(A) ii, iii. (B, E) PCR and PB phase\nresponse of opposite chirality incidence corresponding to the left-handed SOI\nsuppression and doubling processes in Fig. 2(B) ii, iii. (C, F) PCR and PB phase\nresponse of opposite chirality incidence corresponding to the right-handed SOI\ninversion process in Fig. 4(B-C).\nDue to space limitations in the manuscript, we have summarized the\ncorresponding section in the supplementary materials for clearer comparison.\nChanges in the manuscript:\nWe have added a supplementary note in the supplementary materials under the\nsubtitle “8. The result of opposite chirality incidence in the wavelength-angle\nspace” to present the comparisons.\n“8. The result of opposite chirality incidence in the wavelength-angle\nspace\nDue to the intrinsic chirality of the EPs, oppositely handed incidences cannot access\nan EP under identical conditions (wavelength and angle shown in Fig. S13), and\nhence result in the conventional PB phase accumulation. This dictates the chiral\nnature of the proposed SOI steering phenomena, which is firmly supported by direct\nexperimental evidence in the main text.\nFig. S13. The result of opposite chirality incidence in the wavelength-angle space.\n(A, D) PCR and PB phase response of opposite chirality incidence corresponding to\nthe right-handed SOI suppression process in Fig. 2(A) ii, iii. (B, E) PCR and PB phase\nresponse of opposite chirality incidence corresponding to the left-handed SOI\nsuppression and doubling processes in Fig. 2(B) ii, iii. (C, F) PCR and PB phase\nresponse of opposite chirality incidence corresponding to the right-handed SOI\ninversion process in Fig. 4(B-C).”\nComments3: One of the core ideas and key contributions of this work is the\ncomplex plane perspective for reinterpreting the PB phase accumulation.\nHowever, the authors place this discussion in the Supplementary Materials,\nwhich significantly weakens the readability and impact of the manuscript. I\nstrongly recommend presenting this key concept and content in the main text,\nwith concise formulas and clear descriptions, while leaving the detailed\nderivations in the Supplementary Materials.\nResponses:\nWe greatly appreciate your insightful comment. The complex plane approach is\nfoundational to our work, as it provides a clear and intuitive physical picture for\ninterpreting the PB phase accumulation. We fully agree that presenting this key\nconcept in the main text will significantly enhance the readability and impact of the\nmanuscript.\nIn response to your suggestion, we have moved the core discussion of the\ncomplex plane perspective into the main text, along with a concise formula and a\nclear description of the underlying principles. In specific, by acting the rotation\noperation on the non-Hermitian Jones matrix, we can eventually derive the equations\nbelow.\n \n−\n−\n+\n+\n =\n=\n−\n−\n+\n+\n−\n+\n−\n−\n+\n+\n  \n  \nR\nI m\ne T\nT\n(\n(\n)\n)\nR\nI m\ne (T\n( T\n)\n)\nc\nc\no\no\ns\ns\n2\nR\nm\ne\n( T\n(T\n)\n)\ns\ns\nin\nin\n2\n2\n \n+\n+\n−\n−\n =\n=\n+\n+\n−\n−\n+\n−\n+\n+\n−\n−\n  \n  \nR\nI m\ne T\nT\n(\n(\n)\n)\nR\nI m\ne (T\n( T\n)\n)\nc\nc\no\no\ns\ns\n2\nR\nm\ne\n( T\n(T\n)\n)\ns\ns\nin\nin\n2\nij\n“+” represents LCP, while “-” represents RCP. “θ” stands for the rotation angle. Such\nshifted perspective reveals that the accumulation of the PB phase is equivalent to\ndynamically encircling an EP, with the direction of encirclement determined by the\nchirality of the input circular polarization.\nFull derivations are provided in supplementary note 1, which also includes a\ndiscussion on how to disrupt the topological protection with chirality. We believe this\nadjustment will improve the clarity of the manuscript while ensuring that the essential\nconcepts are highlighted in the main text.\nChanges in the manuscript:\nWe add the simplified description of the complex plane perspective to the main\ntext before the subtitle “Topological Reconstruction Controlled by a Single EP\nunder Fixed Wavelength” to improve the readability.\n“These discovered phenomena cannot be thoroughly interpreted in conventional\nPoincaré sphere formalism. To resolve this problem, we adopted a complex plane\nperspective, reinterpreting PB phase accumulation.\nWe first reexamine the mathematics behind the PB accumulation process, which can\nbe described by acting the rotation operation on the Jones matrix. It is described by\nthe following expression.\ncos −sinT T  cos sin\nJ()= R(−)JR()=   xx xy  \nsin cos T\nyx\nT yy−sin cos\n(1)\nSuppose the transmission Jones matrix of the meta-atom is described by J.\nJ =\n T\nT\nx\ny\nx\nx\nT\nT\nx\ny\ny\ny\n\n=\n t\nt\nx\ny\nx\nx\ne\ne\ni\ni\nxx\nyx\nt\nt\nx\ny\ny\ny\ne\ne\ni\ni\nxy\nyy\n\n=\n a\nc\n+\n+\ni b\ni d\nc\ne\n+\n+\ni\ni\nd\nf\n  \n \n(2)\nT\nWhere ijis the transmission from j-polarized input light to i-polarized output light.\nThe meta-atom can impart amplitude and phase modulation, represented by\nt\nij and ij\n\n.\nDue to the reciprocity and the anisotropy,\nT\nx y\n= T\ny x and\nT\nx x\n T\ny y , respectively. In\naddition, a-f are real values obtained from the complex amplitude modulation. After\nbeen rewritten in the basis of circular polarization, it is easy to derive equation (3).\n R\nI m\ne\nTT\n+\n+\n−\n−\n\n=\n=\n((\na\nb\n−\n−\ne\nf\n+\n−\n2\n2\nd\nc\n))\n2\n2\n R\nI m\ne\nTT\n−\n−\n+\n+\n\n=\n=\n((\na\nb\n−\n−\ne\nf\n−\n+\n2\n2\nd\nc\n))\n2\n2\n(3)\nWhere “+” represents LCP, while “-” represents RCP. By taking the non-Hermitian\nform Jones matrix into equation (1), the Jones matrix\n( )  J\ncan be obtained after\nrotation, which is described by a-f. By thinking\n( )  J\nas a new Jones matrix, based\non equation (3), equation (4) can be derived.\n \n+\n+\n−\n−\n =\n=\n+\n+\n−\n−\n+\n−\n+\n+\n−\n−\n  \n  \nR\nI m\ne T\nT\n(\n(\n)\n)\nR\nI m\ne (T\n( T\n)\n)\nc\nc\no\no\ns\ns\n2\nR\nm\ne\n( T\n(T\n)\n)\ns\ns\nin\nin\n2\n2\n \n−\n−\n+\n+\n =\n=\n−\n−\n+\n+\n−\n+\n−\n−\n+\n+\n  \n  \nR\nI m\ne T\nT\n(\n(\n)\n)\nR\nI m\ne (T\n( T\n)\n)\nc\nc\no\no\ns\ns\n2\nR\nm\ne\n( T\n(T\n)\n)\ns\ns\nin\nin\n2\n2\n(4)\nFrom these expressions, the rotation operation is shown to drive an encirclement\naround the origin of the complex plane, which corresponds to an EP of the\nnon-Hermitian Jones matrix, as established in supplementary note 1. This shift in\nperspective reveals that the accumulation of the PB phase is equivalent to\ndynamically encircling an EP, with the direction of encirclement determined by the\nchirality of the input circular polarization. As shown in Fig. 1(F,G), when under LCP\nT ()\nincidence, −+ encircles the LEP in counterclockwise (CCW) manner, whereas\nT ()\nunder RCP incidence, +− encircles the REP in clockwise (CW) manner, while\n continuously increases from 0 to π. Building upon this point of view, we first\ndissect chiral SOI suppression and its evolution in the complex plane.”\nComments4: It will be helpful if the authors provide the resonance analysis at\nthe EP points and offer a clear explanation of the physical origin of the EP\nphenomena.\nResponses:\nWe appreciate your constructive comment. The physical origin of the EP\nphenomena has been explained in previous studies (Science 373.6559: 1133-1137\n(2021), Nat. Commun. 15.1: 232 (2024)), which states that the EP arises when\nopposite and equal surface current distribution occur along one direction, while the\nsurface current of the other orthogonal direction exists. By changing the geometric\nparameters or input wavelength, such near field behavior results in a zero\ncross-polarization conversion. However, the structure that we employ have never been\nscrutinized. Here, we offer a near field analysis of our structure by examining the\nsurface current distribution.\nFig. R3. Near-field analysis of the EP origin. (A) Schematic of the meta-atom with\na 51º rotated main structure. The red and black lines mark the cross sections analyzed\nat the top and bottom layers, respectively. (B) Cross-polarization conversion rate\nversus input wavelength. The five analyzed wavelengths are marked by solid circles,\nwith the EP state highlighted in light-red. (C, D) Surface current distributions on the\ncorresponding cross sections at each wavelength. The insets show the effective\ncurrent (black arrows). (E) Evolution of the net surface current, where the EP\nemergence is indicated by the light-red highlight.\nBy examining the surface currents on the top and bottom layers (marked in red\nand black in Fig. R3(A), respectively), we elucidate the physical origin of the\nexceptional points (EPs) through the mechanism consistent with prior studies. As\nshown in Fig. R3(B), an EP can be accessed by tuning the wavelength. We analyze\nthe five representative wavelengths marked in solid circles, with the corresponding\nsurface current behaviors displayed in Fig. R3(C–D), i-v.\nAs the wavelength increases continuously, the top layer current along the\nlower-left direction weakens and eventually reverses, while its orthogonal component\nremains nearly constant. The effective surface current for this process is indicated by\nblack arrows in the insets of Fig. R3(C, i–v). In contrast, the bottom-layer current\nevolves differently: with increasing wavelength, the surface current component along\nthe upper-right direction strengthens while the orthogonal component diminishes. The\nresulting effective current is similarly denoted by black arrows in Fig. R3(D, i–v).\nThe net surface current is summarized in Fig. R3(E). As the wavelength rises, one\ndirectional component of the net current gradually vanishes, leaving only the\northogonal component. Such state signals the emergence of an EP and is highlighted\nin light red in Fig. R3(B, E-iii). With further increase in wavelength, the net current\ngrows along the opposite (upper-right) direction, and the EP vanishes.\nChanges in the manuscript:\nWe have added one sentence to the first paragraph of the section titled\n“Topological Reconstruction Controlled by dual EPs under Fixed Wavelength” in\nthe main text.\n“Two EPs are observed at 51° and 141° under the wavelength of 748.12 nm, with\nsimilar EPs identified at 60° and 150° under 691.73 nm incidence. See supplementary\nnote 3 for EP verification, and also see supplementary note 10 for the near-field\nanalysis of the EP origin.”\nWe have added a supplementary note in the supplementary materials under the\nsubtitle “10. Near-field analysis of the EP origin” to present the comparisons.\n“10. Near-field analysis of the EP origin\nThe physical origin of the EP phenomena has been explained in previous studies [1,\n2], which states that the EP arises when opposite and equal surface current\ndistribution occur along one direction, while the surface current of the other\northogonal direction exists. By changing the geometric parameters or input\nwavelength, such near field behavior results in a zero cross-polarization conversion.\nHowever, the structure that we employ have never been scrutinized. Here, we offer a\nnear field analysis of our structure by examining the surface current distribution.\nFig. S16. Near-field analysis of the EP origin. (A) Schematic of the meta-atom with a\n51°rotated main structure. The red and black lines mark the cross sections analyzed\nat the top and bottom layers, respectively. (B) Cross-polarization conversion rate versus\ninput wavelength. The five analyzed wavelengths are marked by solid circles, with the\nEP state highlighted in light-red. (C, D) Surface current distributions on the\ncorresponding cross sections at each wavelength. The insets show the effective current\n(black arrows). (E) Evolution of the net surface current, where the EP emergence is\nindicated by the light-red highlight.\nBy examining the surface currents on the top and bottom layers (marked in red and\nblack in Fig. S16(A), respectively), we elucidate the physical origin of the exceptional\npoints (EPs) through the mechanism consistent with prior studies. As shown in Fig.\nS16(B), an EP can be accessed by tuning the wavelength. We analyze the five\nrepresentative wavelengths marked in solid circles, with the corresponding surface\ncurrent behaviors displayed in Fig. S16(C–D), i-v.\nAs the wavelength increases continuously, the top layer current along the lower-left\ndirection weakens and eventually reverses, while its orthogonal component remains\nnearly constant. The effective surface current for this process is indicated by black\narrows in the insets of Fig. S16(C, i–v). In contrast, the bottom-layer current evolves\ndifferently: with increasing wavelength, the surface current component along the\nupper-right direction strengthens while the orthogonal component diminishes. The\nresulting effective current is similarly denoted by black arrows in Fig. S16(D, i–v).\nThe net surface current is summarized in Fig. S16(E). Clearly, as the wavelength rises,\none directional component of the net current gradually vanishes, leaving only the\northogonal component. Such state signals the emergence of an EP and is highlighted\nin light red in Fig. S16(B, E-iii). With further increase in wavelength, the net current\ngrows along the opposite (upper-right) direction, and the EP vanishes.”\nComments5: Can the authors talk about the underline mechanism of why the\nemergence of such an EP can steer the PB phase response? Besides, it seems that\nthe SOI suppression likely emerges between the wavelength range of two EPs,\nand the two EPs have opposite “handedness” in the wavelength-angle space.\nCould the authors also check this point, since it determines whether the SOI\nsuppression effect is induced by single or dual EPs?\nResponses:\nThank you very much for your comment. To provide a clearer and more\norganized response to your comment, we have broken it down into two parts and will\naddress each point separately.\n(1) “Can the authors talk about the underline mechanism of why the emergence of\nsuch an EP can steer the PB phase response?”\nThe underlying mechanism behind the change in the winding number is closely\nrelated to the emergence of the EP. In the absence of a zero cross-polarization\nconversion point, the PB phase accumulation process remains topologically protected,\nmeaning the PB phase response stays conventional and cannot be steered or altered.\nFig. R4. The schematics for demonstrating the underline mechanism of TR-PB\nphase steering. (A) The PB-mediated SOI process is topologically protected. No\nmatter how the rotation angle and wavelength vary, the winding number remain fixed.\n(B) After inducing EP during rotation, the topological protection is disrupted, and the\nPB phase acquisition cannot cover the scope of 2π. (C) The winding number is\nchanged to 0 by EP induction, and hence leading to SOI suppression.\nFor example, Fig. R4(A) shows that a single EP emerges at a specific wavelength\nduring rotation of the main structure. As illustrated in Fig. R4(B), traversing this EP\nchanges the winding number from 1 to 0. As a result, the shaded areas cannot be\nreached by the encirclement as shown in Fig. R4 (B, C). Thus the maximum phase\nmanipulation caused by rotation is limited to a rather small range, which renders the\nPB phase largely insusceptible to rotation. Consequently, it leads to the suppression of\nthe spin-orbit interaction, as shown in Fig. R4(C). Further discussion is provided in\nsupplementary note 1.3.\n(2) “it seems that the SOI suppression likely emerges between the wavelength range\nof two EPs, and the two EPs have opposite “handedness” in the wavelength-angle\nspace. Could the authors also check this point, since it determines whether the SOI\nsuppression effect is induced by single or dual EPs?”\nThe two EPs that emerge at different wavelengths share the same handedness,\nwith one serving to “close” and the other to “re-open” the TR-PB phase response.\nAlternatively, they participate in the SOI suppression process through opposite\ntopological manipulation. To enhance clarity, we differentiate the chirality of the EPs\nusing distinct line types: the solid circle represents the REP (right-handed EP),\nwhile the dashed circle denotes the LEP (left-handed EP).\nFig. R5. EPs in the wavelength-angle space. (A, D) PCR and PB phase response\nduring the SOI suppression process. (B, E) PCR and PB phase response during the\nSOI suppression and doubling processes. (C, F) PCR and PB phase response during\nthe SOI inversion process. Solid circles denote REPs, while the dashed circles\nindicate LEPs. Red EPs increase the winding number, whereas black EPs decrease it\nas the wavelength decreases.\nThe opposite “handedness” of the EPs in the wavelength-angle space reflects\nchanges in the winding number. For instance, in Fig. R5(E), the winding number is +1\nabove 819.55 nm. During the transition from longer to shorter wavelengths, after\npassing the first EP at 819.55 nm, which shows a clockwise phase increment, the\nwinding number drops to 0, corresponding to SOI suppression. This value persists\nuntil the introduction of the second EP at 704.26 nm, where a counterclockwise phase\nincrement restores the winding number to +1, effectively “re-opening” the SOI\nprocess. Subsequently, after traversing the third EP at 691.73 nm, which also exhibits\na counterclockwise phase increment, the winding number increases to +2, leading to\nSOI doubling. Finally, passing through the fourth EP at 681.7 nm, characterized by a\nclockwise phase increment, returns the winding number to +1, thereby terminating the\ndoubling effect. Thus, a counterclockwise phase increment at an EP increases the\nwinding number, while a clockwise increment decreases it.\nIt is worth emphasizing that although EPs may exhibit opposite phase-increment\ndirections and perform opposing topological manipulations, they share the same\nhandedness, as each occurs at a point where t\n− +\n= 0 or t\n+ −\n= 0 . To facilitate the\nidentification of EPs, they are color-coded based on their role in the topological\nreconstruction of PB phase: as wavelength decreases, red indicates EPs that\nincrease the winding number by +1, while black represents those that decrease it\nby -1.\nChanges in the manuscript:\nWe add some words and sentences to the last paragraph before Fig. 3 to highlight\nthe fact that, despite sharing the same handedness, the EPs may exhibit opposite\nphase-increment directions. This reflects the opposing topological manipulations\ninvolved.\n“It is worth pointing out that EPs despite sharing the same handedness, they may\nexhibit opposite phase-increment in the wavelength-angle space, which reflects the\nopposing topological manipulations. For instance, in Fig. 2B, ii-iii, the SOI\nsuppression is caused by the black dashed EP at 823 nm, and is closed by the red\ndashed EP at 705 nm. The SOI doubling is caused by the red dashed EP at 691 nm,\nand is closed by the black dashed EP at 681 nm.”\nComments6: Are the laser spots in the middle parts of Fig. 5c caused by the\n0t-order diffraction of the co-polarized transmission? If yes, why its influence on\nthe vortex beam generations is hard to observe?\nResponses:\nThank you very much for your comment. The middle parts in Fig. 5c are caused\nby the 0th order diffraction of the co-polarized transmitted light. The primary reason\nfor this is the generally low polarization conversion efficiency, which results in a\nrelatively strong co-polarized transmission component that remains detectable after\npassing through the linear polarizer.\nIt is important to recognize that this portion of the 0th-order diffraction does not\nengage in the spin-orbit interaction and therefore does not acquire the\nPancharatnam-Berry (PB) phase—a phenomenon well-documented in prior studies\n(e.g., Adv. Mater. 37, 2417183 (2025); Nanophotonics 9, 4327–4335 (2020)).\nConsequently, when utilizing the PB phase for vortex beam generation, the 0th-order\ncomponent associated with the photonic spin Hall effect does not contribute to the\nhelical phase front of the target vortex beam. Rather, it forms an orthogonally\npolarized background. The influence of this orthogonally polarized background is\nundetectable in our interference fringe patterns, since the reference beam shares the\npolarization of the vortex beam, preventing significant contrast modulation by the\nbackground.\nChanges in the manuscript:\nWe have added several sentences under the subtitle “Optical setup” in the main\ntext to explain the phenomenon of the laser spots in the middle parts of Fig. 5c caused\nby the 0th order diffraction of the co-polarized transmission.\n“Then, the beam passes through a quarter waveplate (QWP2), which has its fast axis\noriented at 90°with respect to QWP1, and a linear polarizer (LP2) to eliminate the\nundesired polarization. Eventually, spin dependent splitting is captured by the camera\nafter passing through a focusing lens (L1, f=150 mm). It is worth mentioning that the\ngenerally low polarization conversion efficiency of our sample leads to a relatively\nstrong co-polarized transmission component, which remains detectable after passing\nthrough the linear polarizer and contributes to the 0th-order diffraction observed in\nthe photonic spin-Hall effect.”\nComments7: In Fig. 3Eiii and 3Fiii, the experimental results for chiral SOI\ndoubling seems not clear without the aid of the white dashed guidelines. It may\nbe more straightforward to be demonstrated using photonic spin-Hall effect,\nwhere the deflection angle will be enlarged under LCP incidence.\nResponses:\nThank you for your valuable suggestion. Your proposal to use the photonic\nspin-Hall effect as a more intuitive demonstration of the chiral spin-orbit interaction\ndoubling is highly insightful. We have conducted theoretical simulations to investigate\nthis phenomenon, and as shown in Fig. R6. The results reveal enlarged deflection\nangle under LCP incidence at 689.22 nm (Fig. R6(C)). Compared to the case under\nRCP incidence at the same wavelength (Fig. R6(D)), this result clearly demonstrates a\ndoubled deflection angle in the photonic spin-Hall effect. It corresponds well with the\ndoubled PB phase response shown in Fig. 2(D), i, ⅳ in the main text. Furthermore, a\nsignificant increase in the deflection angle is also observed under the same LCP\nincidence when wavelength decreases from 843.36 nm (Fig. R6, A) to 689.22 nm (Fig.\nR6, (C)).\nFig. R6. Simulated photonic spin-Hall effects demonstrating chiral SOI doubling.\n(A, B) Phase response at 843.36 nm under LCP and RCP incidence respectively. (C, D)\nPhase response at 689.22 nm under LCP and RCP incidence respectively. The arrow\nhighlighted in yellow demonstrates the doubled deflection angle under LCP incidence,\nwhich corresponds well with the doubled PB phase demonstrated in Fig. 2D, ⅳ.\nTo present the experimental findings more clearly, we have recaptured the result\nof SOI doubling in experiment, as shown in Fig. 3E(iii). By comparing it with Fig.\n3E(i) and 3F(iii), the doubling of the topological charge can be clearly discerned.\nDespite the photonic spin-Hall effect can also present the SOI doubling, it serves\nprimarily as a validation method rather than the focus of our study. Given that the\nrecaptured interference pattern already provides direct and straightforward evidence\nof the SOI doubling process, further experimental confirmation via the photonic\nspin-Hall effect can be demonstrated in future works.\nFig. R7. The revised Fig. 3 in the main text. The red dashed box highlights the\nnewly replaced interference pattern.\nChanges in the manuscript:\nWe have added a supplementary note in the supplementary materials under the\nsubtitle “9. The additional discussion of chiral SOI doubling” to present the\ncomparisons.\n“9. The additional discussion of chiral SOI doubling\nTo demonstrate the chiral spin-orbit interaction doubling in a more intuitive way, we\nadd the simulation results of the doubled photonic spin-Hall effect in correspondence\nwith the doubled PB phase results shown in Fig.2D, ⅳ. The results are shown in Fig.\nS14. The results reveal enlarged deflection angle under LCP incidence at 689.22 nm\n(Fig. S14,C). Compared to the case under RCP incidence at the same wavelength (Fig.\nS14,D), this result clearly demonstrates a doubled deflection angle in the photonic\nspin-Hall effect. Furthermore, a significant increase in the deflection angle is also\nobserved under the same LCP incidence when wavelength decreases from 843.36 nm\n(Fig. S14,A) to 689.22 nm (Fig. S14,C).\nFig. S14. Simulated photonic spin-Hall effects demonstrating chiral SOI doubling.\n(A, B) Phase response at 843.36 nm under LCP and RCP incidence respectively.(C, D)\nPhase response at 689.22 nm under LCP and RCP incidence respectively. The arrow\nhighlighted in yellow demonstrates the doubled deflection angle under LCP incidence,\nwhich corresponds well with the doubled PB phase demonstrated in Fig. 2D, ⅳ.”\nWe also have replaced the original Fig.3E(iii) with a clearer experimental result\nto demonstrate the SOI doubling.\n“Fig. 3. The experimental results of chiral SOI suppression and doubling. (A and B)\nScanning electron micrographs (SEMs) of fabricated metasurface for demonstrating\nchiral SOI suppression and doubling. (C and D) Corresponding to Fig. 3A, i to ⅱ are\nthe measured diffraction patterns of spin-Hall effect, under the excitation of LCP and\nRCP light with wavelengths of 790 nm and 710 nm. (E and F) Corresponding to Fig.\n3B, i to ⅲ are the interference patterns under excitation of LCP and RCP light with\nwavelengths of 850 nm, 800 nm, and 730 nm. The red (blue) shaded area represents\nLCP (RCP) incidence.”\nComments8: In the conclusion, the authors state their method can be used in\ninformation-encryption. However, related information in the main text and\nSupplementary Materials is quite not enough, which reads confusing and\nweakens the narrative. It is meaningful if the authors give an metasurface design\nexample to show the encrypt ability, either in simulation or experiment.\nResponses:\nWe sincerely thank you for this insightful suggestion. In response, we have\nincluded a design example of information encryption based on TR-PB phase\nmodulation in Fig. R7. Six distinct meta-atoms are designed, each providing one of\nfive different PB phase responses, as color-coded in Fig. R7(A). Their PB phase\nresponses are catalogued by incident polarization and wavelength in a TR-PB phase\nlibrary, shown in Fig. R7(B). From such library the PB phase acquisition rule of any\nstructure under specific polarization and input wavelength can be directly extracted.\nFig. R7(C) illustrates the encryption design workflow. A modified GS algorithm\nfirst computes the near-field phase distributions of the decoy image and the hidden\nimage. The two phase-profiles are linked through the rules of reversed, doubled, and\nconventional PB phase accumulation. By comparing the two distributions, a ternary\nmask consisting only of values “-1”, “+1”, and “+2” is obtained. By consulting the\nTR-PB library, every specific meta-atom type of the metasurface and the rotation\nangle of each main structure are determined, together with the correct decryption key\nK (σ, λ) that encodes the required polarization and wavelength. As shown in\nat 730 nm, where both phase inversion and doubling occur, RCP illumination switches\nthe reconstructed hologram from “cat” to “dog,” whereas illumination with opposite\nhandedness merely yields an inverted “cat” (Fig. R7(D)i-ii).\nFig. R7. Information encryption scheme. (A) Schematics of the six available\nmeta-atoms. (B) Library of TR-PB phase responses, sorted by polarization and input\nwavelength. (C) Design example of the encryption scheme. Using a modified GS\nalgorithm, the near-field phase distributions for the decoy image and the hidden image\nare generated. By comparing the two phase distributions and referencing the TR-PB\nphase library, the final arrangement adopting the six available meta-atoms (including\nthe rotation angle of each main structure) is obtained. (D) Decryption demonstration.\nWrong key leads to decoy results, and only by using the correct key, can the hidden\nmessage be retrieved, as shown in the red dotted box in Fig.R7(D), ⅳ.\nChanges in the manuscript:\nWe have deleted the original discussion of information encryption in the\nsupplementary note 5, and replaced it with another title “The potential topology that\nleads to advanced SOI”.\n“5. Information encryption\n5. The potential topology that leads to advanced SOI\nWe propose a spatial-wavelength-polarization encryption paradigm leveraging\nmetasurface spatial reconfigurability. By encoding keys into perturbation distribution\nD, …… and PB phase is robust against wavelength variations, the message can still\nbe safe when the encrypted metasurface is cloned during transmission.\nOur work suggests that there are still rich physics to be discovered for SOI process:\nWhen the topological protection of PB phase-based SOI is disrupted, the winding\nnumber exists the possibility for further evolution, as shown below.\nFig S10. Complex plane representation showing PB phase accumulation can be\ninfluenced to possess different winding number. (A-C) The topology we have verified\nby our experiment. (D) The potential topology that can lead to tripled SOI. The w\nnotation stands for winding number.\nHere we just show one of the many possibilities of winding. In order to reach such\nmanipulation, it is necessary to engineer the meta-atoms to precisely control the PB\nphase accumulation trajectory in the complex plane by inducing EPs at the\nappropriate rotation angle and wavelength. Such control over the winding number not\nonly enables advanced SOI processes (such as tripled SOI), but also opens pathways\ntoward higher order optical wavefront shaping and topologically active photonic\ndevices.”\nWe have added a new section in the main text under the subtitle “TR-PB phase\nused for information encryption” to present design example of information\nencryption based on TR-PB phase modulation.\n“TR-PB phase used for information encryption\nFig. 6. Information encryption scheme. (A) Schematics of the six available\nmeta-atoms. (B) Library of TR-PB phase responses, sorted by polarization and input\nwavelength. (C) Design example of the encryption scheme. Using a modified GS\nalgorithm, the near-field phase distributions for the decoy image and the hidden image\nare generated. By comparing the two phase distributions and consulting the TR-PB\nphase library, the final arrangement adopting the six available meta-atoms (including\nthe rotation angle of each main structure) is obtained. (D) Decryption demonstration.\nWrong key leads to decoy image, only by using the correct key, can the hidden message\nbe retrieved, as shown in the red dotted box in Fig. 6(D), ⅳ.\nIn this work, six distinct meta-atoms are designed, each providing one of five different\nPB phase responses, as color coded in Fig. 6(A). Their PB phase responses are\ncatalogued by incident polarization and wavelength in a TR PB phase library, shown in\nFig. 6 (B), from which the PB phase acquisition rule of any structure under specific\npolarization and input wavelength can be directly extracted.\nFig. 6(C) illustrates the encryption design workflow. A modified GS algorithm first\ncomputes the near field phase distributions of the decoy image and the hidden image.\nThe two phase-profiles are linked through the rules of reversed, doubled, and\nconventional PB phase accumulation. By comparing the two distributions, a ternary\nmask consisting only of values “-1”, “+1”, and “+2” is obtained. By consulting the TR\nPB library, every specific meta atom type of the metasurface and the rotation angle of\neach main structure are determined, together with the correct decryption key K (σ, λ)\n4\nthat encodes the required polarization and wavelength. As shown in Fig. 6(D), only\nwith this precise key can the hidden image be retrieved. For example, at 730 nm, where\nboth phase inversion and doubling occur, RCP illumination switches the reconstructed\nhologram from “cat” to “dog”, whereas illumination with opposite handedness merely\nyields an inverted “cat” (Fig. 6(D), i ii).”\nComments9: It will be helpful if the authors present the corresponding\nmeta-atom schematics in Fig. 2 and Fig. 4, and give out their dimensions in the\nmain text. This can make a clearer guidance to the readers. It is quite easy to be\nconfused that Fig. 2Ai and Fig. 2Bi are the two cross-polarized transmission\nresults of the same rotation meta-atom.\nResponses:\nWe sincerely appreciate your constructive suggestion. Incorporating schematic\nillustrations of the meta-atoms into Figs. 2 and 4 will undoubtedly help readers better\nunderstand the relationship between the simulation results and the corresponding\nstructures. In the revised manuscript, we have revised Figs. 2 and 4 accordingly, as\nshown in the newly provided Fig. R8 and Fig. R9. These updated figures now include\ndetailed meta-atom schematics, along with their key dimensions, providing clearer\nguidance for interpreting the TR-PB phase manipulation discussed in the main text.\nFig. R8. The revised Fig. 2 in the main text. The red dashed box highlights the\nnewly added schematic of the meta-atom schematics.\nFig. R9. The revised Fig. 4 in the main text. The red dashed box highlights the\nnewly added schematic of the meta-atom schematics.\nChanges in the manuscript:\nWe have revised the Fig.2 and Fig.4 in the main text, and added sentences to\nspecify their geometric dimensions.\n“In Fig. 2A the chiral SOI suppression was demonstrated under RCP incidence within\nthe wavelength range of 700-750 nm. Fig. 2A, i depicts the meta-atom we adopted for\nsimulation. The meta-atom has a period of 650 nm with the following geometric\ndimensions: dx = 80 nm, dy = 45 nm, w = 75 nm, L1 = 263 nm, and L2 = 145 nm.\nThe perturbation ellipse has axes of 75 nm and 37.5 nm. The centers of the two\nreference points (red dots) are separated by a displacement of (245 nm, 245 nm). See\nsupplementary note 2 for specific discussion.”\n“As shown in Fig.2B i, the meta-atom has a period of 663 nm with the following\ngeometric dimensions: dx=80 nm, dy=47 nm, w=75 nm, L1=285 nm, L2=160 nm,\nL3=185 nm. The perturbation ellipse has axes of 100 nm and 65 nm. The centers of\nthe two reference points (red dots) are separated by a displacement of (236 nm, 268\nnm). See specific discussion in supplementary note 2.”\n“Fig. 4 presents numerical simulations and evolution analyses of chiral SOI inversion.\nThe meta-atom adopts the configuration in Fig. 4A. It has a period of 700 nm with the\nfollowing geometric dimensions: dx =80 nm, dy=45 nm, w=75 nm, L1=263 nm,\nL2=145 nm. The perturbation ellipses all possess axes of 75 nm and 37.5 nm. The\ncenters of the four perturbation ellipses and the main structure are separated by an\nabsolute displacement of (220 nm, 220 nm).”\n“Fig. 2. Numerical investigations on chiral SOI suppression and doubling. (A to B)\nI are the schematics of the corresponding meta-atom, ii-iii are the cross-polarization\nconversion rate and PB phase acquisition (as a function of rotation angle and\nwavelength) of the meta-atom designed to experience EP at 750nm and 700nm or\n823nm and 706nm. The purple and orange dashed lines in Fig. 2B, ii, represent\nwavelengths under further inspection in C and D. The solid circle represents the REP,\nwhile the dashed circle denotes the LEP. The red circles denote EPs that increase the\nwinding number by +1, and the black circles denote EPs that decrease it by -1, as\nwavelength decreases. Using photonic spin Hall effect and spin to vortex conversion\nwith opposite chirality and distinct response wavelengths as examples, (A to B)\niⅴ-ⅵi\nare the simulated results of metasurfaces designed to demonstrate chiral SOI\nsuppression. (C to D) I-ⅲ are the cross-polarization conversion evolution trajectory of\nchiral SOI suppression effect on the complex plane, and PB phase response at\ncorresponding wavelength. (C to D), ⅳ respectively show the complex-plane\ncross-polarization conversion trajectory and PB phase response of the chiral SOI\ndoubling effect at wavelength of 689.22 nm.”\n“Fig. 4. The process of chiral SOI inversion is investigated by simulation. (A) The\nschematic of the adopted meta-atom. (B and C) The cross-polarization conversion rate\nand PB phase acquisition while changing input wavelength and rotation angle. The\nblack dashed lines in Fig. 4(B,C) represent wavelengths under further inspection in D\nand E. The solid circle represents the REP, the red circles denote EPs that increase the\nwinding number by +1, and the black circles denote EPs that decrease it by -1, as\nwavelength decreases. (D) The cross-polarization conversion evolution trajectory on\nthe complex plane. (E) The PB phase response at corresponding wavelength. (F to H)\nThe simulated results of metasurfaces designed to demonstrate chiral SOI inversion\nwith spin-orbit conversion and photonic spin Hall effect.”\nComments10: Some other small suggestions:\na) The definitions of the signs of “+ and –”, the abbreviations of CCW and CW,\nshould be defined in the main text.\nResponses:\nWe greatly appreciate your suggestions. We have added the definitions of “+”\nand “-” as well as the meanings of the abbreviations CCW and CW in the main text.\nIn specific, the “+” represents LCP, while “-” represents RCP. The CCW represents\ncounterclockwise and the CW represents clockwise.\nChanges in the manuscript:\nWe have added one sentence after the equation (3) to define the signs of “+ and –”\nin the main text.\n“After been rewritten in the basis of circular polarization, it is easy to derive that:\n R\nI m\ne\nTT\n+\n+\n−\n−\n\n=\n=\n((\na\nb\n−\n−\ne\nf\n+\n−\n2\n2\nd\nc\n))\n2\n2\n R\nI m\ne\nTT\n−\n−\n+\n+\n\n=\n=\n((\na\nb\n−\n−\ne\nf\n−\n+\n2\n2\nd\nc\n))\n2\n2\n(3)\nWhere “+” represents LCP, while “-” represents RCP. By taking the non-Hermitian\nform Jones matrix into equation (1), we can derive the Jones matrix ( )  J after\nrotation, which is described by a-f.”\nWe have also added some words before the section subtitled “Topological\nReconstruction Controlled by a Single EP under Fixed Wavelength” to define the\nabbreviations of CW and CCW.\n“This shift in perspective reveals that the accumulation of the PB phase is equivalent to\ndynamically encircling an EP, with the direction of encirclement determined by the\nchirality of the input circular polarization. as shown in Fig.1(F,G), when under LCP\nincidence, ( )  T\n− +\nencircles the LEP in counterclockwise (CCW) manner, whereas\nunder RCP incidence, ( )  T\n+ −\nencircles the REP in clockwise (CW) manner, while\ncontinuously increases from 0 to π. Building upon this point of view, we first dissect\nchiral SOI suppression and its evolution in the complex plane.”\nComments11: b) The authors state that “PB phase are widely used in\nmetasurfaces…to exhibit SOI phenomena, such as the spin Hall effect…”\nHowever, the term of “spin Hall effect” should be used more precisely as\n“photonic spin Hall effect”.\nResponses:\nWe greatly appreciate your comment. The term \"spin Hall effect\" should be used\nwith precision. In the context of our manuscript, we should refer to it as the \"photonic\nspin Hall effect\" to clearly distinguish it from the original spin Hall effect observed in\nelectronic systems. We have made the necessary corrections in the main text.\nChanges in the manuscript:\nTo clarify, we have used the precise terminology \"photonic spin Hall effect\"\nthroughout the main text, instead of \"spin Hall effect\", to accurately describe the\nphenomenon under our study. The following is an example from the Introduction\nsection, where \"photonic\" has been added.\n“PB phase are widely used in metasurfaces, a platform composed of periodically\narranged sub-wavelength nanostructures 11-13, to exhibit SOI phenomena, such as the\nphotonic spin Hall effect 14,15, spin-orbit conversion 16-18, spin-orbit mapping 19, and\nsupport applications including optical holography 20-22 and lenses 23.”\nComments12: c) The thickness of the ITO film mentioned in the main text (160\nnm) and the methods (150 nm) are different. The material parameters of the\nmetal structures and the ITO film should also be presented.\nResponses:\nThank you very much for comment. We apologize for the inconsistency\nregarding the ITO film thickness. The thickness value of the ITO film presented in the\nmain text is correct. Therefore, we have corrected the thickness of the ITO film to 160\nnm in the Methods section. Additionally, we have now included the material\nparameters of both the metal structures and the ITO film used in our work. These\nparameters are provided to ensure clarity and consistency throughout the manuscript.\nChanges in the manuscript:\nWe have revised the value of ITO thickness in the Methods section.\n“Device fabrication\nThe substrate used here is 160 nm ITO-coated silica glass. The substrates were\ncleaned by sequential ultrasonication for 5min in acetone, methanol, and ultrapure\nwater. The fabrication involved electron beam lithography, sequential deposition of a\n6nm titanium (Ti) layer and a 40nm gold (Au) film, and liftoff in butanone.”\nWe have also included additional details on the material parameters for Au and\nITO in the second paragraph of the section titled “Topological Reconstruction\nControlled by a Single EP under Fixed Wavelength” in the main text.\n“For these simulations, metasurfaces are set to the size of 10 μm*10 μm, while\nperfectly matched layers are applied at the x, y and z directions. We adopted the\nmaterial parameters from “Au (gold) Johnson and Christy” for simulating the\nstructures and the refractive index is set to 1.6 for ITO. Such configuration was\napplied throughout our work.”\nComments13: d) The tick marks of the axes in Fig. 2A(i,ii), Fig. 2B(i,ii), and Fig.\n4A,4B are not clear, and the label of the horizontal axes are also not presented.\nBesides, the axes of all the simulated and measured results, as well as\ncorresponding scalebars should be presented.\nResponses:\nThank you very much for your comments. We fully understand your concern\nregarding the clarity and completeness of the data presentation. Here, we have\nupdated Figs. R8 and R9 (corresponding to Figs. 2 and 4 in the main text) by adding\nclear tick marks and labels to all axes, including the previously missing horizontal\naxes. Additionally, scale bars and axes have been included for all simulated results to\nimprove readability and interpretation. The scale bars for the experimental data have\nalso been added, as shown in Figs. R10 and R11 (corresponding to Figs. 3 and 5 in the\nmain text), ensuring that both simulated and measured results are fully and\nconsistently presented.\nFig. R8. The revised Fig. 2 in the main text. The red dashed box highlights the\nnewly added tick marks and axes.\nFig. R9. The revised Fig. 4 in the main text. The red dashed box highlights the newly\nadded tick marks, axes and scale bars.\nFig.R10. The revised Fig. 3 in the main text. The red dashed box highlights the\nnewly added tick marks, axes and scale bar.\nFig. R11. The revised Fig. 5 in the main text. The red dashed box highlights the\nnewly added tick marks, axes and scale bar.\nChanges in the manuscript:\nWe have revised the Figs.2-5 in the main text to add the axes and scales of all the\nsimulated and measured results.\n“Fig. 2. Numerical investigations on chiral SOI suppression and doubling.”\n“Fig. 3. The experimental results of chiral SOI suppression and doubling.”\n“Fig. 4. The process of chiral SOI inversion is investigated by simulation.”\n“Fig. 5. The experimental results of chiral SOI inversion.”\nComments14: e) In Fig. 2Ai and 2Bi, the mapping of t−+ and t+− are not\npresented in an intuitive manner. I suggest that the authors provide, in the\nSupplementary Materials, all line plots versus wavelength at the angles where\nthe EP points occur.\nResponses:\nThank you for the comment. We agree that line plots offer a more intuitive\nrepresentation of the EPs. Furthermore, considering that the main text classifies the\nphenomena based on the number of EPs introduced at fixed wavelengths, presenting\nline plots versus rotation angles precisely at the wavelengths where the EPs occur\nmay offer more advantage. Since it not only presents the results in an intuitive manner\nbut also aligns directly with the narrative of the manuscript. Therefore, we have\nplotted all line plots of t\n−+\nand t\n+−\nversus rotation angles at the wavelength where the\nEPs occur, as shown in Fig. R12.\nFig. R12. All the line plots versus rotation angle at wavelengths where the EP\npoints occur.\nIt is important to note that the low-PCR point at 62° in Fig. R12(D), ⅴ, is not an\nEP, but rather evolves from the EP shown in Fig. R12(D), ⅵ. This interpretation is\nsupported by the PB phase response in Fig. 2B, ⅲ, and by Supplementary Movie 8,\nwhich, from the complex-plane perspective, confirms that the system contains only a\nsingle EP at 691.73 nm. These additions in the supplementary note 7 provide a clearer\nand more intuitive visualization of the EPs, directly addressing your concern.\nChanges in the manuscript:\nWe have also added supplementary note in the supplementary materials with the\nsubtitle “7. Line plots at the wavelengths where the EP points occur”,\n“7. Line plots at the wavelengths here the EP points occur\nFig. S12. All line plots versus rotation angle at the wavelengths where the EP points\noccur. (A) PCR response for the right-handed SOI suppression phenomenon in the\nwavelength-angle space, the red and orange dashed lines are plotted in (D), i-ii. (B)\nPCR response for the left-handed SOI suppression and SOI doubling phenomenon in\nthe wavelength-angle space, the yellow, green, dark green and blue dashed lines are\nplotted in (D), iii-ⅴi. (C) PCR response for the right-handed SOI inversion\nphenomenon in the wavelength-angle space, the deep blue and purple dashed lines\nare plotted in (D), ⅴii-ⅴiii.\nIt should be noted that the low-PCR point at 62° in Fig. S12(D), ⅴ is not an EP but\nevolved from the blue EP shown in Fig. S12(D), ⅵ. This conclusion is supported by\nthe PB phase response in Fig. 2B, ⅲ and Supplementary Movie 8. Both of them\ndemonstrate that the system contains only a single EP at 691.73 nm.”\nComments15: f) In Fig. 3 and Fig. 5, the authors present the interference pattern\nto characterize the vortex beam features. It will be good if authors also include\nthe corresponding intensity distributions without interference from the reference\nbeam.\nResponses:\nThank you for your comment. During the revision, we re-measured the intensity\npatterns of SOI suppression, SOI doubling, and SOI reversal, along with their\ncorresponding intensity patterns under orthogonal polarization. Here we present the\ncorresponding intensity distributions (without the reference beam) in Fig.R13 as\nshown below. The interference patterns are also shown below to enhance the\ncorrespondence.\nFig. R13. (A,B) The intensity distributions without the reference beam that\ncorrespond to the interference pattern shown in Fig.R13(E,F). (C,D) The intensity\ndistributions without the reference beam that correspond to the interference pattern\nshown in Fig.R13(G,H).\nBy comparing Fig.R13,(A) i to ii, we can see that when the SOI is suppressed,\nthe phase singularity is disappeared. However, the SOI doubling and inversion\nphenomenon impose little change in the intensity pattern, as shown in Fig.\nR13(A,B)iii and Fig. R13(C,D)iii-ⅳ.\nChanges in the manuscript:\nWe have added one sentence before the subtitle “Topological Reconstruction\nControlled by dual EPs under Fixed Wavelength” in the main text to remind the\nintensity distributions correspond to the interference pattern in Fig. 3(E,F).\n“The interference pattern at 730 nm shows that topological charge has changed from\n+3 to +6, while the topological charge under RCP excitation remains ℓ= −3 (Fig.3\nC-D, iii). The corresponding intensity distribution are provided in Fig.S16. The\nspectral shift of chiral SOI doubling phenomenon from 692 nm (simulated in Fig. 2D,\niv) to 730 nm in experiment arises from the interplay of two factors.”\nWe have also added one sentence before the subtitle “TR-PB phase used for\ninformation encryption” in the main text to remind the intensity distributions\ncorrespond to the interference pattern in Fig. 5(A,B).\n“Supplementary movie 5-7 captures experimental inversion process for both\ntopological charge transitions and spin-dependent splitting under RCP incidence.\nWhile the corresponding intensity distribution are provided in Fig.S16. Control\nexperiments confirm these effects originate solely from TR-PB phase modulation,\nwhich is achieved through anisotropic coupling between main structure and\nperturbation ellipses, rather than perturbation ellipse, rotation of the dual-L\nconfiguration seen from Fig. S9 in the supplementary materials.”\nMoreover, in the supplementary materials, we have added a supplementary note\nunder the subtitle “11. Intensity distributions without the interference from the\nreference beam” to present the intensity distributions.\n“11. Intensity distributions without the interference from the reference\nbeam\nHere we experimentally present the corresponding intensity distributions (without\ninterference from the reference beam) in Fig.S16 as shown below.\nFig.S16. (A,B) The intensity distributions without the reference beam that correspond\nto the interference pattern shown in Fig.3(E,F). (C,D) The intensity distributions\nwithout the reference beam that correspond to the interference pattern shown in\nFig.5(A,B).\nIt can be clearly seen in Fig. S16(A), ii that when the SOI is suppressed, the phase\nsingularity is disappeared. However, the SOI doubling and inversion phenomenon\nimpose little change in the intensity pattern as shown in Fig. S16(A, B), iii and Fig.\nS16(C, D), iii-ⅳ.”\nReply to Reviewer 2:\nGeneral Comment: The author innovatively proposed a design scheme for\nachieving nonlinear PB phase by introducing perturbations to construct EP\npoints in chiral metasurfaces, and elaborated on the underlying principles. The\nselection of the relative position between the EP point and the incident\nwavelength was demonstrated to suppress, reverse, and double the SOI effect,\nand experimental verification was carried out. The theoretical foundation is\nrobust and the approach demonstrates significant innovation. However, the\nfollowing questions require solution prior to publication:\nResponses:\nWe greatly appreciate your insight and positive comments on the potential and\ninnovative concept of our work. We would carefully address the concerns you have\nraised with the necessary revisions, which could meet the expectations for\npublication.\nComments1: Page 7, line 143: Please explain why SOI suppression in Figure 2A\noccurs between 700 and 750 nm. However, SOI suppression occurs from 790 to\n710 nm under RCP incidence. Does SOI suppression occur at wavelengths below\n700 nm? Could you provide the wavelength ranges for which SOI suppression\ndoes and does not occur?\nResponses:\nWe greatly appreciate your comment. The SOI suppression occurs specifically\nwithin the 700-750 nm wavelength range due to the presence of two exceptional\npoints (EPs) located near 700 nm and 750 nm, respectively. As the wavelength\ndecreases from 850 nm, the first EP at 750 nm reduces the winding number to zero,\ntriggering the SOI suppression. The winding number remains unchanged until the\nwavelength reaches the second EP near 700 nm, which resets it to -1, thereby\nterminating the suppression.\nIn Fig. 2(A, iv-vii), we employ the photonic spin-Hall effect in simulation to\nillustrate this suppression. Representative wavelengths are selected to highlight the\neffect. At 790 nm, which is outside the 700-750 nm wavelength range, both RCP and\nLCP incident beams are deflected. In contrast, at 710 nm, which is within the 700-750\nnm wavelength range, the beam deflection disappears under RCP incidence. For\nwavelengths below 700 nm, the suppression is switched off by the second EP, and the\ndeflection behavior resumes.\nA similar mechanism governs the response in Fig. 2(B), except that the distinct\ngeometry and chirality of the structure shift the effective wavelength band to 706-823\nnm, and change the target polarization to LCP. Here, we choose 850 nm and 800 nm\nas representative wavelengths to demonstrate the SOI suppression via spin to vortex\nconversion. The orbital angular momentum (OAM) generation is clearly observed\noutside the 706-823 nm wavelength range, and the OAM generation is suppressed\nwithin the 706-823 nm wavelength range under LCP incidence.\nTo help readers distinguish between the two designs, we have also added\nschematics of the corresponding meta-atoms in Fig. 2(A) and (B), respectively. We\nhave also added explicit descriptions of the suppression and non-suppression\nwavelength ranges in the main text for clarity.\nChanges in the manuscript:\nWe have revised the Fig.2 and Fig.4. to demonstrate the results corresponding to\nthe different meta-atom designs.\n“Fig. 2. Numerical investigations on chiral SOI suppression and doubling. (A to B)\nI are the schematics of the corresponding meta-atom, ii-iii are the cross-polarization\nconversion rate and PB phase acquisition (as a function of rotation angle and\nwavelength) of the meta-atom designed to experience EP at 750nm and 700nm or\n823nm and 706nm. The purple and orange dashed lines in Fig. 2B, ii, represent\nwavelengths under further inspection in C and D. The solid circle represents the REP,\nwhile the dashed circle denotes the LEP. The red circles denote EPs that increase the\nwinding number by +1, and the black circles denote EPs that decrease it by -1, as\nwavelength decreases. Using photonic spin Hall effect and spin to vortex conversion\nwith opposite chirality and distinct response wavelengths as examples, (A to B)\niⅴ-ⅵi\nare the simulated results of metasurfaces designed to demonstrate chiral SOI\nsuppression. (C to D) I-ⅲ are the cross-polarization conversion evolution trajectory of\nchiral SOI suppression effect on the complex plane, and PB phase response at\ncorresponding wavelength. (C to D), ⅳ respectively show the complex-plane\ncross-polarization conversion trajectory and PB phase response of the chiral SOI\ndoubling effect at wavelength of 689.22 nm.”\n“Fig. 4. The process of chiral SOI inversion is investigated by simulation. (A) The\nschematic of the adopted meta-atom. (B and C) The cross-polarization conversion rate\nand PB phase acquisition while changing input wavelength and rotation angle. The\nblack dashed lines in Fig. 4(B,C) represent wavelengths under further inspection in D\nand E. The solid circle represents the REP, the red circles denote EPs that increase the\nwinding number by +1, and the black circles denote EPs that decrease it by -1, as\nwavelength decreases. (D) The cross-polarization conversion evolution trajectory on\nthe complex plane. (E) The PB phase response at corresponding wavelength. (F to H)\nThe simulated results of metasurfaces designed to demonstrate chiral SOI inversion\nwith spin-orbit conversion and photonic spin Hall effect.”\nIn the section titled “Topological Reconstruction Controlled by a Single EP\nunder Fixed Wavelength”, we have added some words to emphasize the operational\nwavelength ranges.\n“In Fig. 2A the chiral SOI suppression was demonstrated under RCP incidence within\nthe wavelength range of 700-750 nm.”\n“Here we take response at 790 nm and 710 nm as an example, as shown in Fig. 2A,\niv-ⅵi.”\n“In Fig.2 B, the chiral SOI suppression was demonstrated under LCP incidence within\nthe wavelength range of 706-823 nm, and we manifested it in spin-to-vortex\nconversion, using metasurface whose meta-atom rotates following Equation (2), with\ntopological charge set to 3 (ℓ = 3). Equations (1-2) are shown below:”\n“As shown in Fig. 2B, iⅴ-ⅵi, the metasurface generates OAM with ℓ = +3 under LCP\nexcitation above 823 nm (here we take the phase distribution at 850 nm as an example),\nand is transformed to a near plane wave phase distribution below 823 nm (here we take\nthe phase distribution at 800 nm as an example).”\nComments2: Could you add |t-+| and arg[t-+] for metasurface 2A, and |t+-| and\narg[t+-] for metasurface 2B in the supplementary material.\nResponses:\nThank you very much for your comment. The |t | and arg[t ] for metasurface 2A,\n-+ -+\nand |t | and arg[t ] for metasurface 2B is essential for a clear comparison in the\n+- +-\nwavelength-angle space, as it allows the chirality of our phenomenon to be clearly\ndemonstrated. The corresponding amplitude and phase distributions in the\nwavelength-angle space under opposite chirality incidence are illustrated in Fig. R1.\nFig. R1 (A) and (D) illustrate the PCR and PB phase response of opposite\nchirality incidence corresponding to the right-handed SOI suppression process in Fig.\n2(A) ii, iii. Fig. R1 (B) and (E) show the PCR and PB phase response of opposite\nchirality incidence corresponding to the left-handed SOI suppression and doubling\nprocesses in Fig. 2(B) ii, iii. Fig. R1 (C) and (F) demonstrate the PCR and PB phase\nresponse of opposite chirality incidence corresponding to the right-handed SOI\ninversion process in Fig. 4(B-C). It is evident that the other cross-polarization channel\nshows a trivial distribution in the wavelength-angle space, with no EP observed.\nDespite the presence of Rayleigh anomalies in Fig. R1, the conventional PB\nphase accumulation remains clearly visible. This is manifested as a linear dependence\non the rotation angle. Under LCP incidence, the PB phase increases by 2θ, while\nunder RCP incidence, it decreases by −2θ, respectively. The θ stands for rotation\nangle.\nFig. R1. The result of opposite chirality incidence in the wavelength-angle space.\n(A, D) PCR and PB phase response of opposite chirality incidence corresponding to\nthe right-handed SOI suppression process in Fig. 2(A) ii, iii. (B, E) PCR and PB phase\nresponse of opposite chirality incidence corresponding to the left-handed SOI\nsuppression and doubling processes in Fig. 2(B) ii, iii. (C, F) PCR and PB phase\nresponse of opposite chirality incidence corresponding to the right-handed SOI\ninversion process in Fig. 4(B-C).\nDue to limited space in the manuscript, we summarized the corresponding charts\nin the supplementary materials for clear comparison.\nChanges in the manuscript:\nWe have added a supplementary note in the supplementary materials under the\nsubtitle “8. The result of opposite chirality incidence in the wavelength-angle\nspace” to present the comparisons.\n“8. The result of opposite chirality incidence in the wavelength-angle\nspace\nFig. S13. The result of opposite chirality incidence in the wavelength-angle space.\n(A, D) PCR and PB phase response of opposite chirality incidence corresponding to\nthe right-handed SOI suppression process in Fig. 2(A) ii,iii. (B, E) PCR and PB phase\nresponse of opposite chirality incidence corresponding to the left-handed SOI\nsuppression and doubling processes in Fig. 2(B) ii,iii. (C, F) PCR and PB phase\nresponse of opposite chirality incidence corresponding to the right-handed SOI\ninversion process in Fig. 4(B-C).\nDue to the intrinsic chirality of the EPs, oppositely handed incidences cannot access\nan EP under identical conditions (wavelength and angle shown in Fig. S13), and\nhence result in the conventional PB phase accumulation. This dictates the chiral\nnature of the proposed SOI steering phenomena, which is firmly supported by direct\nexperimental evidence in the main text.”\nComments3: For metasurface 2A, the phase distribution in Figure 2Aii shows\nthat the phase variation with rotation angle within a 2π range at an incident\nwavelength of 790 nm has little correlation with the EP point at 750 nm, and the\nregion surrounding the EP point does not appear to be the EP point at 750 nm. Is\nit possible to obtain a 2π phase modulation range at approximately 740 nm?\nResponses:\nWe greatly appreciate your comment. At 790 nm, the PB phase still follows the\nconventional accumulation and can achieve a full 2π phase modulation range. This is\nbecause the SOI suppression occurs only within the 700-750 nm band, and at 790 nm\nthe system remains unaffected by the exceptional point (EP) at 750 nm, leaving the\nPB phase response intact. The simulated result at 790 nm is presented as a\nrepresentative wavelength to demonstrate that SOI suppression does not occur outside\nthe 700-750 nm band; the effect is confined to this interval, as exemplified by our\nsimulation at 710 nm. Furthermore, the regions adjacent to an EP are not themselves\nexceptional points. This is evident from the phase modulation in the wavelength-angle\nspace shown in Fig. 2A(ii), where only a single EP appears at 749.37 nm.\nTo further demonstrate such process, we examined the encircling trajectory in the\ncross-polarization conversion complex plane, along with the PB phase response and\nPCR at the relevant wavelength in Fig. R2. Fig. R2(A, B) details the PCR and PB\nphase across the wavelength-angle space, with three specific wavelengths examined:\n756.89 nm (red line), 749.37 nm (yellow line), and 738.10 nm (blue line), as shown in\nFig. R2(C-E). In Fig. R2(C), the cross-polarization amplitude in panel (i) is low but does\nnot reach the EP (marked by the blue circle). The corresponding trajectory in the\ncomplex plane (panel (ii)) encircles the EP, yielding a winding number of -1.\nConsequently, the resulting PB phase accumulation (panel (iii)) follows a\nconventional profile. Fig. R2(D) shows the case where the EP is reached. Here, the\nencircling trajectory (panel (ii)) passes directly through the EP. This condition\nprevents the PB phase from achieving a full 2π modulation range, as shown in panel\n(iii). Despite the similarly low cross-polarization amplitude in panel (i) of Fig. R2(E)\nand (C), their trajectories in panel (ii) differ fundamentally. The trajectory in Fig.\nR2(E) does not enclose the EP, yielding a winding number of 0. Therefore, as panel\n(iii) demonstrates, the PB phase accumulation is suppressed.\nWhen the incident wavelength is approximately 740 nm, the 2π phase modulation\ncannot be obtained, despite there is a phase jump from π to -π observed at\napproximately 740 nm in Fig. 2(A),iii. It is because when the encircling trajectory\npasses through the EP at 750 nm, the winding number of the encircling trajectory is\ndiminished to zero. The evolution trajectory in the complex plane perspective offers a\nclear insight into such phenomenon. It can be clearly seen in Fig. R2(E),iii, the\ntrajectory cannot reach all the four quadrants, thus the PB phase accumulation range is\nless than π, and the winding number is zero. However, after experiencing EP at 749.37\nnm, the winding number is restored to +1, causing regular PB phase response, despite\nlow cross-polarization rate, as shown in Fig. R2(C),iii.\nFig.R2. The influence of low PCR on TR-PB phase. (A and B) The PCR and PB\nphase response at wavelength-angle space. We fixed the wavelength at the positions\nof the red (756.89 nm), yellow (749.37) and blue (738.10 nm) dotted lines to\ninvestigate the influence of PCR on the desired functionalities. (C-E) The\ncross-polarization conversion rate, encircling trajectory in the cross-polarization\nconversion complex plane and corresponding PB phase response at different\nwavelength.\nChanges in the manuscript:\nWe add supplementary note in the supplementary materials with the subtitle “6.\nThe influence of low PCR on TR-PB phase” to demonstrate the PB phase at around\nthe wavelength of 740 nm.\n“6. The influence of low PCR on TR-PB phase\nDuring the rotation process, reaching the EP inevitably involves passing through\nstates with a low cross-polarization amplitude. However, based on our analysis, such\nlow amplitude does not affect the desired functionalities. Specifically, the winding\nnumber of the PB phase acquisition process is influenced only by the zero\ncross-polarization conversion point, which corresponds to the EP. Therefore, a low\ncross-polarization conversion rate (PCR) does not steer the PB phase-mediated SOI\nprocesses. It simply indicates a low cross-polarization conversion efficiency at the\ncorresponding rotation angle.\nThis can be verified by examining the encircling trajectory in the cross-polarization\nconversion complex plane, alongside the PB phase response and the\ncross-polarization conversion rate at the relevant wavelength. Fig. S11(A,B) details\nthe PCR and PB phase across the wavelength-angle space, with three specific\nwavelengths examined: 756.89 nm (red line), 749.37 nm (yellow line), and 738.10 nm\n(blue line), as shown in Fig. S11(C-E).\nIn Fig. S11(C), the cross-polarization amplitude in panel (i) is low but does not reach\nthe EP (marked by the blue circle). The corresponding trajectory in the complex plane\n(panel (ii)) encircles the EP, yielding a winding number of -1. Consequently, the\nresulting PB phase accumulation (panel (iii)) follows a conventional profile. Fig.\nS11(D) shows the case where the EP is reached. Here, the encircling trajectory (panel\n(ii)) passes directly through the EP. This condition prevents the PB phase from\nachieving a full 2π modulation range, as shown in panel (iii). Despite the similarly\nlow cross-polarization amplitude in panel (i) of Fig. R1(E) and (C), their trajectories\nin panel (ii) differ fundamentally. The trajectory in Fig. S11(E) does not enclose the\nEP, yielding a winding number of 0. Therefore, as panel (iii) demonstrates, the PB\nphase accumulation is suppressed.\nIn summary, the standard PB phase acquisition is preserved as long as the winding\nnumber is preserved, even when the PCR is low. Only after passing through the EP\ncan the winding number be altered, leading to the desired steering functionalities. The\nlow PCR after passing through the EP is a collateral effect of inducing the EP and\ndoes not cause any changes in the winding number. As a result, low PCR has no\nimpact on the topologically reconstructed PB (TR-PB) phase response or the desired\nfunctionalities. This behavior is consistent across the entire study.\nFig. S11. The influence of low PCR on TR-PB phase. (A and B) The PCR and PB\nphase response at wavelength-angle space. We fixed the wavelength at the positions of\nthe red (756.89 nm), yellow (749.37) and blue (738.10 nm) dotted lines to investigate\nthe influence of PCR on the desired functionalities. (C-E) The cross-polarization\nconversion rate, encircling trajectory in the cross-polarization conversion complex\nplane and corresponding PB phase response at different wavelength.”\nComments4: According to Figure 2Bi, the EP generation point does not seem to\nbe at 819.55 nm, but at least exceeds 825 nm. Please correct the wavelength of the\nEP point or the drawing method.\nResponses:\nWe sincerely appreciate this suggestion. The tick marks in Fig. 2 are visually too\nfaint, which has led to misinterpretation of the wavelength positions. In Fig. 2, the\nactual tick position is aligned above the center of its corresponding numerical label.\nThis positional offset creates the visual impression that the EP occurs beyond 825 nm,\nwhile its true location is at 819.55 nm. Here, we have re-plotted the corresponding\nfigures with emphasized tick marks and labels, as shown in Fig. R3. We believe this\nrevision now accurately conveys the wavelength correspondence of the EP response\nand ensures proper interpretation of the data. As shown below, the emphasized tick\nmarks shows that the EP at 819.55 nm in Fig. R3(B), ii exist below 825 nm. We also\nemphasized the tick marks of Fig. R3(A), ii to clarify the wavelength correspondence.\nFig. R3. Numerical investigations on chiral SOI suppression and doubling. (The\nrevised Fig. 2 in the main text.)\nChanges in the manuscript:\nWe have revised Fig. 2 in the main text.\n“Fig. 2. Numerical investigations on chiral SOI suppression and doubling.”\nComments5: The two light spots shown in Figures 3C and 3D are rather odd. Is\none of them the center point?\nResponses:\nThank you very much for your comment. The two optical spots correspond to the\n0th-order diffraction and the beam deflected by the PB geometric phase, respectively.\nFollowing the design of our metasurface based on the conventional PB phase\naccumulation rule, LCP light is deflected to the left, while RCP light is deflected to\nthe right. Unlike the case of SOI inversion in Fig. 5, which requires displaying both\ndeflection directions under specific polarizations, here we intentionally omitted the\nopposite deflection channels (LCP deflected to the right and RCP deflected to the left)\nfor clarity. To help the readers identify the center point, we have now marked the\ncomplementary deflection channels to clearly indicate the central position of the effect,\nas shown in the revised Fig. 3 below.\nThe revised Fig. 3 in the main text. The red dashed box highlights the improved photonic\nspin-Hall effect results.\nThe center points in Fig. 3C,D are caused by the 0th order diffraction of the\nco-polarized transmitted light. The primary reason for this is the generally low\npolarization conversion efficiency, which results in a relatively strong co-polarized\ntransmission component that remains detectable after passing through the linear\npolarizer.\nIt is important to recognize that this portion of the 0th-order diffraction does not\nengage in the spin-orbit interaction and therefore does not acquire the\nPancharatnam-Berry (PB) phase. Such phenomenon is well-documented in prior\nstudies (e.g., Adv. Mater. 37, 2417183 (2025); Nanophotonics 9, 4327–4335 (2020)).\nConsequently, when utilizing the PB phase for vortex beam generation, the 0th-order\ncomponent associated with the photonic spin Hall effect does not contribute to the\nhelical phase front of the target vortex beam. Rather, it forms an orthogonally\npolarized background. The influence of this orthogonally polarized background is\nundetectable in our interference fringe patterns, since the reference beam shares the\npolarization of the vortex beam, preventing significant contrast modulation by the\nbackground.\nChanges in the manuscript:\nWe have added several sentences under the subtitle “Optical setup” in the main\ntext to explain the two light spots shown in Figures 3C and 3D.\n“Then, the beam passes through a quarter waveplate (QWP2), which has its fast axis\noriented at 90°with respect to QWP1, and a linear polarizer (LP2) to eliminate the\nundesired polarization. Eventually, spin dependent splitting is captured by the camera\nafter passing through a focusing lens (L1, f=150 mm). It is worth mentioning that the\ngenerally low polarization conversion efficiency of our sample leads to a relatively\nstrong co-polarized transmission component, which remains detectable after passing\nthrough the linear polarizer and contributes to the 0th-order diffraction observed in\nthe photonic spin-Hall effect.”\nWe have added the complementary deflection channels in Fig. 3 to clearly\nindicate the central position of the effect.\n“Fig.3. The experimental results of chiral SOI suppression and doubling.”\nComments6: Figure 2(C-D)iv shows the generation of multiple EP points and\nshould not be placed in this section, and this section does not discuss this figure.\nResponses:\nWe thank the reviewer for this thoughtful suggestion. We agree that in the\nprevious version, the description of the doubling effect was separated from Fig. 2(C–\nD)iv. Here, we have restructured the manuscript to organize the content based on the\nnumber of EP introduced at a fixed wavelength. In particular, we have merged the two\nsubtitles “Single EP induced topological reconstruction” and “Multiple EPs induced\ntopological reconstruction” into a single section titled “Topological Reconstruction\nControlled by a Single EP under Fixed Wavelength”. Correspondingly, the subtitle\n“Dual EPs induced topological reconstruction” has been revised to “Topological\nReconstruction Controlled by Dual EPs under Fixed Wavelength”. This approach\nintegrates both the suppression and doubling phenomena (two key effects arising from\na single elliptical perturbation) into a unified context, thereby offering a more\ncomprehensive and logically consistent narrative. The relevant texts have been revised\naccordingly, as shown in the updated manuscript. The Fig.1 is also modified to match\nthe narrative:\nFig. R4. The revised Fig. 1 in the main text. The red dashed box and red arrows\nhighlight the reformed narrative.\nChanges in the manuscript:\nWe have merged the two subtitles “Single EP induced topological reconstruction”\nand “Multiple EPs induced topological reconstruction” into a single section titled\n“Topological Reconstruction Controlled by a Single EP under Fixed Wavelength”.\nCorrespondingly, the subtitle “Dual EPs induced topological reconstruction” has been\nrevised to “Topological Reconstruction Controlled by Dual EPs under Fixed\nWavelength”. The logic of Fig.1 has been changed accordingly. The parts in black\nitalic with underline are the position shift of the paragraphs, without any change in\ncontent. Deleted sentences are marked in black with a strikethrough.\n“Single EP induced topological reconstruction\nTopological Reconstruction Controlled by a Single EP under Fixed Wavelength\n…… The supplementary movie 1 demonstrates the suppression evolution.\nWhen EPs emerge consecutively while changing wavelength, it exists the possibility of\nchiral SOI doubling. As shown in Fig. 2B, ii, multiple EP responses emerged below 730\nnm under LCP incidence, resulting in the doubled PB phase accumulation (+4θ) within\nthe spectral window from 682 to 692 nm, as shown in Fig. 2B, iii. The multiple EP\nemergence signifies the topological protection is further disrupted, leading to a\nwinding number of 2, as shown in Fig. 2C, iv, under wavelength marked by yellow\ndashed lines in Fig. 2B, ii, for demonstration. The supplementary movie 8 reveals its\nevolution. Such a status results in a 4π geometric phase accumulation over 180°\nrotation (Fig. 2D, iv) and is verified in experiments shown in Fig. 3E-F, iii. See more\ndiscussion of chiral in supplementary note 9.\nExperiments shown in Fig.3 validate chiral SOI suppression via optical setups in Fig.\nS1 in the supplementary materials. ……, while changing the wavelength. The\ninterference pattern at 730 nm shows that topological charge has changed from +3 to\n+6, while the topological charge under RCP excitation remains ℓ= −3 (Fig.3\nC-D, iii).\nThe corresponding intensity distribution are provided in Fig.S16. The spectral shift of\nchiral SOI doubling phenomenon from 692 nm (simulated in Fig. 2D, iv) to 730 nm in\nexperiment arises from the interplay of two factors. First, disrupting the topological\nprotection of PB phase accumulation with one EP (leading to a winding number of 0)\ndestabilizes EP encircling evolution on the polarization-conversion complex plane. It\nmeans the actual evolution trajectory in experiment is not the same as the simulation.\nSecond, fabrication errors and inhomogeneities of meta-atoms also amplify such\ndeviations from simulated trajectories in Fig. 2C, iv.\nDual EPs induced topological reconstruction\nTopological Reconstruction Controlled by dual EPs under Fixed Wavelength\nThe introduction of two exceptional points (EPs) during rotation at a fixed wavelength\nenables chiral SOI inversion, reversing the PB phase accumulation principle relative\nto rotation angle under target spin. ……\nMultiple EPs induced topological reconstruction\nWhen EPs emerge consecutively while changing wavelength, it exists the possibility of\nchiral SOI doubling……. Second, fabrication errors and inhomogeneities of\nmeta-atoms also amplify such deviations from simulated trajectories in Fig. 2C, iv.”\nFig. 1. Novel chiral SOI phenomena. ……”\nComments7: Why are Au and ITO materials chosen for the metasurface in the\npaper? Can the emergence of EP and the introduction of interference be\nachieved with all-dielectric materials such as Si or Ge?\nResponses:\nThank you for raising the point regarding material selection. The emergence of\nexceptional points (EPs) is fundamentally a phenomenon arising from the degeneracy\nof eigenvalues and eigenstates in the Jones matrix, and is not exclusive to metallic\nstructures. Indeed, all-dielectric materials such as silicon (Si) or germanium (Ge) can\nalso support EPs, as has been demonstrated in recent literature (Light: Science &\nApplications 14, 262 (2025)).\nIn our work, the operational wavelength was designed for the visible to\nnear-infrared range (corresponding to our laser source Ti: Sapphire ultrafast\nfemtosecond laser centered at 800 nm). The Au-ITO configuration was therefore\nselected to ensure optimal resonance matching and experimental feasibility within our\ntarget spectral regime. We fully agree that all-dielectric platforms offer a promising\nalternative, particularly at longer wavelengths or in low-loss applications, and\nappreciate the reviewer's suggestion for potential extensions of this work.\nChanges in the manuscript:\nWe changed the 47th reference to demonstrate such EP is not exclusive to\nmetallic structures：\n“47.J. Kim, et al, Electro-Active Metasurfaces Controlling Exceptional Topological\nPhase Through Low-Voltage Operation on Conductive Polymer. Adv. Funct. Mater. ,\n2501916 (2025).\n47. Yi, C., Wang, Z., Shi, Y. et al. Creating topological exceptional point by on-chip\nall-dielectric metasurface. Light-Sci Appl 14, 262 (2025).”\nComments8: For the metasurface in Figure 2B, at an incident wavelength of\n689.22 nm, two circles around the EP point exhibit a phase variation of 4π.\nHowever, in Figure 3Eiii, the wavelength at which the doubled PB phase appears\nis 730 nm. The meta-atom corresponding to the every results in Figures 3 should\nbe clarified in the figure legend.\nResponses:\nThank you very much for your comment. The spectral shift in the doubling\nphenomenon is explained in the last section of our original manuscript. The spectral\nshift of chiral SOI doubling phenomenon from 690 nm (simulated in Fig. 2D, iv) to\n730 nm in experiment arises from the interplay of two factors. First, disrupting the\ntopological protection of PB phase accumulation with one EP (leading to a winding\nnumber of 0) destabilizes EP encircling evolution in an uncontrollable manner on the\npolarization- conversion complex plane. It means the actual evolution trajectory in\nexperiment is not the same as the simulation. Second, fabrication errors and\ninhomogeneities of meta- atoms also amplify such deviations from simulated\ntrajectories in Fig. 2C, iv.\nIn addition, the meta-atoms corresponding to the results in Fig. 3 are illustrated\nin Fig. 3A-B. Specifically, the results shown in Fig. 3C–D were obtained from a\nmetasurface composed of the meta-atom depicted in Fig. 3A, while those in Fig. 3E–F\noriginate from the structure shown in Fig. 3B. Although the meta-atoms shown in the\nSEM pictures differ in orientation, they share identical geometric parameters.\nChanges in the manuscript:\nWe have revised the figure legend of Figure 3.\n“Fig. 3. The experimental results of chiral SOI suppression and doubling. (A and B)\nScanning electron micrographs (SEMs) of fabricated metasurface for demonstrating\nchiral SOI suppression and doubling. (C and D) Corresponding to Fig. 3A, i to ⅱ are\nthe measured diffraction patterns of spin-Hall effect, under the excitation of LCP and\nRCP light with wavelengths of 790 nm and 710 nm. (E and F) Corresponding to Fig.\n3B, i to ⅲ are the interference patterns under excitation of LCP and RCP light with\nwavelengths of 850 nm, 800 nm, and 730 nm. The red (blue) shaded area represents\nLCP (RCP) incidence.”\nComments9: Is the doubled PB phase in Figure 3E induced by the presence of\ntwo EPs in Figure 2B? If so, would this structure be better addressed in the\n\"Dual EPs induced topological reconstruction\" section? On the other hand, are\nthe opposite PB phases due to the presence of two EPs at different frequencies?\nIf so, could you please explain in the manuscript the specific EPs induce the PB\nphase doubling in each of these three phenomena?\nResponses:\nThank you for this insightful comment. The doubled PB phase observed in Fig.\n3E is not induced by the presence of two EPs at different frequencies, but rather\ninduced by a single EP with counterclockwise phase increment at 691 nm (in red\ndashed circle) in the wavelength angle space, and is restored to normal by a single EP\nwith clockwise phase increment at 681 nm (in black dashed circle). Using winding\nnumber analysis, we can clearly see that one EP activates the SOI doubling and the\nother deactivates it. It is analogous to how SOI suppression is turned on and off by the\ntwo EPs at different wavelengths.\nSimilarly, the reversed PB phase are caused by the dual EPs under a fixed\nwavelength, and not by the two EPs at different wavelength. At 748.12 nm, both of\nthe two EPs increase the winding number, thus the winding number is changed from\n-1 to +1. At 691.73 nm, both of the two EPs decrease the winding number, thus the\nwinding number is changed from +1 back to -1.\nTo help the readers better distinguish the EPs, each EP is assigned a specific\ncolor according to their role in the topological reconstruction of PB phase: red for\nthose increasing the winding number by +1, and black for those decreasing it by\n-1, as wavelength decreases. In addition, the chirality of the EPs is distinguished\nusing different linetypes: The solid circle represents the REP (right-handed EP),\nwhile the dashed circle denotes the LEP (left-handed EP).\nFig. R5. EPs in the wavelength-angle space. (A, D) PCR and PB phase response\nduring the SOI suppression process. (B, E) PCR and PB phase response during the\nSOI suppression and doubling processes. (C, F) PCR and PB phase response during\nthe SOI inversion process. Solid circles denote REPs, while the dashed circles\nindicate LEPs. Red EPs increase the winding number, whereas black EPs decrease it\nas the wavelength decreases.\nFor instance, in Fig. R5(E), the winding number is +1 above 819.55 nm. During\nthe transition from longer to shorter wavelengths, after passing the first EP at 819.55\nnm, which shows a clockwise phase increment, the winding number drops to 0,\ncorresponding to SOI suppression. This value persists until the introduction of the\nsecond EP at 704.26 nm, where a counterclockwise phase increment restores the\nwinding number to +1, effectively “re-opening” the SOI process. Subsequently, after\ntraversing the third EP at 691.73 nm, which also exhibits a counterclockwise phase\nincrement, the winding number increases to +2, leading to SOI doubling. Finally,\npassing through the fourth EP at 681.7 nm, characterized by a clockwise phase\nincrement, returns the winding number to +1, thereby terminating the doubling effect.\nChanges in the manuscript:\nWe add some sentences to the last paragraph before Fig. 3 to emphasize the\nidentification of EP types.\n“It is worth pointing out that EPs despite sharing the same handedness, they may\nexhibit opposite phase-increment in the wavelength-angle space, which reflects the\nopposing topological manipulations. For instance, in Fig. 2B, ii-iii, the SOI\nsuppression is caused by the black dashed EP at 823 nm, and is closed by the red\ndashed EP at 705 nm. The SOI doubling is caused by the red dashed EP at 691 nm,\nand is closed by the black dashed EP at 681 nm. Since the winding number only\nchanges upon traversing an EP, the corresponding low PCR around such transition\nhas no impact on the TR-PB phase response. See supplementary note 6 for a complete\ndiscussion.”\nComments10: Why can elliptical metal columns introduce interference behavior,\nand why can the elliptical particles maintain their original nonlinear\ncharacteristics when the main structure rotates while the main structure does not\nrotate? Please briefly explain this principle in the paper.\nResponses:\nThank you for raising this question regarding the working mechanism of the\nperturbation ellipses. The fixed elliptical structure introduces perturbation because its\ncoupling with the main structure varies with rotation of the main structure. Since EP\nitself is sensitive against perturbation, such coupling modulation can effectively tailor\nthe EP response, enabling the proposed SOI steering processes.\nIt should be clarified that the elliptical structures in our design function as a\nlinear perturbation element and do not exhibit nonlinear properties in this context.\nThey serve primarily as a geometric tool to modulate rotation symmetry and introduce\norientation-dependent coupling, without involving nonlinear optical effects. We will\ninclude a concise explanation of this principle in the revised manuscript to improve\nclarity.\nChanges in the manuscript:\nWe add some words and sentences to the last paragraph of introduction in the\nmain text to emphasize the working mechanism of the perturbation ellipses.\n“By introducing static elliptical perturbations, a rotational anisotropy is induced, as\ntheir coupling to the main structure changes with rotation. This ensures that the\nencircling trajectory avoids full EP collapse, encountering EPs only at discrete points.\nSuch encounters disrupt topological protection, change the winding number, and\ntrigger three novel steering mechanisms:”\nComments11: The adopted meta-atoms in the manuscript involves many\nparameters. How to determine the structural size parameters during the design\nprocess? Are there any theoretical reference values?\nResponses:\nThank you very much for your insightful comment. The determination of\ngeometric parameters for the meta-atoms is indeed a key step in achieving the desired\nTR-PB phase response and SOI steering functionalities.\nIn our design workflow, the structural parameters are obtained through a\nsystematic series of parameter sweeps using finite-difference time-domain (FDTD)\nsimulations. Specifically, we first fix the distances between the two “L” structures (dx\nand dy), and then sweep the arm lengths l1 and l2 to identify the emergence of\ncross-polarization conversion singularities across the wavelength range. These\nsingularities correspond to the formation of EPs. If no EP appears within this initial\nparameter space, we then adjust dx and dy and repeat the l1–l2 sweeps. By iteratively\nrefining this two-stage parameter search, EPs with the desired handedness and\nwavelength position can be reliably located and selected as the main strucutre’s\ngeometric parameters. After determining the main structure, we introduce the\nperturbation element and explore its relative displacement and geometric size.\nThrough sweeping these parameters, we construct the wavelength–angle space maps\n(e.g., Fig. 2A(i)) and identify configurations that exhibit clear PB-phase manipulation\nand SOI steering characteristics. It is important to emphasize that this workflow is\nprimarily simulation-driven and does not rely on predetermined theoretical reference\nvalues. Instead, the EP centered design principle serves as the guiding criterion for\nparameter selection, enabling efficient identification of structures that support the\ndesired topological reconstruction of the PB phase.\nMoreover, our configuration that realize EP response can be explained by a near\nfield analysis via examining the surface current distribution.\nBy examining the surface currents on the top and bottom layers (marked in red\nand black in Fig. R6(A), respectively), we elucidate the physical origin of the\nexceptional points (EPs) through the mechanism consistent with prior studies. As\nshown in Fig. R6(B), an EP can be accessed by tuning the wavelength. We analyze\nthe five representative wavelengths marked in solid circles, with the corresponding\nsurface current behaviors displayed in Fig. R6(C–D), i-v.\nAs the wavelength increases continuously, the top layer current along the\nlower-left direction weakens and eventually reverses, while its orthogonal component\nremains nearly constant. The effective surface current for this process is indicated by\nblack arrows in the insets of Fig. R6(C, i–v). In contrast, the bottom-layer current\nevolves differently: with increasing wavelength, the surface current component along\nthe upper-right direction strengthens while the orthogonal component diminishes. The\nresulting effective current is similarly denoted by black arrows in Fig. R6(D, i–v).\nThe net surface current is summarized in Fig. R6(E). As the wavelength rises, one\ndirectional component of the net current gradually vanishes, leaving only the\northogonal component. Such state signals the emergence of an EP and is highlighted\nin light red in Fig. R6(B, E-iii). With further increase in wavelength, the net current\ngrows along the opposite (upper-right) direction, and the EP vanishes.\nFig. R6. Near-field analysis of the EP origin. (A) Schematic of the meta-atom with\na 51º rotated main structure. The red and black lines mark the cross sections analyzed\nat the top and bottom layers, respectively. (B) Cross-polarization conversion rate\nversus input wavelength. The five analyzed wavelengths are marked by solid circles,\nwith the EP state highlighted in light-red. (C, D) Surface current distributions on the\ncorresponding cross sections at each wavelength. The insets show the effective\ncurrent (black arrows). (E) Evolution of the net surface current, where the EP\nemergence is indicated by the light-red highlight.\nChanges in the manuscript:\nWe have added some words to the first paragraph in the main text of the subtitle\n“Topological Reconstruction Controlled by a Single EP under Fixed Wavelength”\nto emphasize the process of our design.\n“Topological Reconstruction Controlled by a Single EP under Fixed Wavelength\nWhen there is only one EP induced during rotation at a fixed wavelength, chiral SOI\nsuppression occurs, decoupling the photonic SOI from structural rotation. Here, we\ndesigned two metasurfaces to demonstrate chiral SOI suppression under different\nchirality and response wavelength using finite-difference time-domain (FDTD)\nparameter sweep simulations.”\nWe have also added one sentence to the first paragraph of the section titled\n“Topological Reconstruction Controlled by dual EPs under Fixed Wavelength” in\nthe main text.\n“Two EPs are observed at 51° and 141° under the wavelength of 748.12 nm, with\nsimilar EPs identified at 60° and 150° under 691.73 nm incidence. See supplementary\nnote 3 for EP verification, and also see supplementary note 10 for the near-field\nanalysis of the EP origin.”\nWe have added a supplementary note in the supplementary materials under the\nsubtitle “10. Near-field analysis of the EP origin” to present the comparisons.\n“10. Near-field analysis of the EP origin\nThe physical origin of the EP phenomena has been explained in previous studies [1,\n2], which states that the EP arises when opposite and equal surface current\ndistribution occur along one direction, while the surface current of the other\northogonal direction exists. By changing the geometric parameters or input\nwavelength, such near field behavior results in a zero cross-polarization conversion.\nHowever, the structure that we employ have never been scrutinized. Here, we offer a\nnear field analysis of our structure by examining the surface current distribution.\nBy examining the surface currents on the top and bottom layers (marked in red and\nblack in Fig. S15(A), respectively), we elucidate the physical origin of the exceptional\npoints (EPs) through the mechanism consistent with prior studies. As shown in Fig.\nS15(B), an EP can be accessed by tuning the wavelength. We analyze the five\nrepresentative wavelengths marked in solid circles, with the corresponding surface\ncurrent behaviors displayed in Fig. S15(C–D), i-v.\nAs the wavelength increases continuously, the top layer current along the lower-left\ndirection weakens and eventually reverses, while its orthogonal component remains\nnearly constant. The effective surface current for this process is indicated by black\narrows in the insets of Fig. S15(C) i–v. In contrast, the bottom-layer current evolves\ndifferently: with increasing wavelength, the surface current component along the\nupper-right direction strengthens while the orthogonal component diminishes. The\nresulting effective current is similarly denoted by black arrows in Fig. S15(D), i–v.\nThe net surface current is summarized in Fig. S15(E). As the wavelength rises, one\ndirectional component of the net current gradually vanishes, leaving only the\northogonal component. Such state signals the emergence of an EP and is highlighted\nin light red in Fig. S15(B, E), iii. With further increase in wavelength, the net current\ngrows along the opposite (upper-right) direction, and the EP vanishes.\nFig. S15. Near-field analysis of the EP origin. (A) Schematic of the meta-atom with a\n51° rotated main structure. The red and black lines mark the cross sections analyzed\nat the top and bottom layers, respectively. (B) Cross-polarization conversion rate\nversus input wavelength. The five analyzed wavelengths are marked by solid circles,\nwith the EP state highlighted in light-red. (C, D) Surface current distributions on the\ncorresponding cross sections at each wavelength. The insets show the effective current\n(black arrows). (E) Evolution of the net surface current, where the EP emergence is\nindicated by the light-red highlight.”\nComments12: The descriptions of the horizontal and vertical axes in Figure 2 are\nmissing. The vertical axis in Figure 2D ranges from -3 to 3. Are the units in\nradians? The horizontal axis is in degrees. It is recommended that the units used\nbe clearly indicated in the descriptions of the horizontal and vertical axes.\nResponses:\nThank you for your constructive comment. The unit of the vertical axis in Fig.2D\nis in radians, while the horizontal is in degrees. Here we have distinguished the\nradians from the degrees, by adding descriptions “rad” and “°” into the labels, as\nshown below.\nFig. R7. The revised Fig. 2 in the main text. The red dashed box highlights the\nnewly added tick marks and axes.\nChanges in the manuscript:\nWe have revised the Fig. 2 in the main text.\n“Fig. 2. Numerical investigations on chiral SOI suppression and doubling.”\nComments13: Figure S7 and legend are separated in the supporting materials.\nResponses:\nThank you very much for your comment. We fully agree that separating Figure\nS7 from its legend may affect readability and may cause unnecessary confusion for\nreaders. In the revised supporting materials, we have merged Fig. S7 with its\ncorresponding legend into a single, integrated figure panel. This modification ensures\nthat the description of the figure is clearly and immediately accessible, thereby\nimproving clarity and consistency throughout the supplementary document.\nAdditionally, we have reviewed all other figures in the supporting materials to\nensure that their legends are properly positioned and formatted, so that similar issues\ndo not occur elsewhere.\nChanges in the manuscript:\nWe have revised the Fig. S7 in the supplementary materials by deleting a few\nspaces.\n“2. Meta-atom parameters\nFig. S7. The parameters for the three major meta-atoms.”\nDear reviewer:\nThank you for taking the time to review our manuscript entitled “Topologically\nreconstructing Pancharatnam-Berry phase via encircling exceptional point for chiral\nspin-orbit interaction steering” (Manuscript ID: NCOMMS-25-67039A).\nWe greatly appreciate your valuable comments and suggestions, which have been\ncarefully considered in our revisions. Below, we have provided a point-by-point\nresponse, along with the corresponding changes made in the revised manuscript and\nsupplementary materials for your reference. We believe that all the comments have\nbeen addressed, the revised manuscript meets the standard of Nature Communicaitons\nfor publication.\nReply to Reviewer 1:\nGeneral Comment: The authors have addressed all my comments and carefully\nrevised their manuscript. I recommend this manuscript for publication in Nature\nCommunications.\nResponses:\nWe thank the reviewer for the positive assessment of our work. We are pleased to\nconfirm that our detailed responses and revisions have duly addressed the previous\nconcerns, and we appreciate the acknowledgement that the manuscript now fulfills the\ncriteria for publication in Nature Communications.\nReply to Reviewer 2:\nGeneral Comment: After the author provide individual explanations and made\ndetailed revisions, my problems have been resolved accordingly. The quality of\nthe manuscript has improved significantly at present. I believe that fulfills\nNature Communications' criteria for innovation, high impact, and scholarly\nrigor. Therefore, I suggest it for publication. As some optional suggestions, the\nauthors could further strengthen the text by incorporating discussion of the\nfollowing aspect.\nResponses:\nWe thank the reviewer for the positive assessment of our work. We are pleased to\nconfirm that our detailed responses and revisions have addressed the previous\nconcerns, and we appreciate the acknowledgement that the manuscript now fulfills\nNature Communications' criteria. We have also carefully considered the optional\nsuggestions provided and incorporated relevant discussion to further strengthen the\nmanuscript.\nBelow is a more detailed reply and revision. The revised sentences are\nhighlighted in red and added to the revised manuscript, while the sentences in\nblack remain unchanged. Deleted sentences are marked in black with a\nstrikethrough.\nComments1: In the legend of Fig.5, please clarify which linearly polarized\ncomponent's diffraction pattern is represented by Fig.5Ci and Fig.5Cii?\nResponses:\nThank you for your comment. Fig. 5Ci and Fig. 5Cii both show the diffraction\npatterns under left-circularly polarized (LCP) incidence. After polarization conversion,\nthe right-circularly polarized (RCP) component carries the corresponding PB phase\nand thus exhibits leftward deflection. Therefore, the left-deflected beams in these\npanels correspond to the RCP component.\nChanges in the manuscript:\nWe add two sentences in the caption of Fig.5 to clarify the polarization of the\nincidence and the deflected light.\n“Fig. 5. The experimental results of chiral Spin-Orbit-interaction (SOI) inversion.\n(a and b) The interference patterns with OAM of 1and 2, under the excitation of light\nwith wavelength of 795nm and 715nm. (c) The measured diffraction patterns of\nphotonic spin Hall effect, under the excitation of light with wavelength of 800nm and\n720nm. Red and blue shaded backgrounds indicate left circularly polarized (LCP) and\nright circularly polarized (RCP) incidence, respectively. The deflected component\nwithin the white dashed circle corresponds to the cross polarized light. (d) Scanning\nelectron micrographs (SEMs) of fabricated device for demonstrating chiral SOI\ninversion, with the scale bars set to 1μm. ”\nComments2: A brief analysis can be provided in the manuscript on how to\nincrease the cross-polarization conversion rate under the EP mechanism.\nResponses:\nThank you for the reviewer for the insightful suggestion. Indeed, under the EP (exceptional\npoint) mechanism, the cross-polarization conversion rate (PCR) inevitably exhibits\ncertain local minimum during the rotation process. However, as the reviewer correctly\npoints out, the overall modulation depth of the PCR can be systematically enhanced\nthrough several approaches.\nFirst, material optimization plays a crucial role. By selecting materials with lower\noptical losses at the target wavelength, the intrinsic polarization conversion efficiency\ncan be improved. For instance, using low-loss dielectric materials or optimizing the\nthickness of metallic layers can further boost the PCR baseline.\nSecond, structural design refinement offers additional degrees of freedom. The\ngeometry of the meta-atoms can be tailored to maximize the polarization conversion\nat the desired operating wavelength or rotation angle. Finite-difference time-domain\n(FDTD) simulations can be employed to optimize the parameters, thereby enhancing\nthe PCR.\nThird, the manner in which the EP is introduced during rotation can be further\nengineered. Rather than relying on perturbation ellipses, one may design the\nmetasurface such that the EP is dynamically accessed only at specific rotation angles,\nwhile elsewhere the structure operates away from the EP to maintain high PCR. This\ncould be achieved by incorporating multiple resonant modes or by exploiting\ndispersion engineering to shift the EP wavelength as a function of rotation.\nBy combining these strategies it is possible to significantly elevate the overall PCR\nresponse of the proposed scheme. We have added a brief discussion of these aspects in\nthe revised manuscript to address the reviewer's valuable suggestion.\nChanges in the manuscript:\nWe add one sentences in the discussion to discuss the possible ways to enhance the\noverall PCR response.\n“In addition, while researches have focused on incorporating materials such as PCMs\nto introduce additional degrees of freedom for modulation, our method leverages\ntopological control with widely used PB phase to introduce TR-PB phase, thereby\nintroducing a new degree of freedom for metasurface design. Furthermore, the overall\nPCR response and modulation depth can be further enhanced through material\nselection, structural optimization, and the EP excitation method. For instance, this can\nbe achieved by employing low-loss dielectric materials, refining geometric parameters,\nand engineering the EP introduction mechanisms. Our work suggests that there are\nstill rich physics to be discovered for SOI process: When the topological protection of\nPB phase-based SOI is disrupted, the winding number exists the possibility for further\nevolution, as discussed in supplementary note 5. ”\nComments3: It would be beneficial to elaborate further in the section “TR-PB phase\nused for information encryption” how this work achieves the spin-decoupling effect at\nthe wavelength of 730 nm. Specifically, how is the phase of the corresponding pixel in\nFig.6Div determined?\nResponses:\nThank you for your comment. The spin-decoupling effect originates from the chiral\nnature of the proposed TR-PB phase. Specifically, under the EP response, taking\nright-handed EP as an example, the PB phase accumulation rule can be modified from\nthe conventional −2θ to either −4θ or +2θ under RCP incidence. In contrast, for LCP\nincidence, where no EP is excited, the phase accumulation remains the conventional\n+2θ rule.\nBased on this chiral disparity, we incorporate the distinct phase accumulation rules\ninto a modified Gerchberg-Saxton (GS) algorithm. The algorithm simultaneously\ngenerates two target near-field phase distributions, one corresponding to the decoy\nimage and the other to the hidden image, while enforcing that their relationship\nfollows the above TR-PB phase constraints under RCP and LCP respectively. Through\niterative optimization, the algorithm converges to a metasurface phase profile that\nsimultaneously satisfies the imaging requirements for both circular polarizations. The\nresulting phase distribution for each pixel in Fig. 6(D) iv is therefore determined by\nthis optimization process, ensuring that the spin-decoupled response is encoded into\nthe metasurface design.\nChanges in the manuscript:\nWe add further explanation in the last paragraph in the section “TR-PB phase\nused for information encryption” to elaborate the information encryption details.\n“As shown in Fig. 6(d), only with this precise key can the hidden image be retrieved.\nFor example, at 730 nm, where both phase inversion and doubling occur, RCP\nillumination switches the reconstructed hologram from the cat image to the dog image,\nwhereas illumination with opposite handedness merely yields an inverted cat image\n(Fig. 6(d), i-ii). the hidden image is retrieved only with the correct key. For instance,\nunder LCP illumination, the PB phase remains conventional (+2θ) across the entire\nspectrum, and the reconstructed hologram consistently displays the cat image (Fig. 6(d),\ni-ii). In contrast, when the illumination switches to the polarization corresponding to\nthe REP, the PB phase exhibits inversion and doubling at 730 nm in our design. As the\nwavelength changes from 850 nm to 730 nm, the reconstructed hologram accordingly\nswitches from an inverted cat image (Fig. 6(d), iii) to the dog image (Fig. 6(d), iv). This\nspin-decoupling effect, arising from the chiral nature of the proposed TR-PB phase,\nenables the observed wavelength and polarization dependent image switching, thereby\ndemonstrating the information encryption capability.”"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have addressed all my comments and carefully revised their manuscript. I recommend this manuscript for publication in Nature Communications.", "reviewer_2": "After the author provide individual explanations and made detailed revisions, my problems have been resolved accordingly. The quality of the manuscript has improved significantly at present. I believe that fulfills Nature Communications' criteria for innovation, high impact, and scholarly rigor. Therefore, I suggest it for publication. As some optional suggestions, the authors could further strengthen the text by incorporating discussion of the following aspect:\n1. In the legend of Fig.5, please clarify which linearly polarized component's diffraction pattern is represented by Fig.5Ci and Fig.5Cii?\n2. A brief analysis can be provided in the manuscript on how to increase the cross-polarization conversion rate under the EP mechanism.\n3. It would be beneficial to elaborate further in the section “TR-PB phase used for information encryption” how this work achieves the spin-decoupling effect at the wavelength of 730 nm. Specifically, how is the phase of the corresponding pixel in Fig.6Div determined?"}, "author_response": "Dear reviewer:\nThank you for taking the time to review our manuscript entitled “Topologically\nreconstructing Pancharatnam-Berry phase via encircling exceptional point for chiral\nspin-orbit interaction steering” (Manuscript ID: NCOMMS-25-67039A).\nWe greatly appreciate your valuable comments and suggestions, which have been\ncarefully considered in our revisions. Below, we have provided a point-by-point\nresponse, along with the corresponding changes made in the revised manuscript and\nsupplementary materials for your reference. We believe that all the comments have\nbeen addressed, the revised manuscript meets the standard of Nature Communicaitons\nfor publication.\nReply to Reviewer 1:\nGeneral Comment: The authors have addressed all my comments and carefully\nrevised their manuscript. I recommend this manuscript for publication in Nature\nCommunications.\nResponses:\nWe thank the reviewer for the positive assessment of our work. We are pleased to\nconfirm that our detailed responses and revisions have duly addressed the previous\nconcerns, and we appreciate the acknowledgement that the manuscript now fulfills the\ncriteria for publication in Nature Communications.\nReply to Reviewer 2:\nGeneral Comment: After the author provide individual explanations and made\ndetailed revisions, my problems have been resolved accordingly. The quality of\nthe manuscript has improved significantly at present. I believe that fulfills\nNature Communications' criteria for innovation, high impact, and scholarly\nrigor. Therefore, I suggest it for publication. As some optional suggestions, the\nauthors could further strengthen the text by incorporating discussion of the\nfollowing aspect.\nResponses:\nWe thank the reviewer for the positive assessment of our work. We are pleased to\nconfirm that our detailed responses and revisions have addressed the previous\nconcerns, and we appreciate the acknowledgement that the manuscript now fulfills\nNature Communications' criteria. We have also carefully considered the optional\nsuggestions provided and incorporated relevant discussion to further strengthen the\nmanuscript.\nBelow is a more detailed reply and revision. The revised sentences are\nhighlighted in red and added to the revised manuscript, while the sentences in\nblack remain unchanged. Deleted sentences are marked in black with a\nstrikethrough.\nComments1: In the legend of Fig.5, please clarify which linearly polarized\ncomponent's diffraction pattern is represented by Fig.5Ci and Fig.5Cii?\nResponses:\nThank you for your comment. Fig. 5Ci and Fig. 5Cii both show the diffraction\npatterns under left-circularly polarized (LCP) incidence. After polarization conversion,\nthe right-circularly polarized (RCP) component carries the corresponding PB phase\nand thus exhibits leftward deflection. Therefore, the left-deflected beams in these\npanels correspond to the RCP component.\nChanges in the manuscript:\nWe add two sentences in the caption of Fig.5 to clarify the polarization of the\nincidence and the deflected light.\n“Fig. 5. The experimental results of chiral Spin-Orbit-interaction (SOI) inversion.\n(a and b) The interference patterns with OAM of 1and 2, under the excitation of light\nwith wavelength of 795nm and 715nm. (c) The measured diffraction patterns of\nphotonic spin Hall effect, under the excitation of light with wavelength of 800nm and\n720nm. Red and blue shaded backgrounds indicate left circularly polarized (LCP) and\nright circularly polarized (RCP) incidence, respectively. The deflected component\nwithin the white dashed circle corresponds to the cross polarized light. (d) Scanning\nelectron micrographs (SEMs) of fabricated device for demonstrating chiral SOI\ninversion, with the scale bars set to 1μm. ”\nComments2: A brief analysis can be provided in the manuscript on how to\nincrease the cross-polarization conversion rate under the EP mechanism.\nResponses:\nThank you for the reviewer for the insightful suggestion. Indeed, under the EP (exceptional\npoint) mechanism, the cross-polarization conversion rate (PCR) inevitably exhibits\ncertain local minimum during the rotation process. However, as the reviewer correctly\npoint out, the overall modulation depth of the PCR can be systematically enhanced\nthrough several approaches.\nFirst, material optimization plays a crucial role. By selecting materials with lower\noptical losses at the target wavelength, the intrinsic polarization conversion efficiency\ncan be improved. For instance, using low-loss dielectric materials or optimizing the\nthickness of metallic layers can further boost the PCR baseline.\nSecond, structural design refinement offers additional degrees of freedom. The\ngeometry of the meta-atoms can be tailored to maximize the polarization conversion\nat the desired operating wavelength or rotation angle. Finite-difference time-domain\n(FDTD) simulations can be employed to optimize the parameters, thereby enhancing\nthe PCR.\nThird, the manner in which the EP is introduced during rotation can be further\nengineered. Rather than relying on perturbation ellipses, one may design the\nmetasurface such that the EP is dynamically accessed only at specific rotation angles,\nwhile elsewhere the structure operates away from the EP to maintain high PCR. This\ncould be achieved by incorporating multiple resonant modes or by exploiting\ndispersion engineering to shift the EP wavelength as a function of rotation.\nBy combining these strategies it is possible to significantly elevate the overall PCR\nresponse of the proposed scheme. We have added a brief discussion of these aspects in\nthe revised manuscript to address the reviewer's valuable suggestion.\nChanges in the manuscript:\nWe add one sentences in the discussion to discuss the possible ways to enhance the\noverall PCR response.\n“In addition, while researches have focused on incorporating materials such as PCMs\nto introduce additional degrees of freedom for modulation, our method leverages\ntopological control with widely used PB phase to introduce TR-PB phase, thereby\nintroducing a new degree of freedom for metasurface design. Furthermore, the overall\nPCR response and modulation depth can be further enhanced through material\nselection, structural optimization, and the EP excitation method. For instance, this can\nbe achieved by employing low-loss dielectric materials, refining geometric parameters,\nand engineering the EP introduction mechanisms. Our work suggests that there are\nstill rich physics to be discovered for SOI process: When the topological protection of\nPB phase-based SOI is disrupted, the winding number exists the possibility for further\nevolution, as discussed in supplementary note 5. ”\nComments3: It would be beneficial to elaborate further in the section “TR-PB phase\nused for information encryption” how this work achieves the spin-decoupling effect at\nthe wavelength of 730 nm. Specifically, how is the phase of the corresponding pixel in\nFig.6Div determined?\nResponses:\nThank you for your comment. The spin-decoupling effect originates from the chiral\nnature of the proposed TR-PB phase. Specifically, under the EP response, taking\nright-handed EP as an example, the PB phase accumulation rule can be modified from\nthe conventional −2θ to either −4θ or +2θ under RCP incidence. In contrast, for LCP\nincidence, where no EP is excited, the phase accumulation remains the conventional\n+2θ rule.\nBased on this chiral disparity, we incorporate the distinct phase accumulation rules\ninto a modified Gerchberg-Saxton (GS) algorithm. The algorithm simultaneously\ngenerates two target near-field phase distributions, one corresponding to the decoy\nimage and the other to the hidden image, while enforcing that their relationship\nfollows the above TR-PB phase constraints under RCP and LCP respectively. Through\niterative optimization, the algorithm converges to a metasurface phase profile that\nsimultaneously satisfies the imaging requirements for both circular polarizations. The\nresulting phase distribution for each pixel in Fig. 6(D) iv is therefore determined by\nthis optimization process, ensuring that the spin-decoupled response is encoded into\nthe metasurface design.\nChanges in the manuscript:\nWe add further explanation in the last paragraph in the section “TR-PB phase\nused for information encryption” to elaborate the information encryption details.\n“As shown in Fig. 6(d), only with this precise key can the hidden image be retrieved.\nFor example, at 730 nm, where both phase inversion and doubling occur, RCP\nillumination switches the reconstructed hologram from the cat image to the dog image,\nwhereas illumination with opposite handedness merely yields an inverted cat image\n(Fig. 6(d), i-ii). the hidden image is retrieved only with the correct key. For instance,\nunder LCP illumination, the PB phase remains conventional (+2θ) across the entire\nspectrum, and the reconstructed hologram consistently displays the cat image (Fig. 6(d),\ni-ii). In contrast, when the illumination switches to the polarization corresponding to\nthe REP, the PB phase exhibits inversion and doubling at 730 nm in our design. As the\nwavelength changes from 850 nm to 730 nm, the reconstructed hologram accordingly\nswitches from an inverted cat image (Fig. 6(d), iii) to the dog image (Fig. 6(d), iv). This\nspin-decoupling effect, arising from the chiral nature of the proposed TR-PB phase,\nenables the observed wavelength and polarization dependent image switching, thereby\ndemonstrating the information encryption capability.”"}]

Decision letter