# Higher-order Poincaré Spheres and Spatio-Spectral Poincaré Beams

R. Fickler  Tampere University, Photonics Laboratory, Physics Unit, Tampere,
FI-33720, Finland<EMAIL_ADDRESS>L. Kopf  Tampere University,
Photonics Laboratory, Physics Unit, Tampere, FI-33720, Finland  M. Ornigotti
Tampere University, Photonics Laboratory, Physics Unit, Tampere, FI-33720,
Finland

###### Abstract

The study of fundamental optics effects has been stimulated through the
increasing ability to structure light in all its degrees of freedom (DOFs) in
sophisticated but simple experimental settings. However, with such an increase
in experimental capabilities, it has also become important to study
theoretical descriptions for a more intuitive understanding of the underlying
concepts. Here, we introduce a visual representation of light that is
structured in its transverse space, frequency, and polarization in the form of
a higher-order Poincaré sphere and discuss interesting links to its
fundamental counterpart. We further leverage this connection to discuss and
experimentally generate light possessing all possible polarization states
across its spatio-spectral shape, which we term _spatio-spectral Poincaré
beams_. By invoking all DOFs of light in the powerful description of higher-
order Poincaré spheres, our work can pave the way for a deeper understanding
and beneficial application of structured light as a powerful tool in optics.

††preprint: APS/123-QED

Increasing the complexity of light fields by shaping their amplitude in space,
time, and polarization, i.e., all their degrees of freedom (DOFs) is advancing
research in various different fields. Driven by increasingly advanced
technologies to control light, ideas from different optics fields have been
adapted, combined, and extended leading to growing interest in fundamental
studies of linear, nonlinear, and quantum optics, as well as applications in
imaging, sensing, computing, and communications Forbes  _et al._ (2021); He
_et al._ (2022). A lot of attention has been attributed to form complex
spatial polarization states by combining polarization with different spatial
mode structures, leading to so-called spatial vector beams Chen  _et al._
(2018). Spatial vector beams are used in applications such as tight focusing
Dorn  _et al._ (2003), the study of complex entanglement pattern Fickler  _et
al._ (2014), high-speed imaging schemes Berg-Johansen  _et al._ (2015), and
the observation of three-dimensional polarization structures like knots and
links Larocque  _et al._ (2018). Mathematically, they have been described
through a generalization of the fundamental Poincaré sphere (PS), the so-
called higher-order PS for spatial vector beams Milione  _et al._ (2011);
Holleczek  _et al._ (2010), which enables an intuitive visual representation
of these vectorial light fields and helps describing complex evolutions, as
for example light fields obtained through a high-order Pancharatnam–Berry
phase Milione  _et al._ (2012). Additionally, the concept of higher-order PS
can be linked to different applications in communications Ndagano  _et al._
(2017), metamaterial design Jiang  _et al._ (2020), and extensions to more
complex spatially structured light Shen  _et al._ (2020).

A particularly interesting set of spatial vector beams describes light fields
containing all possible polarization states, so-called Poincaré beams Beckley
_et al._ (2010a). For such beams, the transverse polarization pattern can be
mapped to the entire fundamental PS through a stereographic projection. The
patterns can further be linked to interesting topological features, such as
complex polarization singularities Dennis (2002), polarization Möbius stripes
Bauer  _et al._ (2015), and optical skyrmions in the classical Shen  _et al._
(2024) and quantum domain Ornelas  _et al._ (2024) to highlight a few.

In this letter, we extend the idea of high-order PSs to states of light with a
complex polarization structure over its spatial as well as spectral DOF, and
present interesting links to its fundamental counterpart. We first demonstrate
that single points on higher-order PSs of spatial as well as spectral vector
beams can be linked to latitudes and meridians on the fundamental PS,
respectively. We then introduce a more complex higher-order PS describing
spatio-spectral vector beams, i.e., light with a spatially and spectrally
imhomogenous polarization state. We show that such light fields can exhibit
all possible polarization states over the spatio-spectral domain and how the
patterns can be mapped onto the entire surface of the fundamental PS. As such,
we term these states of light _Spatio-Spectral Poincaré beams_ (SSPBs). We
then generate SSPBs using a simple experimental setup and verify that all
polarization states of the PS are found using spatio-spectrally resolving
polarization tomography.

The vectorial nature of light, i.e., its polarization, can be described as a
two-dimensional state using the circular polarisation basis
$\\{\ket{R}$,$\ket{L}\\}$, so that any state can be constructed as a
superposition with

$\displaystyle\ket{\Psi}=\cos\left(\frac{\theta}{2}\right)\ket{R}+\sin\left(\frac{\theta}{2}\right)e^{i\phi}\ket{L},$
(1)

where $\theta\in[0,\pi]$ and $\phi\in[0,2\pi]$. These states are conveniently
represented through the surface of the PS, where $\ket{R}$ and $\ket{L}$ are
located on the north- and south-pole of the PS, respectively (see
supplementary for more information). With a similar argument, one can also
describe other DOFs of a paraxial light field, i.e., the transverse spatial
Padgett and Courtial (1999); Agarwal (1999); Dennis and Alonso (2017) and the
spectral (temporal) Brecht  _et al._ (2015) DOF, or, in a more general
framework, two-dimensional quantum states in the form of the Bloch sphere
Nielsen and Chuang (2010).

Figure 1: Higher-order Poincaré spheres (PS) in the space and frequency
domain. (a) Higher-order spatial PS constructed by two states at the poles
with orthogonal circular polarization and two spatial Laguerre-Gaussian modes
$LG_{0,\ell}$ with $\ell=\pm 1$. A single point on the sphere describes a beam
with a transverse spatial polarization pattern, which itself can be visualized
as a line along a latitude on the fundamental PS (see inset). The states on
opposing sides are graphically depicted by their transverse intensity in grey
scale overlaid by the polarization ellipses. (b) Higher-order spectral PS
built by the two spectral Hermite-Gaussian modes $HG_{n}$ with $n=0,1$, each
with the opposite circular polarization. Here, a single point corresponds to a
beam with a spectrally varying polarization pattern, whose distribution
follows a meridian of the fundamental PS (see inset). Graphically, the states
are depicted by their spectral intensity in rainbow colors with the
polarization ellipses underneath. $\ket{H}$, $\ket{V}$, $\ket{D}$, $\ket{A}$,
$\ket{R}$, and $\ket{L}$ stand for horizontal, vertical, diagonal, anti-
diagonal, and left- and right-handed circular polarization states,
respectively. Both inset examples, a) and b), are shown for $\theta=2\pi/3$
and $\phi=\pi/4$. For all polarization ellipses, linear polarization is
depicted by black lines, right- and left-handed polarization by red and green
ellipses, respectively.

When polarization is correlated with one of the other DOFs, a higher-order PS
can be defined Milione  _et al._ (2011); Holleczek  _et al._ (2010). In the
spatial domain, any set of two orthogonal transverse spatial modes
$T_{n}(\vec{r})$ is sufficient, where $n$ denotes the mode number, and
$\vec{r}$ the transverse spatial coordinate. Thus, Eq. (1) is transformed to

$\ket{\Psi}_{T}=T_{1}(\vec{r})\cos\left(\frac{\theta_{T}}{2}\right)\ket{R}+T_{2}(\vec{r})\sin\left(\frac{\theta_{T}}{2}\right)e^{i\phi_{T}}\ket{L},$
(2)

with $\theta_{T}\in[0,\pi]$ and $\phi_{T}\in[0,2\pi]$, as before. The complex-
valued spatial mode functions $T_{n}(\vec{r})$ introduce a spatially varying
polarisation pattern in the transverse plane, such that these beams are termed
_spatial vector beams_. For the spatial higher-order PS, the two poles are
described by two orthogonal polarization states, where each state has an
orthogonal spatial mode function and where $\theta_{T}$ and $\phi_{T}$ span
the whole surface of the sphere. Note, that for $T_{1}=T_{2}$, the higher-
order PS reduces to the fundamental PS.

Arguably, the most popular spatial vector beams are realized when the spatial
mode functions are implemented by higher-order Laguerre-Gaussian (LG) modes,
i.e.,
$T_{n}(\vec{r})=LG_{p,\ell}=G(r)L_{p}^{\absolutevalue{\ell}}(\rho)e^{-i\ell\varphi}$,
with $r$ being the radius, $\rho=2r^{2}/w_{0}^{2}$ with $w_{0}$ being the beam
waist, $G(r)$ describing a Gaussian profile,
$L_{p}^{\absolutevalue{\ell}}(\rho)$ are the generalized Laguerre polynomials,
and $\varphi$ the azimuthal angle (simplified description of the transverse
field for clarity). In the case of no radial structure
($L_{0}^{\absolutevalue{\ell}}(\rho)=1$), we can choose the two first order LG
modes carrying opposite OAM ($\ell=\pm 1$), such that $T_{\pm
1}(\vec{r})=G(r)e^{\pm i\varphi}$. Then, Eq. (2) becomes

$\ket{\psi}_{T}=G(r)e^{i\varphi}\left[\cos\left(\frac{\theta_{T}}{2}\right)\ket{R}+\sin\left(\frac{\theta_{T}}{2}\right)e^{i(\phi_{T}-2\varphi)}\ket{L}\right].$
(3)

The resulting higher-order PS is shown in Fig. 1a, where the two poles of the
sphere are described by two donut-shaped light fields with two orthogonal
transverse uniform circular polarizations. Any point on the higher-order PS,
except the two poles, corresponds to one combination of
$\\{\theta_{T},\phi_{T}\\}$ forming a unique spatial vector beam. It contains
a transverse varying polarization state, where the angular coordinate
$\varphi$ relates to different phases between the two circular polarization
components and cycles through all possible phases twice. Thus, the higher-
order PS and the fundamental PS are linked through $\theta=\theta_{T}$ and
$\phi=\phi_{T}+2\phi$. Interestingly, this results in each spatial vector beam
containing all polarization states of a latitude (horizontal line) on the
fundamental PS (see example and inset in Fig. 1a and supplementary material
for a more detailed discussion).

Figure 2: Spatio-spectral Poincaré sphere with the North (South) pole
describing a right-circular (left-circular) polarization state with a Gaussian
(first order Hermite-Gaussian) spectral mode and a transverse spatial
Laguerre-Gaussian mode with $\ell=1$ ($\ell=-1$). The inset on the right shows
a single spatio-spectral Poincaré beam, where half of the spectrum covers the
whole surface of a fundamental PS. Linear polarization is depicted by black
lines, right- and left-handed circular polarization by red and green ellipses,
respectively.

Although less studied, a similar vector beam can be constructed by combining
polarization and the spectral or temporal DOF of light. As before, we can
extend Eq. (1) by including two orthogonal spectral mode functions
$F_{n}(\omega)$ such that we obtain a so-called _spectral vector beam_ Kopf
_et al._ (2021) of light described by the state

$\ket{\Psi}_{F}=F_{1}(\omega)\cos\left(\frac{\theta_{F}}{2}\right)\ket{R}+F_{2}(\omega)\sin\left(\frac{\theta_{F}}{2}\right)e^{i\phi_{F}}\ket{L},$
(4)

with $\omega$ being the angular frequency of the light field,
$\theta_{F}\in[0,\pi]$, and $\phi_{F}\in[0,2\pi]$. The states are forming the
surface of a higher-order spectral PS, where the poles are describing two
orthogonal circular polarization states, each having a different orthogonal
spectral mode. As before, for $F_{1}=F_{2}$, the higher-order PS reduces to
the fundamental one. Note, that although here the higher-order PS is discussed
in the spectral domain, all results hold also for the temporal domain as both
are linked via the Fourier relation.

Commonly, the spectral mode functions of choice are Hermite-Gaussian (HG)
spectral modes with $F_{n}(\omega)=HG_{n}=G(\omega)H_{n}(\Omega)$, where
$G(\omega)$ describes a Gaussian profile, $H_{n}$ are the Hermite polynomials,
and $\Omega=(\omega_{0}-\omega)/\sigma$ with $\sigma$ labelling the width of
the spectrum and $\omega_{0}$ its central frequency Brecht  _et al._ (2015).
For the two lowest order modes, i.e., the first two Hermite polynomials
$H_{0}(x)=1$ and $H_{1}(x)=2x$, we obtain $F_{1}(\omega)=G(\omega)$ and
$F_{2}=G(\omega)2\Omega$ and Eq. (4) can be rewritten as

$\displaystyle\ket{\Psi}_{F}=$ $\displaystyle
G(\omega)\Omega^{\prime}\left[\frac{1}{\Omega^{\prime}}\cos\left(\frac{\theta_{F}}{2}\right)\ket{R}+\Omega^{\prime}\sin\left(\frac{\theta_{F}}{2}\right)e^{i\phi_{F}}\ket{L}\right],$
(5)

with $\Omega^{\prime}=\sqrt{2\Omega}$. The resulting higher-order PS is shown
in Fig. 1b, where all but the two states on the poles have a polarization that
varies depending on the frequency component. In contrast to the previously
introduced spatial vector beams, these spectral vector beams show a different
connection to the fundamental PS. Eq. (5) shows that for $\Omega^{\prime}=0$,
i.e., $\omega=\omega_{0}$ the beam is in state $\ket{L}$. For
$\Omega^{\prime}\rightarrow\pm\infty$ with a fixed phase $\pm\phi_{F}$ between
the right- and left-handed component, the polarization continuously evolves to
$\ket{R}$. The correct mapping of these beams onto the fundamental PS is then
given by $\phi=\phi_{F}$ and
$\theta=\arctan\left(\Omega^{\prime}\,\tan\theta_{F}/2\right)$. Thus, any
point $\\{\theta_{F},\phi_{F}\\}$ on the higher-order PS contains all
polarization states of a meridian (vertical line) on the fundamental PS (see
inset in Fig. 1b, and the supplementary material).

To fully describe a complex vectorial light field in space and spectrum
(time), we combine all DOFs, such that one can write

$\displaystyle\ket{\Psi}_{TF}=$ $\displaystyle
T_{1}(\vec{r})F_{1}(\omega)\cos\left(\frac{\theta_{TF}}{2}\right)\ket{R}$
$\displaystyle+$ $\displaystyle
T_{2}(\vec{r})F_{2}(\omega)\sin\left(\frac{\theta_{TF}}{2}\right)e^{i\phi_{F}}\ket{L}.$
(6)

Because the polarization state changes across the transverse beam profile as
well as its frequency spectrum, these beams can be termed _spatio-spectral
vector beams_ Kopf  _et al._ (2023). As before, for $T_{1}=T_{2}$ and
$F_{1}=F_{2}$ the description reduces to simple homogeneous polarization
states. If only one of the two cases hold, the reduction instead leads to a
spatial or spectral vector beam.

In the following, we use $T_{1,2}(\vec{r})=LG_{0,\pm 1}$ and
$F_{1,2}(\omega)=HG_{0,1}$, to align with the examples of higher-order PSs
described above. As before, it is possible to define a more complex higher-
order PS (see Fig. 2), where the two poles have a uniform polarization.
However, all other states now contain all possible polarization states of the
fundamental PS when sampled across space and spectrum, i.e., they are spatio-
spectral Poincaré beams (SSPB). Analogue to spatial and spectral vector beams,
all SSPBs states, excluding those on the poles, have a frequency dependent
amplitude ratio between their $\ket{L}$ and $\ket{R}$ contributions. At the
same time, the relative phase between $\ket{L}$ and $\ket{R}$ cycles from 0 to
2$\pi$ twice dependent on the transverse angle $\varphi$. In other words, the
polarization pattern of such a beam is a complex stereographic projection of
the PS onto the three-dimensional space of the spatio-spectral domain (see
supplementary information for more details).

Figure 3: Scheme to generate a Spatio-Spectral Poincaré beam (SSPB). By using
a half-wave plate (HWP), a beam from a fs-laser is diagonally polarized with
respect to the optical axis of the birefringence crystal (BBO). The generated
spectral vector beam is then sent through an S-vortex plate (SP) to generate a
SSPB. The SSPB is characterized through spatio-spectrally resolved
polarization tomography using polarization optics (quarter-wave plate (QWP),
HWP, polarizing beam splitter (PBS)), a spatial slit mask, and a spectrometer.

Let’s visualize this feature by analysing the exemplary state on the equator
of the higher-order PS with $\\{\theta_{TF}=\pi/2,\phi_{TF}=\pi/2\\}$ (inset
in Fig. 2). The corresponding points $P$ on the fundamental PS are given by

$P_{\frac{\pi}{2},\frac{\pi}{2}}=(\sin\arctan\Omega^{\prime})\\{\sin\,2\varphi,\cos\,2\varphi,1\\}.$
(7)

If we fix the value of $\Omega^{\prime}$, $P_{\frac{\pi}{2},\frac{\pi}{2}}$
evolves as the angle $\varphi$ changes from $0$ to $2\pi$ and creates a doubly
covered great circle in the equatorial plane, If we instead fix a value of
$\varphi$ and vary the frequency $\Omega^{\prime}$ from $-\infty$ to
$+\infty$, $P_{\frac{\pi}{2},\frac{\pi}{2}}$ describes a great circle in the
meridian plane, i.e., from the South pole to the North pole. Notice, that as
$\omega\rightarrow\pm\infty$, $\arctan\Omega\rightarrow\pm\pi/2$, which is
equivalent to $\sin\arctan\Omega\rightarrow\,0,\pi.$

Figure 4: Experimentally reconstructed polarization pattern of a Spatio-
Spectral Poincaré beam. a) Reconstructed polarization ellipses depending on
the angle of the slit mask and wavelength. Right- and left-handed
polarizations are depicted by red and green ellipses, respectively. b) All
measured spatio-spectral polarization states on their respective position on
the fundamental PS. All data points measured at the same wavelength for
different spatial mask angles are connected by a line and colored according to
their wavelength. Ideally, the lines are latitudes, however, measurement
errors lead to slight deformations.

There are well-known experimental techniques generate SSPB’s by using the
first higher-order LG modes in space Bolduc  _et al._ (2013) and HG modes in
frequency Weiner (2011) However, there is a surprisingly simple method to
approximate them well by using only a two optical elements. In the spectral
domain, it was recently shown that a birefringence material can be used to
create a polarization-dependent temporal splitting of a laser pulse and
thereby producing a spectral varying polarization pattern, i.e., a spectral
vector beam Kopf  _et al._ (2021). This technique does not generate two
orthogonal spectral modes, i.e. the overlap between the two resulting spectral
distributions is small but non-vanishing, however, over a certain wavelength
range, the obtained state approximately translates to Eq. (5) with
$\\{\theta_{F}=\pi/2,\phi_{F}=\pi/2\\}$. In the spatial domain, the key
component to generate a polarization-dependent azimuthal phase distribution
between the two LG modes of opposite handedness is a segmented half-wave plate
know as an S-vortex plate Biener  _et al._ (2002). More details on these
approximations along with the individual experimental generation of a spectral
and spatial vector beam can be found in the supplementary.

In our experiment, we use Fourier-limited laser pulses with approximately 220
fs pulse duration centered at a wavelength of around 804.5 nm. At first, we
generate a spectral vector beam by modulating the beams polarization with a
half-wave plate which is aligned at 45∘ with respect to the optical axis of a
2 mm thick birefringence BaB2O4 crystal (BBO) with a cut-angle of 23.4∘. After
propagation through the BBO crystal, the laser pulses are coherently split
into two trailing pulses of half its initial amplitude, which results in the
required spectral polarization pattern. Subsequently, we sent the laser
through an S-vortex plate, which imprints the polarization-dependent OAM
($\ell=\pm 1$) thereby completing the generation scheme. See Fig. 3 for a
simplified scheme of the setup.

To characterize the generated SSPB, we perform spatio-spectrally-resolved
polarization tomography. We first filter the beam for different polarizations
with a set of wave plates and a polarizing beam splitter and for angular
positions with a slit mask (with an opening angle of approximately 14∘) and
then measure its wavelength-resolved intensity with a spectrometer. From these
measurements, we reconstruct the polarization ellipses for different
wavelengths and angular positions of the slit mask as shown in Fig. 4a (see
supplementary information for more details). We find, that the reconstructed
polarizations states nicely match the expected patterns described above,
despite some experimental imperfections, such as the non-vanishing overlap of
the spectral modes, and small optical misalignments. To show that the spatio-
spectral polarization patterns of the generated beams indeed cover the full
PS, we show the measured polarization states on the surface of the fundamental
PS and, again, find a nice agreement apart from minor experimental
imperfections (Fig. 4b).

In conclusion, we introduce an extension to the concept of higher-order PSs by
including all DOFs of light and discuss how light can exhibit every possible
polarization state. Furthermore, we experimentally generate a SSPB and verify
that its polarization ellipses indeed cover the whole PS. Following this
initial study, a multitude of different research opportunities could arise. A
direct next step can be to investigate the SSPBs in connection to the
Pancharatnam–Berry phase, for which the introduced higher-order PS will be
essential Milione  _et al._ (2012). Another promising path will be to examine
SSPBs in terms of complex polarization topologies, which have mostly been
studied in the spatial domain Larocque  _et al._ (2018); Shen  _et al._
(2024); Dennis (2002), and see how they relate to polarization states of
polychromatic light Pisanty  _et al._ (2019); Sugic  _et al._ (2020).
Moreover, having access to a higher-order PS representation for all DOFs of
light fields could provide new insight into the geometrical properties of
multipartite systems, as higher-order PSs allow an easier representation of
such systems. This scheme, in fact, can be easily generalised to a system
possessing $N$ DOFs, thus giving access to a geometrical representation of
multipartite systems beyond the present schemes based on fibrations of
spheres, which are limited to tripartite states Lévay (2004); Bengtsson and
Życkowski (2020). Finally, as all DOFs of the light field are strongly
correlated in SSPBs, they can also serve as a valuable system to explore the
similarities of classical and quantum non-separable states of light Kopf  _et
al._ (2023). Here, an extension of the recently discussed connection between
spatial vector beams and entangled photon pairs Peters  _et al._ (2023);
Ornelas  _et al._ (2024) could be extended to an analogy between SSPBs and
tripartite quantum entangled system leading to interesting non-local features.

###### Acknowledgements.

R. F. acknowledges the support of the Research Council of Finland through the
Academy Research Fellowship (decision 332399). L. K. acknowledges the support
by the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of
Science and Letters. All authors acknowledge the support of the Research
Council of Finland through the Photonics Research and Innovation Flagship
(PREIN - decision 320165).

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##  I Supplementary

This Supplementary material is structured as follows: in Sec. I.1 we briefly
recall the conventions and definition that allow one to represent an arbitrary
polarization state on the Poincaré sphere (PS) and derive the expression for
the Stokes parameters of an arbitrarily polarised beam within the framework of
the PS representation. Sec. I.2 and I.3 are dedicated to briefly recap the
extension of the concept of the PS to spatial and spectral modes,
respectively, and the subsequent definition of the higher-order PS. These
concepts are then combined in Sec. I.4 to introduce the spatio-spectral PS and
calculate the Stokes parameters within this formalism. Finally, Sec. I.5 gives
more details about the experimental generation of spatio-spectral beams.

###  I.1 Poincaré sphere

In this section, we introduce the convention used in the manuscript for
defining the PS. Any polarization state of light can be described as a two-
dimensional state using the circular polarisation states $\ket{R}$ and
$\ket{L}$, i.e.,

$\ket{\Psi}=\cos\left(\frac{\theta}{2}\right)\ket{R}+\sin\left(\frac{\theta}{2}\right)e^{i\phi}\ket{L}.$
(8)

Here, $\theta\in[0,\pi]$ and $\phi\in[0,2\pi]$, and the orthonormal circular
polarization basis is defined as

$\displaystyle\ket{L}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\vec{x}+i\vec{y}\right),$ (9)
$\displaystyle\ket{R}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\vec{x}-i\vec{y}\right),$ (10)

where $\vec{x}$ and $\vec{y}$ are the transverse Cartesian coordinates of the
light field. Notice, that with this choice of parametrization, $\ket{R}$ is
located on the north-pole of the PS, while $\ket{L}$ is on the south-pole. The
equator of the sphere is spanned by the linear polarization states

$\displaystyle\ket{H}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\ket{R}+\ket{L}\right)=\vec{x},$ (11)
$\displaystyle\ket{V}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\ket{R}-\ket{L}\right)=\vec{y},$ (12)
$\displaystyle\ket{D}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\ket{R}-i\ket{L}\right)=\frac{1}{\sqrt{2}}\left(\vec{x}+\vec{y}\right),$
(13) $\displaystyle\ket{A}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\ket{R}+i\ket{L}\right)=\frac{1}{\sqrt{2}}\left(\vec{x}-\vec{y}\right),$
(14)

corresponding to horizontal (H), vertical (V), diagonal (D) and anti-diagonal
(A) polarisation, respectively. The Stokes parameters are then defined as
usual, i.e.,

$\displaystyle S_{0}$
$\displaystyle=|\innerproduct{R}{\Psi}|^{2}+|\innerproduct{L}{\Psi}|^{2}=|\innerproduct{H}{\Psi}|^{2}+|\innerproduct{V}{\Psi}|^{2}=|\innerproduct{D}{\Psi}|^{2}+|\innerproduct{A}{\Psi}|^{2},$
(15a) $\displaystyle S_{1}$
$\displaystyle=|\innerproduct{H}{\Psi}|^{2}-|\innerproduct{V}{\Psi}|^{2},$
(15b) $\displaystyle S_{2}$
$\displaystyle=|\innerproduct{D}{\Psi}|^{2}-|\innerproduct{A}{\Psi}|^{2},$
(15c) $\displaystyle S_{3}$
$\displaystyle=|\innerproduct{R}{\Psi}|^{2}-|\innerproduct{L}{\Psi}|^{2}.$
(15d)

For a general state as described in Eq. (8), the above definitions result in

$\displaystyle S_{0}$ $\displaystyle=1,$ (16a) $\displaystyle S_{1}/S_{0}$
$\displaystyle=2\cos(\frac{\theta}{2})\sin(\frac{\theta}{2})\cos(\phi)=\sin(\theta)\cos(\phi),$
(16b) $\displaystyle S_{2}/S_{0}$
$\displaystyle=2\cos(\frac{\theta}{2})\sin(\frac{\theta}{2})\sin(\phi)=\sin(\theta)\sin(\phi),$
(16c) $\displaystyle S_{3}/S_{0}$
$\displaystyle=\cos^{2}\left(\frac{\theta}{2}\right)-\sin^{2}\left(\frac{\theta}{2}\right)=\cos(\theta).$
(16d)

###  I.2 Higher-Order Poincaré sphere in space

Let us now consider not only polarisation as an accessible degree of freedom
(DOF) of the field, but also its transverse spatial structure. Let’s focus in
particular on the first-order modes of Laguerre-Gaussian (LG) beams, which
show a twisted phase front and are known to carry orbital angular momentum
(OAM) Andrews and (editors). A detailed analysis of the resulting PS, i.e.,
the so-called higher-order PS can be found, for example, in Refs. Milione  _et
al._ (2011) and Holleczek  _et al._ (2010). Here, we just briefly recap the
relevant results for our discussion in Sec. I.4.

In general, one can write a hybrid space-polarization state by adding two
orthogonal transverse spatial mode functions $T_{n}(\vec{r})$ into the
expression of an arbitrarily polarised state (8) such that

$\displaystyle\ket{\Psi}_{T}=\cos\left(\frac{\theta_{T}}{2}\right)T_{1}(\vec{r})\ket{R}+\sin\left(\frac{\theta_{T}}{2}\right)e^{i\phi_{T}}T_{2}(\vec{r})\ket{L},$
(17)

with $\vec{r}$ being the transverse spatial coordinate and, as before,
$\theta_{T}\in[0,\pi]$ and $\phi_{T}\in[0,2\pi]$. Analogue to fundamental
polarization states, $\ket{\Psi}_{T}$ can be considered as a state in a so-
called higher-order PS, where the two poles are orthogonal polarization
states, each with an orthogonal spatial mode function attached to them. The
angles $\theta_{T}$ and $\phi_{T}$ can then be understood as the angular
positions on the surface of this higher-order PS. Note, that in this picture
one can define mutually unbiased basis states as well as Stokes parameters in
an analogous way as described above.

$T_{n}(\vec{r})$ can be implemented with the first-order LG modes, i.e., modes
with $\\{p=0,\ell=\pm 1\\}$, which carry one quanta of OAM. For this
particular choice, the transverse field is easiest described in cylindrical
coordinates with the radial position $r=\absolutevalue{x+iy}$ and the angle
$\varphi=\arg(x+iy)$ such that

$\displaystyle T_{\pm 1}(r,\varphi)=G(r)e^{\pm i\varphi},$ (18)

with $G(\vec{r})$ describing a transverse Gaussian profile. By replacing
$T_{1}$ and $T_{2}$ in Eq. (17) with $T_{-1}$ and $T_{1}$, respectively, one
obtains

$\displaystyle\ket{\Psi}_{T}=G(r)e^{-i\varphi}\left[\cos\left(\frac{\theta_{T}}{2}\right)\ket{R}+\sin\left(\frac{\theta_{T}}{2}\right)e^{i\phi_{T}}e^{i2\varphi}\ket{L}\right],$
(19)

Note that, for $T_{1}=T_{2}$ or considering a plane wave (as often done), the
higher-order PS reduces to the fundamental one.

The Stokes parameters for a state on the higher-order PS are then given by

$\displaystyle S_{0}$ $\displaystyle=|G(r)|^{2},$ (20a) $\displaystyle
S_{1}/S_{0}$
$\displaystyle=2\cos(\frac{\theta_{T}}{2})\sin(\frac{\theta_{T}}{2})\cos(\phi_{T}-2\varphi),$
(20b) $\displaystyle S_{2}/S_{0}$
$\displaystyle=2\cos(\frac{\theta_{T}}{2})\sin(\frac{\theta_{T}}{2})\sin(\phi_{T}-2\varphi),$
(20c) $\displaystyle S_{3}/S_{0}$
$\displaystyle=\cos^{2}\left(\frac{\theta_{T}}{2}\right)-\sin^{2}\left(\frac{\theta_{T}}{2}\right),$
(20d)

which can be simplified to

$\displaystyle S_{1}/S_{0}$
$\displaystyle=\sin(\theta_{T})\cos(\phi_{T}-2\varphi)$ (21a) $\displaystyle
S_{2}/S_{0}$ $\displaystyle=\sin(\theta_{T})\sin(\phi_{T}-2\varphi)$ (21b)
$\displaystyle S_{3}/S_{0}$ $\displaystyle=\cos(\theta_{T}).$ (21c)

From the expression above, we see that for every point on this higher-order PS
(with exception of the two poles, i.e., $\theta_{T}\neq 0,\pi/2$), the two
normalized Stokes parameters $S_{1}/S_{0}$ and $S_{2}/S_{0}$ are cyclically
changing. This means that for varying values of the angular coordinate
$\varphi$ the polarization changes along a parallel line (horizontal circle)
on the fundamental PS. The higher-order PS for spatial vector beams is shown
in Fig.5 along with two explicit examples on how the transverse varying
polarization relates to the fundamental PS.

Figure 5: Higher-order Poincaré Sphere (PS) for spatial vector beams. Every
point of the higher-order PS corresponds to a spatial mode for which the
polarisation vector changes with the transverse angle $\varphi$, such that it
follows a latitude on the fundamental PS. This is depicted for two examples.
The upper example (highlighted in blue) is shown for $\theta_{F}=2\pi/3$ and
$\phi_{F}=\pi/4$, while the lower example (highlighted in red) shows a state
on the equator which corresponds to a radially polarized beam with
$\theta_{F}=\pi/2$ and $\phi_{F}=0$.

###  I.3 Higher-Order Poincaré sphere in frequency (time)

A link similar to the one shown above can be made for the polarisation and
mode functions by combining the polarization DoF with the spectral or temporal
DoF of light. Note, that in the following this novel type of a higher-order PS
is described by linking spectral modes to polarization and thereby
constructing so-called spectral vector beams Kopf  _et al._ (2021). The same
argument holds for linking temporal modes to polarization Brixner and Gerber
(2001), as both domains are linked via a Fourier transform.

Similarly as before, we can extend the initial arbitrary polarization state
(8) by correlating it to two orthogonal spectral mode functions
$F_{n}(\omega)$ such that we obtain a hybrid frequency-polarization state

$\displaystyle\ket{\Psi}_{F}=\cos\left(\frac{\theta_{F}}{2}\right)F_{1}(\omega)\ket{R}+\sin\left(\frac{\theta_{F}}{2}\right)e^{i\phi_{F}}F_{2}(\omega)\ket{L},$
(22)

with $\omega$ being the angular frequency of the light field,
$\theta_{F}\in[0,\pi]$, and $\phi_{F}\in[0,2\pi]$. Again, a higher-order PS
can be constructed, where the poles are given by two orthogonal polarization
states, each with a different orthogonal spectral mode. The other states on
the surface of the sphere are described by spectral vector fields, where the
polarization state varies across the frequency spectrum.

To improve the understanding of such states, it is instructive to discuss them
in terms of two specific orthogonal frequency mode functions, such as Hermite
Gaussian (HG) spectral modes Brecht  _et al._ (2015)

$\displaystyle F_{n}(\omega)=G(\omega)H_{n}(\omega),$ (23)

where $G(\omega)$ describes a Gaussian profile and $H_{n}(\omega)$ are the
Hermite polynomials. For later convenience, we can introduce the spectrum-
scaled frequency, $\Omega=\sqrt{2(\omega_{0}-\omega)/\sigma}$, where
$\omega_{0}$ is the central frequency of the pulse and $\sigma$ its spectral
bandwidth. With $F_{1}(\omega)=H_{0}(\omega)=1$,
$F_{2}(\omega)=H_{1}(\omega)=2\omega$, and the definition of $\Omega$ given
above, one can then rewrite Eq. (22) to

$\displaystyle\ket{\Psi}_{F}=G(\Omega)\left[\frac{1}{\Omega}\cos\left(\frac{\theta_{F}}{2}\right)\ket{R}+\Omega\sin\left(\frac{\theta_{F}}{2}\right)e^{i\phi_{F}}\ket{L}\right],$
(24)

where $G(\Omega)=\Omega\,G(\omega)$. Note again, that for $F_{1}=F_{2}$ or
monochromatic light fields (plane waves), the higher-order PS reduces to the
fundamental one.

In contrast to spatial vector beams constructed with twisted light fields,
these hybrid polarization-frequency states have a different connection to the
fundamental PS. Any point on the higher-order spectral PS contains all
polarization states of a meridian (vertical circle) on the fundamental PS,
rather than along a parallel line as for their spatial counterpart described
in Sec. I.2. This feature is demonstrated by Eq. (24) where a point on the
higher-order PS is described by one combination of $\theta_{F}$ and
$\phi_{F}$. The resulting spectral spectral vector beam contains a
polarization state that is changing across the frequency spectrum from
$\ket{L}$ at $\omega\rightarrow-\infty$ to $\ket{R}$ at $\omega=\omega_{0}$,
and back again to $\ket{L}$ at $\omega\rightarrow\infty$. Hence, the
polarization state consists of all points along a meridian on the fundamental
PS.

Figure 6: Higher-order Poincaré Sphere (PS) for spectral vector beams. Every
point of the higher-order PS corresponds to a meridian on the fundamental PS,
as shown here for two examples. The upper example, highlighted in red, is
shown for $\theta_{F}=2\pi/3$ and $\phi_{F}=\pi/4$, while the lower example
(highlighted in blue) shows a state on the equator with $\theta_{F}=\pi/2$ and
$\phi_{F}=0$.

The Stokes parameters now have the following form

$\displaystyle S_{0}$
$\displaystyle=|G(\omega)|^{2}\,\left[\cos^{2}\left(\frac{\theta_{F}}{2}\right)+|\Omega|^{4}\sin^{2}\left(\frac{\theta_{F}}{2}\right)\right],$
(25a) $\displaystyle S_{1}/S_{0}$
$\displaystyle=4\cos(\phi_{F})\,\left[\frac{|\Omega|^{2}\,\tan\left(\frac{\theta_{F}}{2}\right)}{1+|\Omega|^{4}\tan^{2}\left(\frac{\theta_{F}}{2}\right)}\right],$
(25b) $\displaystyle S_{2}/S_{0}$
$\displaystyle=4\sin(\phi_{F})\,\left[\frac{|\Omega|^{2}\,\tan\left(\frac{\theta_{F}}{2}\right)}{1+|\Omega|^{4}\tan^{2}\left(\frac{\theta_{F}}{2}\right)}\right],$
(25c) $\displaystyle S_{3}/S_{0}$
$\displaystyle=\frac{1-|\Omega|^{4}\tan^{2}\left(\frac{\theta_{F}}{2}\right)}{1+|\Omega|^{4}\tan^{2}\left(\frac{\theta_{F}}{2}\right)}.$
(25d)

If we then define, in analogy to Ref. Beckley  _et al._ (2010b),
$\xi=|\Omega|^{2}\tan\left(\frac{\theta_{F}}{2}\right)\equiv\tan(\theta/2)$,
we can rewrite the Stokes parameters as follows

$\displaystyle S_{1}/S_{0}$
$\displaystyle=\frac{4\xi\cos(\phi_{F})}{1+\xi^{2}}\equiv\cos(\theta)\cos(\varphi),$
(26a) $\displaystyle S_{2}/S_{0}$
$\displaystyle=\frac{4\xi\sin(\phi_{F})}{1+\xi^{2}}\equiv-\sin(\theta)\sin(\varphi),$
(26b) $\displaystyle S_{3}/S_{0}$
$\displaystyle=\frac{1-\xi^{2}}{1+\xi^{2}}\equiv\cos(\theta),$ (26c)

where $\\{\theta,\varphi=-\phi_{F}\\}$ are the polar and azimuthal angles on
the fundamental PS, with the polar angle measured from the equator rather than
the North pole. Using this convention, we can interpret the triplet
$\\{S_{1}/S_{0},S_{2}/S_{0},S_{3}/S_{0}\\}$ as a stereographic projection of a
sphere from the South pole. Notice, moreover, that this definition implies
that
$\theta\equiv\theta(\Omega)=2\arctan\left[|\Omega|^{2}\,\tan\left(\frac{\theta_{F}}{2}\right)\right]$
and therefore $\lim_{\Omega\rightarrow\pm\infty}\theta(\Omega)=\pm\pi$. This
means, that for $\Omega\rightarrow-\infty$, (i.e., at the trailing edge of the
pulse) $\theta=-\pi$ and the polarisation of the beam is $\ket{L}$. Then, when
$\Omega=0$ (meaning $\omega=\omega_{0}$, i.e., at the carrier frequency) we
have $\theta=0$ and the polarisation of the field is $\ket{R}$, as the point
representing the pulse on the PS reaches the North pole from the
$\theta\in[-\pi,0]$ sector of the sphere. Finally, when
$\Omega\rightarrow\infty$ (i.e., at the leading edge of the pulse) the polar
angle is $\theta=\pi$ and the polarisation of the field returns to $\ket{L}$,
i.e., the field reaches the South pole from the $\theta\in[0,\pi]$ sector of
the sphere. The higher-order PS for spectral modes along with two example
points and their relation to the fundamental PS are shown in Fig. 6.

###  I.4 Higher-Order Poincaré sphere in space and frequency (time)

After establishing PSs as a visual representation of correlated polarization
and spatial or spectral modes, it is now possible to go beyond this
description and look at polarization in connection to both spatial and
spectral domains. One can, in fact, simply combine the two descriptions
introduced above, such that

$\displaystyle\ket{\Psi}_{TF}=\cos\left(\frac{\theta_{TF}}{2}\right)T_{1}(\vec{r})F_{1}(\omega)\ket{R}+\sin\left(\frac{\theta_{TF}}{2}\right)e^{i\phi_{F}}T_{2}(\vec{r})F_{2}(\omega)\ket{L}.$
(27)

When using spatial modes carrying $\pm 1\hbar$ OAM quanta and the two lowest
order HG spectral modes, Eq. (27) can be rewritten as

$\displaystyle\ket{\Psi}_{TF}$ $\displaystyle=$ $\displaystyle
G(r)e^{-i\varphi}G(\omega)\Omega\left[\frac{1}{\Omega}\cos\left(\frac{\theta_{TF}}{2}\right)\ket{R}+\Omega\sin\left(\frac{\theta_{TF}}{2}\right)e^{i(\phi_{TF}+2\varphi)}\ket{L}\right].$
(28)

Again, note that for $T_{1}=T_{2}$ and $F_{1}=F_{2}$ (monochromatic plane
waves), the higher-order PS reduces to the fundamental one. If only
$T_{1}=T_{2}$ ($F_{1}=F_{2}$) applies, then the reduction leads to the
spectral (spatial) higher order PS. The most interesting feature, however, is
that light fields described by the state given in Eq. (28) include all
possible polarization states of the PS. These states are commonly known as
Poincaré beams Beckley  _et al._ (2010b).

In contrast to earlier discussions on Poincaré beams, the type of beams
described by Eq. (28) include all polarization states distributed over the
whole spatial-spectral domain, hence the name _Spatio-Spectral Poincaré beams_
(SSPB). Analog to for spatial and spectral vector beams individually, one can
see that for SSPBs the amplitude between left- and right-circular polarization
varies across the frequency spectrum, while the relative phase changes across
the transverse spatial extend.

The Stokes parameters in this case are given by

$\displaystyle S_{0}$
$\displaystyle=|G(r)|^{2}|G(\omega)|^{2}\,\left[\cos^{2}\left(\frac{\theta_{TF}}{2}\right)+|\Omega|^{4}\sin^{2}\left(\frac{\theta_{TF}}{2}\right)\right],$
(29a) $\displaystyle S_{1}/S_{0}$
$\displaystyle=4\cos(\phi_{TF}+2\varphi)\,\left[\frac{|\Omega|^{2}\,\tan\left(\frac{\theta_{TF}}{2}\right)}{1+|\Omega|^{4}\tan^{2}\left(\frac{\theta_{TF}}{2}\right)}\right],$
(29b) $\displaystyle S_{2}/S_{0}$
$\displaystyle=4\sin(\phi_{TF}+2\varphi)\,\left[\frac{|\Omega|^{2}\,\tan\left(\frac{\theta_{TF}}{2}\right)}{1+|\Omega|^{4}\tan^{2}\left(\frac{\theta_{TF}}{2}\right)}\right],$
(29c) $\displaystyle S_{3}/S_{0}$
$\displaystyle=\frac{1-|\Omega|^{4}\tan^{2}\left(\frac{\theta_{TF}}{2}\right)}{1+|\Omega|^{4}\tan^{2}\left(\frac{\theta_{TF}}{2}\right)},$
(29d)

which we can rewrite using
$\xi=|\Omega|^{2}\tan\left(\frac{\theta_{TF}}{2}\right)\equiv\tan(\theta/2)$
to

$\displaystyle S_{1}/S_{0}$
$\displaystyle=\frac{4\xi\cos(\phi_{TF}+2\varphi)}{1+\xi^{2}}\equiv
2\sin(\theta)\cos(\Phi),$ (30a) $\displaystyle S_{2}/S_{0}$
$\displaystyle=\frac{4\xi\sin(\phi_{TF}+2\varphi)}{1+\xi^{2}}\equiv-2\sin(\theta)\sin(\Phi),$
(30b) $\displaystyle S_{3}/S_{0}$
$\displaystyle=\frac{1-\xi^{2}}{1+\xi^{2}}\equiv\cos(\theta).$ (30c)

This again corresponds to a stereographic projection of a sphere from the
North pole, which can be brought back to its original state via the mapping
$\Phi=\phi_{TF}+2\varphi$ and
$\theta=2\arctan\left[|\Omega|^{2}\,\tan\left(\frac{\theta_{F}}{2}\right)\right]$,
such that the back-projection of the stereographic map onto the sphere gives
the point
$P=\\{\sin(\theta)\cos(\Phi),\sin(\theta)\sin(\Phi),\cos(\theta)\\}$.

If we now take as an example the state on the $S_{1}/S_{0}$-axis on the
spatio-spectral PS with $\\{\theta_{TF}=\pi/2,\phi_{TF}=0\\}$, the
corresponding point $P$ on the “normal" PS is given by

$P_{\pi/2,0}=[\sin(\arctan\Omega)]\\{\cos(2\ell\varphi),\sin(2\varphi),0\\}.$
(31)

For a fixed valued of $\Omega$, the term $\sin(\arctan\Omega)$ is just a
numerical prefactor. $P_{\pi/2,0}$ describes a doubly covered great circle on
the sphere at the equator as the spatial coordinate $\varphi$ changes from $0$
to $2\pi$, i.e., the corresponding beam contains all polarisation states on
the equator. If we instead have a constant value of $\varphi$ and let $\Omega$
vary from $-\infty$ to $+\infty$, i.e., we sweep across the whole spectrum of
the beam, $P_{\pi/2,0}$ describes a great circle in the meridian plane, i.e.,
from the South pole to the North pole. Notice, that as
$\omega\rightarrow\pm\infty$, $\sin(\arctan\Omega)\rightarrow\pm\pi/2$, which
implies that in the second case $\theta\rightarrow\pi/2-\theta$, i.e., the
polar angle is measured from the equator instead of from the North pole.

###  I.5 Experimental realization of a spatio-spectral Poincaré beam

As described in the main text, we implement a spatio-spectral Poincaré beam
with a simple setup with which we approximate the states described above. The
initial state of our laser is described by a horizontally polarized beam with
a Gaussian spectrum $G(\lambda)$ over its wavelength with a bandwidth of
approximately 10 nm centered at 804.5 nm. The spatial profile is approximately
described by a Gaussian distribution $G(r)$ of around 2-3 mm beam waist. Thus,
the initial state can be described by

$\displaystyle\ket{\Psi}^{exp}$ $\displaystyle=$ $\displaystyle
G(r)\,G(\lambda)\ket{H}.$ (32)

Using a half-wave plate we adjust the polarization such that it has a 45∘
angle with respect to the optical axis of a subsequent birefringence crystal.
We adjusted the optical axis of the crystal to be aligned with the horizontal
polarization, such that with respect to the crystal the state can now be
described as

$\displaystyle\ket{\Psi}^{exp}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}G(r)\,G(\lambda)\left(\ket{H}+\ket{V}\right).$
(33)

Due to the birefringence of the crystal, the vertical components of the pulse
experience a time delay $\tau$ with respect to the horizontal ones. Here, we
use a BaB2O4 crystal (BBO) with a cut-angle of 23.4∘, such that the two parts
are temporally split by approximately 220 fs, which corresponds to the initial
pulse length. This delay in time corresponds to a constant phase gradient over
wavelength Kopf  _et al._ (2021), such that we can write

$\displaystyle\ket{\Psi}^{exp}_{F}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}G(r)\,G(\lambda)\left(\ket{H}+e^{2i\kappa}\ket{V}\right),$
(34)

where $2\kappa=\pi\tau c/\lambda$. The state above can be conveniently
rewritten in the circular basis as

$\displaystyle\ket{\Psi}^{exp}_{F}$ $\displaystyle=$ $\displaystyle
G(r)\,G(\lambda)e^{i\kappa}\left[\cos(\kappa)\ket{R}-i\sin(\kappa)\ket{L}\right].$
(35)

For the wavelength range where $-\pi/2\leq\kappa\leq\pi/2$ applies, we find
that the polarization oscillates from left-circular to diagonal, to right-
circular, to anti-diagonal, and back to left-circular again. Hence, we obtain
a state that resembles the spectral vector beams described in Eq. (24) and the
main text by Eq. (5) for $\theta_{F}=\pi/2$ and $\phi_{F}=\pi/2$. To verify
the introduced spectral vector beam, we perform spectrally-resolved
polarization tomography using a set of half- and quarter-wave plates, a
polarizing beam splitter, and a spectrometer. The obtain the spectra for each
polarization along with the reconstructed wavelength-dependent polarization
pattern as shown in Fig. 7.

Figure 7: Experimentally measured spectral vector beam. Top: Recorded spectra
for different polarizations (data points connected by solid lines) and overall
spectral distribution of the whole laser pulse (dotted line). $S_{0}$ is
obtained by averaging over the three possible measurements (see Eq. (15a)),
each done by adding up the intensities in each basis. Bottom: Reconstructed
wavelength-dependent polarization pattern.

To shape the polarization in space, i.e., generate a spatial vector beam, we
use a so-called S-vortex plate, which is a half-wave plate where the optical
axis changes along the azimuthal angle $\varphi$. Thus, upon transmission
through the S-vortex plate, circularly polarized light not only flips its
polarization but also experiences an azimuthal phase gradient depending on the
initial polarization of the light.

Figure 8: Experimentally measured spatial vector beam. Left: Recorded
intensity images for different polarizations. Right: Reconstructed spatially
varying polarization pattern. The intensity in the background shows the
averaged $S_{0}$ intensity, i.e. the average over the sum of the intensity
images in each basis.

Here, the S-vortex plate imprints an azimuthal phase gradient up to 2$\pi$
such that we can replace $\ket{R}$ with $e^{-i\varphi}\ket{L}$ and $\ket{L}$
with $e^{i\varphi}\ket{R}$. First, we use the S-vortex plate on the laser beam
without temporal shaping and reconstruct the resulting polarization pattern in
space. We have a diagonal polarization state as described in Eq. (33), which
can be expressed in terms of circular polarizations as

$\displaystyle\ket{\Psi}^{exp}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}G(r)\,G(\lambda)\left(\ket{R}-i\ket{L}\right).$
(36)

After the S-vortex plate, the state is

$\displaystyle\ket{\Psi}^{exp}_{T}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}G(r)\,G(\lambda)\left(e^{-i\varphi}\ket{L}-ie^{i\varphi}\ket{R}\right),$
(37)

which is resembles the state described in Eq. (19) and the main text in Eq.
(3) for $\theta_{T}=\pi/2$ and $\phi_{T}=\pi/2$. The resulting polarization
pattern has an azimuthally varying linear polarization with a spiral
structure, as shown in the main text in Fig. 1 a), where different spatial
vector beams span the higher-order PS. We reconstruct the polarization pattern
with spatially-resolved polarization tomography using a set of half- and
quarter-wave plates, a polarizing beam splitter, and a CMOS camera. All images
obtained for different polarizations and the reconstructed polarization
patterns are shown in Fig. 8. The patterns nicely match the expected spatial
vector beams, with slight imperfections on the lower and upper part of the
beam due to experimental imperfections, which is for example visible by
comparing the projection on left and right circularly polarized light in Fig.
8.

After having demonstrated the two techniques to generate spectral and spatial
polarization patterns, we combine the two and generate a spatio-spectral
vector beam which contains all polarizations, i.e. a spatio-spectral Poincaré
beam. For this, we cascade the birefringent crystal with the S-vortex plate,
such that the state of Eq. (35) becomes

$\displaystyle\ket{\Psi}^{exp}_{TF}$ $\displaystyle=$ $\displaystyle
G(r)\,G(\lambda)e^{i\kappa}\left[\cos(\kappa)e^{-i\varphi}\ket{L}-i\sin(\kappa)e^{i\varphi}\ket{R}\right].$
(38)

After one reflection of a mirror, which induces a flip of the circular
polarizations and a sign-change in the azimuthal phase gradient, the state can
be written as

$\displaystyle\ket{\Psi}^{exp}_{TF}$ $\displaystyle=$ $\displaystyle
G(r)\,G(\lambda)e^{i\kappa}\left[\cos(\kappa)e^{i\varphi}\ket{R}-i\sin(\kappa)e^{-i\varphi}\ket{L}\right].$
(39)

As described in the main text, this state describes the spatio-spectral
Poincaré beam of Eq. (28) or Eq. (5) in the main text for $\theta_{TF}=\pi/2$
and $\phi_{TF}=\pi/2$. To show the resulting polarization pattern, we first
filter the beam’s polarization using a set of half- and quarter-wave plates
and a polarizing beam splitter. Subsequently, we filter for a specific angle
using the angular slit mask shown in Fig. 9 a). The mask is fabricated by
laser-cutting two slits into a black card board. Finally, we use a
spectrometer to measure the spectral intensity for different polarization and
mask setting, such that we can tomographically reconstruct the spatio-spectral
polarization pattern.

Figure 9: Experimental details and results. a) A photograph of the utilized
1-inch slit mask, cut out of black cardboard with a laser cutter. The slits
have a measured width of approximately 340 µm and a length of around 1.10 mm.
The distance between the slits is around 1.65 mm. b) Reconstructed
polarization ellipses depending on the angle of the slit mask and wavelength
which are measured with the setup described in the text. All reconstructed
ellipses are used in Fig. 4 b) of the main text and plotted with their
respective positions on the fundamental PS. Right- and left-handed
polarizations are depicted by red and green ellipses, respectively.

To show the resulting polarization pattern, we filter the beam first for a
specific angle using the angular slit mask shown in Fig. 9 a). The mask is
fabricated by laser-cutting two slits into a black card board. Subsequently,
we perform a spectrally-resolved polarization tomography using a set of half-
and quarter-wave plates, a polarizing beam splitter, and a spectrometer for
different spatial mask rotation angles. The results are given in Fig. 9 b) and
in the main text in Fig. 4. The results nicely demonstrate that it is possible
to generate a spatio-spectral Poincaré beam using minimally with only two
optical elements, a birefringent material and an S-vortex plate (given that
the input beam is linearly polarized in respect to the birefringent crystal
axis).