# High Fidelity Detection of the Orbital Angular Momentum of Light by Time
Mapping

Paul Bierdz, Minho Kwon, Connor Roncaioli, Hui Deng  Department of Physics,
University of Michigan, Ann Arbor, MI 48109, USA<EMAIL_ADDRESS>
###### Abstract

We demonstrate high-fidelity detection of the orbital angular momentum (OAM)
of light using a compact and practical OAM spectrometer that maps the OAM
spectrum to time. The spectrometer consists of a single optical delay loop to
achieve timing mapping, a vortex phase plate that iteratively decreases the
OAM value, and a single mode fibre to distinguish zero from non-zero OAM
states. Light with arbitrarily OAM compositions can be measured. For light
with OAM up to $4\hbar$, we measured an average crosstalk of -21.3 dB, which
is mainly limited by the purity of the input states and optical alignment.

##  1 Introduction

The orbital angular momentum (OAM) of light has been exploited for a wide
range of modern applications. With a discrete, yet unbounded Hilbert space
[1], OAM has been used to increase the capacity of free-space communications
[2, 3], generate high order entanglement [4, 5] and increase security of non-
classical communications [6]. The unique topological properties of OAM states
have been utilized to create optical tweezers that can apply varying degrees
of torque [7, 8], high-sensitivity vortex coronagraphs for extrasolar planet
detection[9] and revealing topological properties of objects[10]. A recent
paper suggests that light emitted nearby rotating black holes is twisted such
that the angular momentum of the black hole can be inferred from the OAM
spectrum [11].

For these applications, it is often crucial to discern different OAM states
with high fidelity. Several methods have been demonstrated for measuring the
OAM spectrum including those using fork diffraction gratings coupled with
single mode fibres (SMF) [4], cascading Mach-Zehnder interferometers [12] and
transformation optics [13]. However, all these methods map different OAM
states to spatially separated modes; as a result, complexity and size of the
experimental setups increase with the highest measurable OAM state. Fidelity
of the detection is often limited by the complexity of the experimental setup
and optical diffraction. There are other techniques that do not spatially
separate the modes. Courtial et. al. proposed a scheme based on OAM-dependent
rotational Doppler shift [14]; yet experimental implementation has not been
possible for optical wavelengths. Recently, we proposed a scheme based on OAM
to time mapping [15]. The scheme utilizes the concept of quantum
counterfactual measurement and quantum Zeno effect to allow non-destructive
sorting of the OAM spectrum with unit efficiency, if optical loss or
misalignment is neglected. Experimental implementation with lossy optics,
however, will significantly reduce the efficiency for high-order OAM state.

Here we demonstrate a simplified, and practical, OAM-to-time mapping scheme,
which achieved a record high extinction ratio among five OAM states at an
operation speed of 80 MHz. Since OAM is mapped to time, the same apparatus can
be used to measure an arbitrary number of unique OAM states. We note that a
similar scheme was also adopted in a recent experiment on time-division
multiplexing of OAM, although the fidelity and repetition rate were orders of
magnitude lower than reported here [16]. We also note that, without employing
a quantum Zeno investigator, time-division schemes are not suitable for
improving channel capacity in communication.

##  2 Principle of the OAM spectrometer

As shown in figure 1a, our OAM spectrometer consists of a single optical loop
to perform time mapping, a vortex phase plate (VPP) of topological charge 1 as
an OAM ladder operator, and an SMF as a filter for the fundamental Gaussian
mode with zero OAM. Consider an incident pulse consisting of a fraction
$\beta_{\ell}$ of OAM components with OAM value $\ell$, where $\ell$ is an
integer and $\sum_{\ell=-\infty}^{\infty}\beta_{\ell}=1$. The optical loop
converts the pulse into a sequence of pulses equally separated by the round
trip propagation time $T$ in the loop. The loop needs an even number of
reflections to maintain the same sign of OAM. Per loop, the VPP decreases the
OAM value of each OAM component by $1$. Hence, after $N$ loops, the fractional
$\beta_{\ell}$ of the original pulse will have OAM value $\ell-N$. Only the
component with zero OAM value in each pulse can pass through the SMF to be
detected. Thus, the OAM component $\beta_{\ell}$ with OAM value $\ell$ in the
original pulse will exit the spectrometer at a time $t=lT$ (figure 1b).

Figure 1: (a) Schematic of the OAM spectrometer, consisting of an optical loop
that converts an input pulse into a sequence of pulses equally spaced in time,
a VPP that decreases the OAM value by 1 per pass, and an SMF to filter out
states with non-zero OAM. (b) Showing how the OAM value changes in time for an
example input state with $\ell_{0}=3$.

###  2.1 Energy distribution

The distribution of the input-pulse energy among the sequence of pulses can be
pre-calibrated and controlled by the beam splitter with polarization optics.
We control the distribution with a polarizing beam-splitter (PBS) and half-
wave plate (labelled as R in figure 1a) in the loop. The incident light is
first set to be linearly polarized at an angle $\phi_{0}$ with respect to the
vertical. The PBS splits off the fraction of $\sin^{2}\\!\phi_{0}$ into the
first time window and sends the rest, now vertically polarized, into the loop.
The wave plate R, rotated at an angle $\theta/2$ with respect to the vertical,
rotates the vertically polarized light by an angle $\theta$. The corresponding
energy distribution is described below:

$\displaystyle P(l)=\cases{\sin^{2}\\!\phi_{0}&\mbox{if $\ell=0$,}\\\
\cos^{2}\\!\phi_{0}\cos^{2}\\!\theta&\mbox{if $\ell=1$,}\\\
\cos^{2}\\!\phi_{0}\sin^{4}\\!\theta\cos^{2(\ell-2)}\\!\theta&\mbox{if
$\ell\geq 2$.}}$ (1)

Here $\ell$ represents the $\ell$th output pulse. Alternatively, the PBS and
wave plate can be replaced by a non-polarizing beam-splitter with a chosen
splitting ratio for polarization-insensitive measurements. This would allow
for information to be encoded in the polarization degree of freedom.

##  3 Experimental implementation

We implemented an OAM spectrometer as illustrated in figure 1(a). To test its
performance, we used OAM eigenstates as input pulses and detected the output
pulses using a Hamamatsu streak camera.

The input pulse was generated by diffracting a pulsed Gaussian laser beam off
fork-diffraction patterns [17] on a Holoeye PLUTO LCoS spatial light modulator
(SLM) with a period of about 10 lines per millimeter and pixel pitch of
8$\mu$m. The initial laser beam was from a Tsunami Ti-Sapphire laser centered
at 730 nm, with a pulse width of 100 fs and repetition rate of 80 MHz, and
collimated with a 10x objective lens from an SMF ensuring an initial $M^{2}$
close to 1 and a spot size within the frame of the SLM.

We first verify the input states by measuring its far-field intensity and
phase distributions, as shown in figure 2. The intensity distributions were
measured with a charge-coupled device. The higher-order OAM beams show a
larger spatial size as expected. Inhomogeneity in the radial intensity
distribution reflects the lack of purity of the OAM value.

We measured the phase distribution by interfering the OAM beam with a
reference Gaussian beam from the original laser. This produces a forked
interference pattern. From this interference pattern we extract the phase
front, as plotted in figure 2(f-j), by performing a Fourier transform,
selecting the first diffracted mode, centring and then applying an inverse
Fourier transform. The phase should twist about by $2\pi\ell$ for an
$\ell^{\mbox{th}}$-order OAM beam. Deviation from this results implies impure
initial OAM states from optical aberrations or phase ripples from modulating
the liquid crystals.

To calibrate the time-mapping and energy distribution of the OAM spectrometer,
we used the zero-OAM Gaussian state as the input and replaced the VPP by a
flat glass plate of the same thickness. The output from the SMF was detected
by a Hamamatsu streak camera with a time resolution of 0.02 ns. As shown in
figure 1(b), the input pulse was mapped into a sequence of pulses separated by
$T=1.03$ ns. The timing and dynamic range of the streak camera allowed us to
measure the first five output pulses, or, the first five OAM states, at the
laser repetition rate of 80 MHz. The experiment can also be redesigned to
allow the measurement of a larger number of pulses by using different laser
repetition rates, loop sizes or measurement devices.

Figure 2: (a-e) The intensity measured with a camera of the different initial
OAM beams from the SLM with $\ell=0-4$. (f-j) The phase fronts of the input
OAM beams. They are calculated from the interference patterns between the OAM
beams and a reference Gaussian beam, as explained in the text.

The energy distribution among the pulses was calculated by integrating the
intensity over each output pulse. The results are shown in figure 3(a) (the
$\ell_{0}=0$ curve). We performed the same measurement with the input state
$\ell_{0}=1$. The ratio of the two energy normalization curves at each $\ell
T$ gives an estimate of the misalignment of the SMF-coupling (figure 3(b)) for
the $\ell^{\mbox{th}}$ output pulse, as will be discussed in detail in Section
4.2 later. The uniformity of ratio among different $\ell$ confirms that the
loop was well aligned.

Finally, we benchmark the performance of the OAM spectrometer by comparing the
correct versus incorrect detection of an input OAM eigenstate. Figure 4(a-e)
show examples of the output pulse sequences for input states $\ell_{0}=0-4$.
The integrated intensity under each peak versus the input state $\ell_{0}$ and
the output time bin $\ell$ is shown in figure 4(f), where each row has been
renormalized to the peak of the correct detection. We define the crosstalk to
be the ratios between the left and right adjacent incorrect detections versus
the correct detection, i.e., $\frac{P(\ell_{0}=\ell,N=\ell\pm
1)}{P(\ell_{0}=\ell,N=\ell)}$ where $P(\ell_{0},N)$ refers to the intensity
measured at time $NT$ with input OAM state $\ell_{0}$. For $\ell_{0}=0\mbox{
to }4$ we measured crosstalk values: $-12.3$, $-18.8$, $-20.0$, $-21.1$,
$-21.6$, $-23.2$, $-24.8$ and $-28.2$ dB. The geometric mean of the crosstalk
is $7.47\times 10^{-3}$ or $-21.3$ dB.

Figure 3: (a) Unnormalized power through the fibre versus the number of passes
through the loop without a VPP and different initial OAM values
($\ell_{0}=0,1$) set by the SLM. (b) Computed misalignment of SMF.

Figure 4: (a-e) Streak camera images for OAM eigenstates ($\ell=0,2,4$)
normalized based on (1). (f) Tabulated results of the OAM spectrometer up to
$\ell=4$.

##  4 Analysis of the spectrometer performance

Below we simulate the performance of the spectrometer and analyze the main
sources of error. We model the laser pulses as a superposition of Laguerre-
Gaussian (LG) modes, which are a complete orthonormal set of solutions to the
paraxial wave equation. Each optical element operating on the pulse maps each
LG mode into a different superposition of other LG modes. In particular, we
focus on SMF, VPP and SLM. The remaining optics: mirrors, beam splitters,
waveplates, etc. are insensitive to the transverse mode of the beam and thus
preserve the LG mode. We then simulate the propagation of an input pulse
through the spectrometer, and the coefficients of superposition obtained after
the SMF corresponds to the measurement results.

###  4.1 Laguerre-Gaussian modes

The electric fields of LG modes can be described in cylindrical coordinates
$(\rho,\phi,z)$ as [18]:

$\displaystyle
u_{\ell,p}(\rho,\phi,z)=\langle\rho,\phi,z|\ell,p\rangle=\frac{1}{w(z)}\sqrt{\frac{2}{\pi}\frac{p!}{(p+|\ell|)!}}\left(\frac{\sqrt{2}\rho}{w(z)}\right)^{|\ell|}\exp\left({-\frac{\rho^{2}}{w^{2}(z)}}\right)L_{p}^{|\ell|}\left(\frac{2\rho^{2}}{w^{2}(z)}\right)\rme^{\rmi\ell\phi}$
$\displaystyle\exp\left(\frac{\rmi k_{0}\rho^{2}z}{2\left(z^{2}+z_{\rm
R}^{2}\right)}\right)\exp\left({-\rmi(2p+|\ell|+1)\tan^{-1}\left(\frac{z}{z_{\rm
R}}\right)}\right)$ (2)

Here $\ell\hbar$ ($\ell\in\mathbb{Z}$) is the OAM per photon, $p\geq 0$ labels
the radial modes, $w(z)$ is the beam waist, $z_{\rm R}$ is Rayleigh range, and
$k_{0}$ is the wave number of the fundamental Gaussian mode.

###  4.2 Single mode fibre

Ideally, an SMF selects only the fundamental Gaussian mode ($\ell=0,p=0$)
while all the other orthogonal spatial modes do not propagate through. In
reality, higher order LG modes may still couple through the fibre due to
finite apertures of optics and the fibre, imperfections of the fibre,
mismatched beam waists between free-space and fibre-modes, and transverse
misalignment between the propagation axes of the free-space and fibre-modes.
In our experiments, the last effect, the transverse misalignment, is by far
the dominant, while the other effects were too small to be measured. Hence we
restrict our discussion to the transverse misalignment only. For a
misalignment of $\Delta$ (in units of beam waist) between the two optical
axes, the coupling efficiency of the LG beam $(\ell,p)$ through the SMF is
given by (A.2):

$|{}_{\Delta}\langle
0,0|\ell,p\rangle|^{2}=\frac{1}{p!(p+|\ell|)!}\left(\frac{\Delta^{2}}{2}\right)^{2p+|\ell|}\rme^{-\Delta^{2}}$
(3)

Experimentally, the above misalignment of the SMF can be estimated using the
data for calibrating the energy distribution, as shown in figure 3(a). Nonzero
output intensity from an input pulse of $\ell_{0}=1$ modes implies a
misaligned (or imperfect) SMF. The amount of misalignment $\Delta$ can be
calculated from the ratio of the two output intensities corresponding to the
two input modes $\ell_{0}=0$ versus $\ell_{0}=1$ via:

$\frac{P(\ell_{0}=1)}{P(\ell_{0}=0)}=\frac{|{}_{\Delta}\langle
0,0|M_{\mbox{VPP1}}|0,0\rangle|^{2}}{|{}_{\Delta}\langle 0,0|0,0\rangle|^{2}}$
(4) $|{}_{\Delta}\langle
0,0|M_{\mbox{VPP1}}|0,0\rangle|^{2}=\frac{\pi}{8}\Delta^{2}\rme^{-\frac{3}{2}\Delta^{2}}\left(I_{0}\left(\frac{\Delta^{2}}{4}\right)+I_{1}\left(\frac{\Delta^{2}}{4}\right)\right)^{2}$
(5)

The left hand side of (4) is measured experimentally. $M_{\mbox{VPP1}}$
represents the operation by the SLM to create the input beam with $\ell_{0}=1$
and in a superposition of different $p$-states [19]. Equation (5) is obtained
using (3) and (6) (See A.3 for details). We plotted the derived misalignments
in figure 3(b). The misalignment is less than 10.1% for all loops and
converges to 3.7% for higher loop numbers, indicating only a minor
accumulative loop misalignment.

###  4.3 VPP/SLM

The VPP and SLM are used to change the OAM state. The SLM, when applying the
forked phase pattern, is mathematically equivalent to VPP for small
diffraction angles [20, 21]. For a VPP of topological charge $\beta$, its
operation on the laser beam can be described by a four-dimensional tensor
$M_{VPP\beta}(z)$. We solve for the tensor elements
$m_{\ell_{1},p_{1};\ell_{2},p_{2};\beta}(z)$ analytically (A.1) in the LG-mode
basis:

$\displaystyle
m_{\ell_{1},p_{1};\ell_{2},p_{2};\beta}(z)=\left[\sqrt{\frac{p_{1}!p_{2}!}{(p_{1}+|\ell_{1}|)!(p_{2}+|\ell_{2}|)!}}\right.\sum_{k=0}^{p_{1}}\sum_{m=0}^{p_{2}}(-1)^{k+m}\frac{(p_{1}+|\ell_{1}|)!}{(p_{1}-k)!(|\ell_{1}|+k)!k!}$
$\displaystyle\frac{(p_{2}+|\ell_{2}|)!}{(p_{2}-m)!(|\ell_{2}|+m)!m!}\left.\Gamma\left(\frac{|\ell_{1}|+|\ell_{2}|}{2}+k+m+1\right)\right]$
$\displaystyle\left[\exp\left(\rmi(2(p_{1}-p_{2})+|\ell_{1}|-|\ell_{2}|)\tan^{-1}\left(\frac{z}{z_{\rm
R}}\right)\right)\right]$
$\displaystyle\left[\frac{\exp[2\pi\rmi(\ell_{2}+\beta-\ell_{1})]-1}{2\pi\rmi(\ell_{2}+\beta-\ell_{1})}\right]$
(6)

This equation consists of a product of three terms denoted by square brackets.
The first term with the double sum comes from the amplitude overlap between
the different LG modes. The second term, consisting of an exponential,
describes the effects of Gouy phases. The third term comes from OAM
conservation and can describe the error in the topological charge of the VPP.
We discuss the effect of each term below.

####  4.3.1 Gouy phase effects

Different LG modes have different Gouy phases which also vary differently as
the beam propagates; therefore, whenever there is a superposition of modes,
there can be interference effects between the modes that varies under free
space propagation [22] and can lead to the loss of efficiency. However, the
effect does not produce crosstalk between different $\ell$-states, as OAM
remain unchanged. In our experiment, the effect of the Gouy phase accounts for
less than $0.05\%$ loss in efficiency. This upper bound of $0.05\%$ was
calculated based on distances between the optics in the experimental setup and
using (6). In general, the effect is negligible when the propagation distance
is much smaller than the Rayleigh range ($z\ll z_{\rm R}$). When the size of
the OAM spectrometer becomes comparable to the Rayleigh range, the use of 4-f
systems between all phase elements would eliminate any Gouy phase effects.

####  4.3.2 Topological charge error

If the topological charge of the VPP is an integer, then the OAM of the beam
is changed by that amount. If the laser’s wavelength is different from the
nominal wavelength of the VPP, or if the beam comes at a skew angle to the
VPP, then the VPP will appear thicker or thinner. As a result, the OAM of the
beam will be changed by a fractional value instead, or equivalently, become a
superposition of LG modes of many OAM values. This leads to a loss in both the
efficiency and fidelity of the OAM spectrometer.

Such topological charge error can be modelled by non-integer $\beta$ in (6).
In our experiment, both the laser wavelength and the angle of incidence are
tightly controlled. Even if we consider a rather generous 0.5% error in the
topological charge, it results in less than 0.05% loss of efficiency and less
than -34dB crosstalk. Therefore the error in the topological charge is
negligible.

####  4.3.3 Lateral misalignment

Although topological charge error will reduce the fidelity, it is, by far, not
the leading cause. If the optical axes of the beam and VPP are displaced
relative to each other, the VPP produces a superposition of not just $p$, but
$\ell$ states [23] and thus reduces the fidelity. Such lateral misalignment of
the VPP affects all three terms in (6). To model it, we first express the VPP
tensor $M_{\beta}=\rme^{\rmi\beta\phi}$ in a coordinate displaced from the
common optical axis of the spectrometer. We then numerically evaluate
$m_{\ell_{1},p_{1};\ell_{2},p_{2};\beta}(z)$ in a subspace of
${\ell,p}=[-7,7]\times[0,10]$. This chosen subspace yields less than 2% error
on theoretical crosstalk.

To compare with the experimental result, we calculated the average crosstalk
versus VPP misalignment, assuming a transverse misalignment of the SMF of 3.7%
to 10.1% as obtained in 4.2. The results are shown in 5(a), where the
x-direction corresponds to the direction of SMF misalignment. The measured
crosstalk of $-21.3$ dB (3) corresponds to a VPP misalignment of 4.0% to 6.1%
of the beam waist $w_{0}$. A larger (smaller) VPP misalignment is allowed when
its direction is aligned with (perpendicular to) that of the SMF misalignment.

###  4.4 Limiting factors of fidelity

In short, we show that misalignments of 3.7% to 10.1% at the SMF and 4.0% to
6.1% at the VPP are the leading causes for the loss of fidelity in our
experiment, with an average crosstalk of -21.3 dB. Gouy phase effects do not
affect fidelity and topological charge error contributes less than -34 dB to
crosstalk.

The fidelity of the spectrometer can be further increased by improving the
optical alignment at the VPP and SMF. We show in figure 5(b) the average
crosstalk as a function of the VPP misalignment alone, neglecting SMF
misalignment, and as a function of the SMF misalignment alone, neglecting VPP
misalignment. It is more sensitive to VPP misalignment than to SMF
misalignment, but decreases superlinearly with the reduction of either
misalignment. With a VPP (SMF) misalignment of $<1\%$, the crosstalk is
reduced to $<-34.4$ dB ($<-43.4$ dB). When including both SMF and VPP
misalignments, the crosstalk varies depending on the relative angle between
the directions of the two misalignments. The crosstalk is the greatest when
the misalignments are in perpendicular directions from each other. We show
this upper bound crosstalk in 5(c). When the misalignments are in opposite
directions, we generally see a reduction in crosstalk, sometimes up to 20 dB.
Curiously, if the SMF is misaligned, crosstalk can be _decreased_ by
introducing some VPP misalignment, and vice versa. This is because with
multiple elements misaligned, they can compensate each other. However, zero
crosstalk is reached only when both misalignments are exactly zero.

Figure 5: Calculated the geometric mean of crosstalk (in dB) as a function of
the lateral misalignment of the SMF and VPP (normalized by $w_{0}$). (a)
Crosstalk (in dB) as a function of the lateral misalignment of the VPP. SMF
misalignment (measured in 3(c)) is taken along the x direction for
convenience, since only the relative angle between VPP and SMF misalignment is
important. The measured crosstalk of $-20.9$ to $-21.6$ dB corresponds to the
values between the two black lines. (b) The crosstalk versus VPP misalignment
with no SMF misalignment (green solid line), and the crosstalk versus SMF
misalignment with no VPP misalignment (blue dashed line). (c) The upper bound
of crosstalk (in dB) as a function of the SMF and VPP misalignments. The upper
bound corresponds to when the directions of the SMF and VPP misalignments are
perpendicular to each other. The crosstalk can be significantly reduced if the
directions of the SMF and VPP lateral displacements are anti-parallel to each
other.

As misalignment is reduced, other subtle effects will need to be taken under
consideration. Such as imperfections in the surface roughness or other
aberrations in the optics, which could further reduce efficiency and fidelity.
Eventually, the fidelity will be limited by overlap in time between adjacent
pulses. We did not include these effects in our model.

##  5 Conclusions

In summary, we present a practical OAM spectrometer that can map OAM to time
up to arbitrarily large OAM values. We have shown an average nearest neighbor
crosstalk of -21.3dB among 5 OAM states, limited mainly by optical
misalignment. The high fidelity of the demonstrated OAM spectrometer may
enable accurate measurements of topological properties of objects such as in
spiral imaging [10] and the angular momentum of black holes, which is encoded
into the OAM spectrum of light from the accretion disc due to strong
gravitational effects predicted by general relativity [11]. We also
demonstrated speeds (80 MHz) orders of magnitude faster than the switching
times of SLMs. Miniaturization of optics could allow for GHz detection rate.

## Acknowledgments

We would like to thank Professor Miles Padgett for his helpful advice. And
also thank Professor Theo Lasser and Matthias Geissbühler for their
_Morgenstemning_ colormap [24].

## Appendix A

In this appendix section we seek to derive the various actions and overlaps
between the LG modes and optics. We first derive the action of the VPP in the
LG basis. Then we derive the overlap of any LG mode with a misaligned SMF.
Lastly, we derive the overlap of OAM states created by a VPP with a misaligned
SMF. This is done by explicit integration and definitions of various special
functions. We use the inner product between Laguerre-Gaussian modes defined in
2 in cylindrical coordinates:

$\displaystyle\langle\ell_{1},p_{1}|\ell_{2},p_{2}\rangle=\int_{0}^{\infty}\rho\rmd\rho\int_{0}^{2\pi}\rmd\phi\langle\ell_{1},p_{1}|\rho,\phi,z\rangle\langle\rho,\phi,z|\ell_{2},p_{2}\rangle$
$\displaystyle\times\int_{0}^{\infty}\rho\rmd\rho\int_{0}^{2\pi}\rmd\phi
u_{\ell_{1},p_{1}}^{*}(\rho,\phi,z)u_{\ell_{2},p_{2}}(\rho,\phi,z)$ (7)

###  A.1 Derivation of VPP tensor

We derive the four dimensional tensor of the action of a VPP in the LG basis
as seen in (6). We start by explicitly writing down the overlap integral:

$\langle\ell_{1},p_{1}|M_{\mbox{VPP}\beta}|\ell_{2},p_{2}\rangle=\int_{0}^{\infty}\rho\rmd\rho\int_{0}^{2\pi}\rmd\phi
LG^{*}_{\ell_{1},p_{1}}(\rho,\phi)\rme^{+\rmi\beta\phi}LG_{\ell_{2},p_{2}}(\rho,\phi)$
(8)

The LG functions are given by (2). Due to separability of variables, we will
solve the $\phi$-integral first:

$\int_{0}^{2\pi}\rmd\phi\rme^{-\rmi\ell_{1}\phi}\rme^{+\rmi\beta\phi}\rme^{+\rmi\ell_{2}\phi}=\frac{\exp\left(2\pi\rmi\left(\ell_{2}+\beta-\ell_{1}\right)\right)-1}{\rmi\left(\ell_{2}+\beta-\ell_{1}\right)}$
(9)

In the limit where $\beta$ is an integer, the $\phi$-integral yields the OAM
conserving solution of $2\pi\delta_{\ell_{1}+\beta-\ell_{2},0}$, where
$\delta_{a,b}$ is the Kronecker delta. The remaining $\rho$-integral is found
by expanding the rest of the LG modes and the generalized Laguerre polynomials
given by (10). This yields a finite polynomial of degree
$|\ell_{1}|+|\ell_{2}|+2p_{1}+2p_{2}+1$ (the +1 is from the Jacobian)
multiplied by the Gaussian function
$\exp\left(\frac{-\rho^{2}}{2w_{0}^{2}}\right)$. This integral can be easily
solved by using one of the definitions of the Gamma function as shown in (11)
to yield the result in the text (6).

$L_{p}^{k}(x)=\sum_{m=0}^{p}(-1)^{m}\frac{(p+k)!}{(p-m)!(k+m)!m!}x^{m}$ (10)

$\int_{0}^{\infty}\rmd
xx^{k}\rme^{-x^{2}}=\frac{1}{2}\Gamma\left(\frac{1+k}{2}\right)$ (11)

###  A.2 Overlap between fundamental Gaussian and Laguerre-Gaussian

We first derive the overlap between a misaligned fundamental Gaussian mode and
generic higher order LG modes. This is (3) in the text. We start with the
fundamental Gaussian mode defined in cylindrical coordinates ($\rho,\phi,z$)
with an origin shifted by $w_{0}\Delta$ along the $\phi=0$ direction. The
propagation direction is along the z-axis.

${}_{\Delta}\langle
0,0|\rho,\phi,z\rangle=\frac{1}{w_{0}}\sqrt{\frac{2}{\pi}}\exp\left({\frac{-\left(\rho^{2}+2\rho
w_{0}\Delta\cos\phi+w_{0}^{2}\Delta^{2}\right)}{w_{0}^{2}}}\right)$ (12)

If the origin were shifted along a different $\phi$ angle, the only change
would be the overall phase which does not affect the intensity. These
calculations are performed at the beam waist since the beam should be focused
on the fibre to have any substantial coupling. With a change of variables
$x=\frac{\sqrt{2}\rho}{w_{0}}$ the complete overlap integral is now:

$\displaystyle{}_{\Delta}\langle
0,0|\ell,p\rangle=\int_{0}^{\infty}\int_{0}^{2\pi}x\rmd
x\rmd\phi\frac{1}{\pi}\sqrt{\frac{p!}{(p+|\ell|)!}}x^{|\ell|}\exp\left(\frac{-x^{2}}{2}\right)L_{p}^{|\ell|}\left(x^{2}\right)\rme^{\rmi\ell\phi}$
$\displaystyle\exp\left(-\left(\frac{x^{2}}{2}+\sqrt{2}x\Delta\cos\phi+\Delta^{2}\right)\right)$
(13)

The $\phi$-integral is related to the well known Bessel integral (14), so
(A.2) becomes (15).

$J_{n}(z)=\frac{1}{2\pi\rmi^{n}}\int_{0}^{2\pi}\rmd\phi\rme^{\rmi
z\cos\phi}\rme^{\rmi n\phi}$ (14)

${}_{\Delta}\langle
0,0|\ell,p\rangle=\sqrt{\frac{p!}{(p+|\ell|)!}}\int_{0}^{\infty}x\rmd
xx^{|\ell|}\rme^{-\left(x^{2}+\Delta^{2}\right)}L_{p}^{|\ell|}\left(x^{2}\right)2\rmi^{-|\ell|}(-1)^{|\ell|}J_{|\ell|}\left(\rmi\sqrt{2}\Delta
x\right)$ (15)

The Bessel function can be converted into an infinite sum of generalized
Laguerre polynomials as given in (16) below with
$\sqrt{w}=\rmi\Delta/\sqrt{2}$. Using (16) and the orthogonality relationship
between generalized Laguerre polynomials (17). Equation (15) magnitude squared
surprisingly reduces to the rather simple expression of (3) in the text.

$\displaystyle\sum_{n=0}^{\infty}\frac{L_{n}^{k}(x)}{\Gamma(n+k+1)}w^{n}=e^{w}(xw)^{\frac{-k}{2}}J_{k}\left(2\sqrt{xw}\right)$
(16)

$\displaystyle\int_{0}^{\infty}\rmd
x\rme^{-x}L_{n}^{k}(x)L_{m}^{k}(x)=\frac{(n+k)!}{n!}\delta_{m,n}$ (17)

###  A.3 Overlap between fundamental Gaussian and vortex phase plate mode

We create states with OAM by passing a Gaussian beam through a VPP. This state
can be expressed as a sum of Laguerre Gaussian modes as given in (18), which
is a special case of the more general equation solved in section A.1. In the
previous section, we calculated the overlap of a shifted fundamental Gaussian
mode and any Laguerre Gaussian mode. Therefore, by combining these two
calculations, we can derive the overlap of a shifted Gaussian mode and an OAM
mode created by a VPP (19), here $z=\frac{\Delta^{2}}{2}$.

$\displaystyle
M_{\mbox{VPPN}}|0,0\rangle=\sum_{p=0}^{\infty}m_{0,0;N,p;N}|N,p\rangle$ (18)
$\displaystyle\mbox{where
}m_{0,0;N,p;N}=\frac{N}{2}\sqrt{\frac{1}{(p+N)!}}\Gamma\left(p+\frac{N}{2}\right)$

$\displaystyle{}_{\Delta}\langle
0,0|M_{\mbox{VPPN}}|0,0\rangle=\frac{e^{-z}z^{\frac{N}{2}}(-1)^{N}N}{2}\sum_{p=0}^{\infty}\frac{(-z)^{p}\Gamma\left(p+\frac{N}{2}\right)}{(p+N)!p!}$
(19)

Equation (19) can be written in the form of a regularized confluent
hypergeometric function (20) which can be evaluated using
(21)111http://functions.wolfram.com/07.21.03.0011.01 to yield the result in
(22), where $I_{\alpha}(x)$ is the modified Bessel function of the first kind.
The expression can be further simplified by noting
$I_{\alpha}(-z)=\rmi^{2\alpha}I_{\alpha}(z)$. In the case of $N=1$ and taking
the magnitude squared, this reduces to the form in the text (5).

${}_{1}\tilde{F}_{1}(a;b;z)=\sum_{k=0}^{\infty}\frac{\Gamma(a+k)}{\Gamma(b+k)\Gamma(a)}\frac{z^{k}}{k!}$
(20)

${}_{1}\tilde{F}_{1}\left(\frac{N}{2};N+1;z\right)=\frac{e^{\frac{z}{2}}\sqrt{\pi}}{2\Gamma\left(\frac{N}{2}+1\right)}z^{\frac{1}{2}-\frac{N}{2}}\left(I_{\frac{N}{2}-\frac{1}{2}}\left(\frac{z}{2}\right)-I_{\frac{N}{2}+\frac{1}{2}}\left(\frac{z}{2}\right)\right)$
(21)

$\frac{\rme^{\frac{-3\Delta^{2}}{4}}\Delta^{2}\sqrt{\pi}}{2\sqrt{2}}\left(I_{\frac{N}{2}-\frac{1}{2}}\left(\frac{\Delta^{2}}{4}\right)+I_{\frac{N}{2}+\frac{1}{2}}\left(\frac{\Delta^{2}}{4}\right)\right)$
(22)

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# Topologically controlled multiskyrmions in photonic gradient-index lenses

Yijie Shen<EMAIL_ADDRESS>Optoelectronics Research Centre, University of
Southampton, Southampton SO17 1BJ, United Kingdom  Chao He
<EMAIL_ADDRESS>Department of Engineering Science, University of Oxford,
Parks Road, Oxford, OX1 3PJ, United Kingdom  Zipei Song  Department of
Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, United
Kingdom  Binguo Chen  Guangdong Engineering Center of Polarization Imaging and
Sensing Technology, Tsinghua Shenzhen International Graduate School, Tsinghua
University, Shenzhen 518055, China  Honghui He  Guangdong Engineering Center
of Polarization Imaging and Sensing Technology, Tsinghua Shenzhen
International Graduate School, Tsinghua University, Shenzhen 518055, China
Yifei Ma  Department of Engineering Science, University of Oxford, Parks Road,
Oxford, OX1 3PJ, United Kingdom  Julian A.J. Fells  Department of Engineering
Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, United Kingdom
Steve J. Elston  Department of Engineering Science, University of Oxford,
Parks Road, Oxford, OX1 3PJ, United Kingdom  Stephen M. Morris  Department of
Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, United
Kingdom  Martin J. Booth  Department of Engineering Science, University of
Oxford, Parks Road, Oxford, OX1 3PJ, United Kingdom  Andrew Forbes  School of
Physics, University of the Witwatersrand, Private Bag 3, Wits 2050, South
Africa

###### Abstract

Skyrmions are topologically protected quasiparticles, originally studied in
condensed-matter systems and recently in photonics, with great potential in
ultra-high-capacity information storage. Despite the recent attention, most
optical solutions require complex and expensive systems yet produce limited
topologies. Here we demonstrate an extended family of quasiparticles beyond
normal skyrmions, which are controlled in confined photonic gradient-index
media, extending to higher-order members such as multiskyrmions and
multimerons, with increasingly complex topologies. We introduce new
topological numbers to describe these complex photonic quasiparticles and
propose how this new zoology of particles could be used in future high-
capacity information transfer. Our compact creation system lends integrated
and programmable solutions of complex particle textures, with potential
impacts on both photonic and condensed-matter systems for revolutionizing
topological informatics and logic devices.

In the current age of information explosion, the pursuit of next-generation
information carriers and data storage is endless, eminently benefiting our
daily lives. Since Tony Skyrme proposed a model to unify a large class of
fundamental particles by methods of topology in the 1960s skyrme1962unified ,
known today as skyrmions, these have emerged as one of the most potential
information carriers due to their topologically robust spin textures localized
in ultra-small regions, in particular, which has revolutionized the high-
density data storage technologies in magnetic materials in recent years
bogdanov2020physical ; fert2017magnetic ; bernevig2022progress ; han2022high .
In order to further expand the capacity in informatic applications, it is
highly desired to manipulate complex quasiparticles with higher-order
topological textures in addition to the fundamental skyrmions gobel2021beyond
. For instance, the meron textures with fractional skyrmion numbers can be
controlled in chiral magnets yu2018transformation , and recently in
antiferromagnets at room temperature jani2021antiferromagnetic . Novel
quasiparticles such as skyrmion bags foster2019two , skyrmion bundles
tang2021magnetic , and skyrmion braids zheng2021magnetic  with large range
control of skyrmion numbers were created in magnetic materials. Moreover,
exotic spatially knotted quasiparticle of hopfions ackerman2017static ;
liu2020three ; liu2022emergent , torons ackerman2017diversity ;
poy2022interaction , and heliknotons tai2019three , characterized by complex
topologies were designed in magnets, colloids, and chiral liquid crystals. The
realization of novel quasiparticles in diversified condensed-matter systems
has triggered the development of static or quasi-static ultra-density data
storage techniques. However, using topological quasiparticles for long-range
secure information transfer remains a considerable challenge.

Optical or photonic skyrmions, which have recently been realized, provide new
degrees of freedom in terms of the construction of topological quasiparticles
shen2022topological . Following the initial demonstrations of the formation of
photonic skyrmions in surface plasma by evanescent electromagnetic fields
tsesses2018optical ; davis2020ultrafast , as well as optical spins du2019deep
; dai2020plasmonic , photonic skyrmions have also been generated in optical
polarization and Stokes fields in free space shen2021generation ;
sugic2021particle ; shen2023topological , electromagnetic fields in space-time
shen2021supertoroidal ; zdagkas2021observation , pseudospins in nonlinear
crystals karnieli2021emulating , and so on. However, previous studies have
demonstrated a very limited number of topological states, often requiring the
use of complex and expensive systems. Moreover, practical information transfer
protocols based upon photonic skyrmions are still to be explored and highly
desired.

Figure 1: GRIN lens arrays for photonic skyrmion generation. a, Schematic of
the GRIN lens and GRIN lens array, with radial and azimuthal distributions of
the fast/slow axes. The radius of the cross-section of the lens can be less
than 500 microns. b,c, The diversified polarization vectorial light fields
corresponding to the different complex photonic quasiparticles generated can
be controlled by cascades of GRIN lenses, represented by (b) polarization
ellipse distributions and (c) Stokes vector distributions, which include
(b1,c1) a skyrmion of $N_{s}=2$, (b1,c1) a skyrmion of $N_{s}=0$, (b2,c2) a
skyrmionium of $N_{s}=0$, (b3,c3) a quadruskyrmion of $N_{s}=4$ comprising
four elementary skyrmions of $N_{s}=1$, and (b2,c2) a quadrumeron of $N_{s}=2$
comprising four elementary merons of $N_{s}=1/2$. d, Conceptual schematic of
the vector texture of a second-order skyrmionium ($N_{r}=2$, $N_{s}=0$), which
includes skyrmion ($N_{r}=1$, $N_{s}=2$) and meron structure ($N_{r}=1/2$,
$N_{s}=1$) in its subspace. e, The topological protection of the photonic
skyrmions, where (e1) a Néel-type skyrmion can be transformed into (e2) a
Bloch-type skyrmion upon propagation and resilient to perturbation, under the
same skyrmion number of $N_{s}=1$. It is, however, impossible to be
transformed into (e3) an anti-skyrmion of different skyrmion number of
$N_{s}=-1$.

Here, we demonstrate both experimentally and theoretically an extended class
of photonic quasiparticles constructed by vectorial structured light in
gradient-index (GRIN) lenses he2019complex ; he2022revealing . The
quasiparticles that are formed using vectorial-structured light exhibit novel
forms of higher-order configurations beyond elementary skyrmions with
increasingly complex topology and geometry. These complex quasiparticles can
be characterized by multiple topological numbers and their diversified
topological states can be controlled by cascaded GRIN lenses he2019complex .
To illustrate the benefit of this zoology of topological states, we propose a
protocol for high-capacity secure encryption, where the multiple topological
indices of diversified quasiparticles in a compact GRIN lens array are used to
encode information that are robust to environmental perturbations, leading to
an alphabet-bit encoded quasiparticle array for high-security optical
encryption. In addition, the information capacity in this scheme can be
flexibly spanned by arranging the GRIN-lens or quasiparticle array. This work
paves the way to the exploitation of topological photonic quasiparticles for
next generation information technology, while our novel compact configuration
offers the potential to be integrated with on-chip solid state storage
devices.

##  I Results

Vectorial optics of GRIN lenses. GRIN lenses are a kind of optical element
that possesses spatially varying refractive index profile as well as
birefringence, which enable focusing, imaging, and shaping of light through a
rod-like structure barretto2009vivo ; kim2012fabrication ; he2019complex ;
forbes2019common , as shown in Fig. 1. GRIN lenses can be fabricated with
diameters on the sub-millimeter scale, which means that GRIN lens arrays can
be formed with dimensions of the order of millimeters (left-bottom insert of
Fig. 1). Each GRIN lens can be comprised by different cascaded segments with
different refractive index gradient designs, together with segments of quarter
wave plates (QWP) and half wave plates (HWP). If we input a plane-wave into
one side of a cascaded GRIN lens, the output will be light with spatially
varying polarization patterns he2019complex ; forbes2019common . Here we will
show that through judicious assembly of the cascades of GRIN lenses combined
with tuning the polarization of the input light, we can obtain a wide variety
of high-order photonic skyrmions with complex topologies.

Topological photonic quasiparticles. A skyrmion refers to a three-component
real vector or spin texture mapped from a 2-sphere to a localized 2D real
space, donated as $\mathbf{s}(x,y)=[s_{x}(x,y),s_{y}(x,y),s_{z}(x,y)]$, with
basic topology characterized by the skyrmion number $N_{s}$ (which counts how
many times the spins can wrap around a 2-sphere with full azimuth):

$N_{s}=\frac{1}{4\pi}\iint_{\sigma}{\mathbf{s}\cdot\left(\frac{\partial\mathbf{s}}{\partial
x}\times\frac{\partial\mathbf{s}}{\partial y}\right)}\text{d}x\text{d}y$ (1)

where $\sigma$ represents the particle confined region. The photonic skyrmions
or quasiparticles can be constructed by the polarization Stokes vectors, i.e.
the vector defined by the three Stokes parameters, of the obtained structured
light, analogous to the magnetic spin used to construct magnetic skyrmions
shen2022topological ; shen2021topological ; gao2020paraxial . Figure 1b1 shows
a theoretical result of polarization ellipse distribution of a skyrmionic
light field generated by a customized GRIN lens, the stokes vector field of
which shows an fundamental skyrmion of $N_{s}=2$ (Fig. 1c1). Note that, for
the fundamental skyrmion, the signal of skyrmion number was decided by its
central spin of spin-up ($+$) or spin-down state ($-$), which is defined as
the polarity, $N_{p}=\text{sgn}(N_{s})$, and the absolute value of skyrmion
number determines the vortex charge of transverse $(x,y)$ component
distribution, defined as the vorticity, $N_{v}=|N_{s}|$. The result of Fig.
1c1 shows and central spin of spin-up and vortex charge of transverse
distribution of 2, thus resulting into $N_{s}=2$.

In addition to the fundamental skyrmion, cascaded GRIN lens can also generate
generalized quasiparticle, for instance, the skyrmionium zhang2016control , a
skyrmion radially nested with another one with opposite topological number,
resulting into skyrmion number of $N_{s}=(+2)+(-2)=0$ (Figs. 1b2 and 1c2). If
the radially nested number, $N_{r}$, is larger than two, the extended
configurations are previously called $N_{r}\pi$-skyrmions or target skyrmions
song2019field ; zheng2017direct , which, we argue, can also be easily realized
in our cascaded GRIN lens system. The versatility of GRIN lenses that are
cascaded to form arrays allows us to access novel forms of photonic
quasiparticles that are challenging to observe in other systems, for instance,
the exotic multiskyrmions and multimerons, comprising multiple elementary
skyrmions and merons in subspace with high symmetry. Figures 1b3 and 1c3
demonstrate a quadruskyrmion with four elementary skyrmions of unit skyrmion
number, resulting into total skyrmion of $N_{s}=4$. Figures 1b4 and 1c4, on
the other hand, demonstrate a quadrumeron with four elementary merons of half
skyrmion number, resulting into total skyrmion of $N_{s}=2$. To describe the
symmetry in multiskyrmions and multimerons, we define a number of centrality,
$N_{c}$, which counts how many spin-up or -down center points there are. The
quadruskyrmion and quadrumeron are both of centrality of $N_{c}=4$. On the
other hand, the skyrmions, skyrmioniums, and target skyrmions are always of
centrality of $N_{c}=1$. Figure d conceptually demonstrate the elementary
relationship among a skyrmionium, a skyrmion, and a meron.

The photonic skyrmions possess the property of topological protection, meaning
that the skyrmion number remains invariant upon propagation evolution and is
robust to environmental perturbations nape2022revealing . Once a skyrmion is
generated by a GRIN lens, its topology will be protected upon further
propagation in isotropic media or coupled in free space shen2021topological ;
gao2021erratum . For instance, an Néel-type skyrmion with hedgehog-like
texture (Fig. 1e1) can evolve into a Bloch-type skyrmion with vortex-like
texture (Fig. 1e2), provided they are of a same skyrmion number $N_{s}=1$,
however, it is impossible for such a skyrmion to be transformed into an
antiskyrmion of opposite skyrmion number $N_{s}=-1$ (Fig. 1e3). The mechanism
of topological protection works for arbitrary extended complex quasiparticles.

Figure 2: Generation and topological transformation of a wide variety of
photonic quasiparticles using cascaded GRIN lenses of different configurations
(see also Supplementary Information): a, a skyrmion of $N_{r}=1$ and $N_{v}=2$
is transformed into a skyrmion of the same radiality and vorticity but
opposite polarity, b, a skyrmionium of $N_{r}=2$ and $N_{v}=0$ is transformed
into the skyrmionium of the same radiality and vorticity but opposite
polarity, c, a quadruskyrmion of $N_{r}=2$ and $N_{v}=4$ is transformed into a
quadruskyrmion of the same radiality and vorticity but opposite polarity, d, a
quadrumeron of $N_{r}=1$ and $N_{v}=2$ is transformed into a quadrumeron of
the same radiality and vorticity but opposite polarity, and, e, a target
quadrumeron of $N_{r}=4$ and $N_{v}=0$ is transformed into a target
quadrumeron of the same radiality and vorticity but opposite polarity, by
tuning the input light polarization from LCP to RCP. Figure 3: Photonic
quasiparticle encryption. Alice holds the cascaded GRIN lens array, which can
create high-density arbitrary photonic quasiparticles with on demand polarity,
radiality, vorticity, and centrality, in a millimetric small volume. Then
Alice relays the photonic quasiparticle array from the Fourier plane to the
far-field by a lens, and in the form of s a ciphertext into a free-space
secure channel, in which the quasiparticles cannot be detected due to the
spatially overlapped superposed state. In parallel, Alice sends a key to Bob
via a classic channel which contains the map of the topological numbers in
relation to alphanumerical characters. Bob then receives the ciphertext using
a Fourier lens to observe the quasiparticle array in the Fourier plane. By
apply the detection method as outlined in Methods to measure the four
topological numbers of each quasiparticle which are then translated into the
plaintext (“arxiv e-print”) using the key only known to Alice and Bob.

Experimental generation and topological control. We have designed a series of
GRIN lens cascades to experimentally generate a diverse set of complex
photonic quasiparticle with controlled multiple topological numbers in
addition to skyrmion number $N_{s}$, which include polarity $N_{p}$, vorticity
$N_{c}$, radiality $N_{r}$, and centrality $N_{c}$. Firstly, we show the
topological control of the fundamental skyrmions. Using a GRIN lens segment
with input light of left-handed circular polarization (LCP), we can create a
2nd-order skyrmion of $N_{p}=-1$ and $N_{v}=2$, see the left of Fig. 2a. To
control the polarity $N_{p}$, we tune the input light polarization from LCP to
linear polarization then to right-handed circular polarization (RCP),
correspondingly, the skyrmion evolves into biskyrmion and quadrumeron
configurations as intermediate states and finally into the skyrmion with
opposite polarity ($N_{p}=1$ and $N_{v}=2$, the right of Fig. 2a). Note that,
for the case of linear polarization input, the output quasiparticle will
exhibit a singular symmetry inducing skyrmion number of zero, $N_{s}=0$, where
the polarity, $N_{p}=\text{sgn}(N_{s})$, cannot be defined.

To control radiality, we apply a cascade of two identical GRIN lens segments,
which results into a skyrmionium, see the left of Fig. 2b. Polarity control by
tuning input polarization works for this case as well, and the corresponding
results are similar to that presented in Fig. 2a but with a layer of radially
nested structure, resulting into a skyrmion number of zero.

To obtain more complex quasiparticle with increased skyrmion number, we show
experimental results of quadruskyrmion by applying a cascaded configuration of
“GRIN+HWP+GRIN”, see the left of Fig. 2c, which comprises four skyrmions with
same unit vorticity. Polarity control can also be applied to the
quadruskyrmion. In addition, by switching to the cascade of “QWP+HWP+GRIN”,
the quadrumeron can be generated comprising four elementary merons of the same
half charge, see Fig. 2d (distinct from the quadrumeron in the middle of Fig.
2a).

By varying the design of the cascaded GRIN lens array, increasingly complex
topologies can be accessed. When we cascade multiple times of the style of
“QWP+HWP+GRIN”, we can generate quasiparticles of multi-radially-nested
quadrumeronium and target quadrumeron. Figure 2e shows an experimental
quadrumeronium and target quadrumeron with radiality of $N_{r}=4$ by a
multiple of four cascades of the configuration described previously, i.e. a
four-times nested matryoshka-like structure based on a basic central structure
of quadrumeron as in Figure 2d. Based on the principle outlined above, the
quasiparticles that can be accessed are far greater than those demonstrated
here. For example, we could easily generate and control the following photonic
skyrmions: quadruskyrmionium, octuskyrmion, octumeron, octuskyrmionium,
octumeronium, etc. Furthermore, there is still a huge space to explore in
terms of the creation of more exotic and complex quasiparticles that have not
existed before.

Topological optical encryption. Based on the complex photonic quasiparticles
characterized by multiple topological numbers and confined in very small
volume, we conceptionally demonstrate a new high-density long-range
information transfer protocol that exploits these quasiparticles as
information carriers, as shown in Fig. 3, Alice holds a GRIN lens array
system, which can generate different photonic quasiparticles with customized
topological numbers in terms of the polarity $N_{p}$, vorticity $N_{c}$,
radiality $N_{r}$, and centrality $N_{c}$, as demonstrated in last section.
Alice is able to prepare a secret key, i.e., a map from quasiparticle
topological numbers to plaintext. Due to the advantage of the vast range of
controllable topological numbers of the quasiparticle, the plaintext can be
encoded by a large number of bits, for instance, the key for 30-bit encoding
is shown in the bottom right of Fig. 3, which covers a full alphabet. All the
quasiparticles corresponding to the bits can be theoretically generated in our
cascaded GRIN lens arrays, see Methods. Note that this scheme can be easily
extended into higher-dimensional bit encoding by exploiting the control of
higher topological numbers of multiskyrmions and the information bandwidth or
capacity (speed) in the channel can also be extended by simply increasing the
GRIN lenses.

An example of the preparation of the photonic quasiparticles for the plaintext
message “arxiv e-print” that Alice would like to send is shown in the top-
middle of Fig. 3, which corresponds to the customized quasiparticles with
different combinations of topological numbers $(N_{p},N_{r},N_{v},N_{c})$ of
$(1,1,2,1)$, $(-1,3,2,4)$, $(1,4,4,4)$, $(-1,4,2,1)$, $(1,2,4,4)$,
$(-1,2,4,4)$, $(1,5,2,1)$, $(-1,4,4,4)$, $(-1,3,2,4)$, $(-1,1,2,4)$,
$(-1,4,2,1)$, $(1,4,2,4)$, $(-1,5,2,4)$. Next, Alice generates the
corresponding quasiparticles array, and couples them into free space as a
secure channel by a Fourier lens and sends to Bob (see Method). In the secure
channel, the light behaves as a ciphertext, i.e. a spatially superposed state
of various quasiparticles due to Fourier lens transformation, where it is
impossible to detect information by a third party without a specific
polarimetry device, so as to ensure security. At the receiver side, Bob need
to prepare another lens to focus the ciphertext light onto the Fourier plane,
where the quasiparticle array is imaged and a CCD camera is used to capture
the image. Between the CCD and lens is a Stokes polarimetry system comprising
a QWP and a polarizer, which is used to measure distributions of all
components of Stokes vector, see details in Methods. After reconstruction of
quasiparticle array, Bob can read the combinations of the four topological
numbers $(N_{p},N_{r},N_{v},N_{c})$, assisted by simple calculations, see
details in Methods. Finally, Alice need to send the key to Bob by another
classic channel, where a third party also cannot get any readable information.
Once the key has been received, Bob can translate the measured results for the
quasiparticle topological numbers into readable plaintext, and this massage is
only known between Alice and Bob. In this example, we have used “arxiv
e-print” as the plaintext just for simplification, which can be replaced with
any arbitrary length texts, exploiting the diversity of quasiparticles and
their range of controlled topological numbers.

##  II Discussion

Here we have created a novel family of topological quasi-particles controlled
by photonic GRIN lens cascades, extending the fundamental skyrmions to exotic
radially nested skyrmioniums, multiskyrmions and multimerons, which possess
sophisticated spin textures characterized by multiple topological numbers. The
scheme presented here could be further extended in terms of diversity of the
photonic quasiparticles by taking advantage of the plethora of potential
designs of the cascades of the GRIN lenses. In our experiment, we repeatedly
cascade the same combination of segments to control the radially nested
structure, but in the future, it would also be interesting to explore
alternative hybrid cascades of different combination segments and to identify
a mechanism to control more complex forms of quasiparticles.

The versatility of cascaded GRIN lens design will open new technological
directions to continually explore and discover new kinds of topological
quasiparticles with sophisticated textures and properties. In addition to the
proposed concept of encryption, the quasiparticles still have great potential
for other informatic applications such as optical ultra-capacity
communications wan2022divergence ; pryamikov2022rising ; willner2021orbital ,
high-dimensional multi-party secret sharing pinnell2020experimental , and
ultra-sensitive ultraprecise sensing and metrology li2022highly ;
yuan2019detecting , and more. Here, the topological photonic quasiparticles
always have the advantage of much higher information or data density in
contrast to other carries, due to their multiple controlled degrees of freedom
in particle-like unit volumes. For instance, using orbital angular momentum
(OAM) of light as a carrier can also achieve same encryption, as OAM can also
possess high-dimensional topological numbers with chirality control,
equivalent to the vorticity with polarity control of skyrmions. While, the
multiskyrmions open additional topological numbers such as centrality and
radiality and controlled in more condensed spatial volume, promising step
towards future ultra-capacity optical communications and informatics. In
addition, in contrast to the scalar OAM light, the vectorial topologies of
photonic multi-skyrmions can provide strong resilience against the
perturbations in complex media such as turbulence nape2022revealing ,
promising the improvement of robustness in communication networks.

Finally, the new photonics quasiparticles proposed and discovered here in
photonic fields are also highly desired for other communities and in other
condensed-matter systems. On the one hand, quasiparticles were transferred to
photonics from magnetic and condensed-matter systems to created
interdisciplinary studies, on the other hand, photonic methods can inspire new
forms of quasiparticles not existed before in any system, leading to new
fundamental physical effects that could be explored and new applications such
as topological optoelectronic devices.

##  III Methods

Stokes polarimetry of structured light. The experimental reconstruction of the
optical quasiparticles is achieved through Stokes polarimetry, more
specifically, through the reconstruction of the distributions of three
normalized Stokes parameters $s_{1}$, $s_{2}$ and $s_{3}$, which were computed
from a set of polarization projection and intensity measurements. Figure 5
demonstrates the setup for Stokes vector field measurements. When the
generated Skyrmionic beams from a GRIN lens or its cascades passing through a
quarter waveplate (QWP) and a polarizer (P), spatial variance intensity
pattens are recorded by a CCD camera. We used the QWP, P and the CCD to form a
Stokes polarimeter to measure the polarization distribution by rotating the
QWP to four different angles. This is a well-known process that can be found
in previous technical reviews he2021polarisation ; he2022revealing . The
principal equations for calculation of the polarization field are given by:

$S^{n}_{\text{out}}=M_{\text{P}}\cdot M^{n}_{\text{QWP}}\cdot
S_{\text{in}}=M_{\text{P}}\cdot M^{n}_{\text{QWP}}\cdot(A^{-1}\cdot I)$ (2)

where $S_{\text{in}}$ is the Stokes vector of the incident light field,
$M_{\text{P}}$ and $M^{n}_{\text{QWP}}$ are Mueller matrices of the
corresponding P and QWP, $M^{n}_{\text{out}}$ is the output Stokes vector for
the $n$-th fast axis orientation state of the QWP. $M^{n}_{\text{QWP}}$ is the
Mueller matrix of the QWP for the $n$-th fast axis orientation, $A$ is an
instrument matrix, which is derived from $M_{\text{P}}M^{n}_{\text{QWP}}$.
$I=A\cdot S_{\text{in}}$ is the intensity information recorded by the CCD
camera. Note that for a 2D distributed Stokes vector field
$S_{\text{in}}(x,y)$, it consists of $[1,s_{1}(x,y),s_{2}(x,y),s_{3}(x,y)]$,
where $s_{1}$, $s_{2}$, and $s_{3}$ are the vector components of Stokes vector
he2022revealing .

Figure 4: Schematic setup of the configuration used for Stokes polarimetry. P: polarizer; QWP: quarter waveplate; CCD: detector. The remaining components relate to the GRIN lens (or cascades of GRIN lenses). The cascades are combined with different types of GRIN lenses and interstitial optical elements (such as a 4f system). Note that although the GRIN lens itself can be an imaging device, we use a 4f system for extended imaging purposes. Table 1: Cascade design for secret key preparation $(N_{c},N_{v})$ $N_{r}$ | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
$(1,2)$ | GRIN | GRIN+GRIN | GRIN+GRIN+GRIN | GRIN$\times$4 | GRIN$\times$5
$(4,2)$ | C1=QWP+HWP+GRIN | C1+C1 | C1+C1+C1 | C1$\times$4 | C1$\times$5
$(4,4)$ | C2=GRIN+HWP+GRIN | C2+C2 | C2+C2+C2 | C2$\times$4 | C2$\times$5
Figure 5: The photonic quasiparticles of controlled topological numbers
$(N_{p},N_{r},N_{v},N_{c})$ as encoding bits for secret key preparation. The
left and right groups represent the quasiparticles with opposite polarity
$N_{p}=\pm 1$. Columns in each group represent the control of different values
of radiality from $N_{r}=1$ to $5$. Three rows are for three different
controls of centrality and vorticity, $(N_{c},N_{v})=(1,2)$, $(4,2)$, and
$(4,4)$.

Detection of quasiparticle topological numbers. After reconstruction of the
Stokes field of a quasiparticle,
$\mathbf{s}(x,y)=[s_{1}(x,y),s_{2}(x,y),s_{3}(x,y)]$, the number of $N_{r}$
can be easily decided by counting the radially nested structures in which. For
the cases of $N_{r}>1$, we always care about the basic structure in the
center, we refer to here as the core structure, and the outer layers are just
the same as the core structure only with staggered polarity. Then, we care
about the number of $N_{c}$ of the core structure, i.e. How many spin-up or
-down center points in the core region exist, which is equivalent to how many
singularities of the transverse component $[s_{1}(x,y),s_{2}(x,y)]$ are
contained in the core region. This singularity counting problem can be solved
by an algorithm for singularity searching of a complex multi-singularity light
field wang2022deep . To further characterize structural details, the skyrmion
density distribution of the quasiparticle is calculated by
$\rho_{s}(x,y)=\mathbf{s}(x,y)\cdot[\partial_{x}\mathbf{s}(x,y)\times\partial_{y}\mathbf{s}(x,y)]$.
If $N_{r}>1$, the skyrmion density distribution will also correspondingly show
radially nested structure. Next, we integrate the skyrmion density to the core
region and the skyrmion number $N_{s}$ of the core structure is obtained.
Finally, the polarity and vorticity can be obtained by
$N_{p}=\text{sgn}(N_{s})$ and $N_{p}=|N_{s}|$, and all the four numbers
$(N_{p},N_{r},N_{v},N_{c})$ required to characterize the photonic
quasiparticles (multiskyrmions and multimerons) are measured, see detailed
analysis and calculation results of experiments in Supplementary Information.
Secret key preparation. To achieve the functionality of the key shown in Fig.
3, we need to build the channel to controllably emit corresponding photonic
quasiparticle. In this channel, the sender hold multiple segments of HWP, QWP,
and GRIN lenses. The cascade designs corresponding to the required topological
numbers of $(N_{r},N_{v},N_{c})$ are shown in Table 1. The polarity $N_{p}=\pm
1$ is controlled by the input light with polarization of either RCP or LCP.
The corresponding quasiparticles are shown in Fig. 5.

Acknowledgments
C.H. would like to thank the support of the John Fell Fund from University of
Oxford, and the Junior Research Fellowship from St John’s College, University
of Oxford.
Author contributions
Y.S. conceived the idea and wrote the paper, C.H. designed the GRIN lens
cascades, performed simulation and experiment of the GRIN lens based
skyrmionic beam generation, with the input of Z.S., B.C., H.H., Y.M.,
J.A.J.F., S.J.E., S.M.M., and M.J.B. together. Y.S, C.H., S.M.M., and A. F.
contributed to data analysis and manuscript revision.
Competing interests
The authors declare no competing interests.

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