plus1sp

[2]Benjamin Perez-Garcia

# Parabolic-accelerating vector waves

Bo Zhao  Wang Da-Heng Collaborative Innovation Center, Heilongjiang Provincial
Key Laboratory of Quantum Manipulation and Control, Harbin University of
Science and Technology, Harbin 150080, China  Valeria Rodríguez-Fajardo
School of Physics, University of the Witwatersrand, Private Bag 3,
Johannesburg 2050, South Africa  Xiao-Bo Hu  Wang Da-Heng Collaborative
Innovation Center, Heilongjiang Provincial Key Laboratory of Quantum
Manipulation and Control, Harbin University of Science and Technology, Harbin
150080, China  Raul I. Hernandez-Aranda  Photonics and Mathematical Optics
Group, Tecnologico de Monterrey, Monterrey 64849, Mexico, e-mail:
<EMAIL_ADDRESS>Carmelo Rosales-Guzmán† Wang Da-Heng Collaborative Innovation
Center, Heilongjiang Provincial Key Laboratory of Quantum Manipulation and
Control, Harbin University of Science and Technology, Harbin 150080, China
Centro de Investigaciones en Óptica, A.C., Loma del Bosque 115, Colonia Lomas
del campestre, 37150 León, Gto., Mexico, e-mail<EMAIL_ADDRESS>
###### Abstract

Complex vector light fields have become a topic of late due to their exotic
features, such as their non–homogeneous transverse polarisation distributions
and the non-separable coupling between their spatial and polarisation degrees
of freedom. In general, vector beams propagate in free space along straight
lines, being the Airy-vector vortex beams the only known exception. Here, we
introduce a new family of vector beams that exhibit novel properties that have
not been observed before, such as their ability to freely accelerate along
parabolic trajectories. In addition, their transverse polarisation
distribution only contains polarisation states oriented at exactly the same
angle but of different ellipticity. We anticipate that these novel vector
beams might not only find applications in fields such as optical manipulation,
microscopy or laser material processing, but extend to others.

##  1 Introduction

The ingrained notion that light travels along a straight line was first defied
in 2007 by Siviloglou et al. [1], who introduced a novel kind of light beam
with the ability to self-accelerate along a parabolic trajectory upon free
space propagation [2]. Such light beams, known as Airy beams, are natural
solutions of the normalised paraxial wave equation. Crucially, even though
they seem to propagate along parabolic trajectories, their centroid propagates
along straight lines, in accordance to the electromagnetic momentum
conservation law. Along with the discovery of Airy beams, their fascinating
properties prompted the development of novel applications, which have impacted
a wide diversity of fields, such as, optical manipulation, microscopy, laser
material processing, among others (see for example [3] for an extensive
review). More importantly, the experimental demonstration of Airy beams
ignited the quest for other kinds of accelerating beams [4, 5, 6, 7, 8, 9, 10,
11]. Of particular interest is the case of accelerating parabolic beams, which
form a complete and infinite orthogonal family of solutions of the normalised
paraxial wave equation [4, 12]. Such beams also propagate in free space in a
non diffracting way describing parabolic trajectories.

Noteworthy, most of the work carried out with accelerating beams has only
considered the case of homogeneously polarised beams, while the manipulation
of other degrees of freedom is gaining popularity, giving rise to a more
general class of beams generally known as structured light fields. This is the
case of complex vector light beams, classically-entangled in their spatial and
polarisation degrees of freedom, which feature a non-homogeneous polarisation
distribution across the transverse plane[13, 14, 15]. Such beams have gained
popularity in recent time not only due to their unique traits, such as their
quantum-like non separability [16, 17, 18, 19, 20], but also due to the many
applications they are pioneering [21, 22, 23, 24, 25, 26]. In vector beams,
their spatial and polarisation degrees of freedom are coupled in a non-
separable way, which generates the non-homogeneous polarisation distribution.
Importantly, while the polarisation degree of freedom is restricted to a two-
dimensional space, the spatial one is not, as any of the unbounded solution
sets of the wave equation, either in its exact or paraxial version, can be
used. Examples of vector beams that have been experimentally demonstrated are
Bessel, Laguerre-, Ince- and Mathieu-Gauss beams, amongst others, all of which
propagate along straight trajectories [27, 28, 29, 30, 31]. Along this line,
previous works have demonstrated the acceleration of vectorial fields, in
which case, their polarisation structures rotates around the optical axis,
while still propagate along straight lines [32]. Perhaps the only known case
of a vector beam capable to propagate along a parabolic trajectory is the
Airy-vortex vector beam [33].

We propose and experimentally demonstrate a new family of vector beams, which
we term Accelerating Vector Waves (AVWs), that are non-separable weighted
superpositions of the polarisation and spatial degrees of freedom encoded in
the orthogonal set of accelerating waves. These beams exhibit two interesting
properties, namely, that their non-homogeneous polarisation distributions
propagate in free space along parabolic trajectories maintaining a maximum
degree of coupling, and that, even though the non-homogeneous transverse
polarisation distribution of an individual AVW contains different states of
elliptical polarisation, all of them are located on a great circle on the
Poincaré sphere representation for polarisation. Here, we start by describing
these beams theoretically, then move to their implementation in the
laboratory, and finally show experimental results to showcase their novel
features. Due to their intriguing properties, we expect AVWs will attract the
wide interest of the optical community, stemming not only from their potential
applications but also from their fundamental aspects.

##  2 Theory

Accelerating parabolic waves (APWs) are solutions of the paraxial wave
equation in parabolic coordinates. They are non-diffracting beams that
accelerate during free-space propagation. Their experimentally realisable
finite–energy form is given by [4]

$\displaystyle\phi_{n}(\eta,\xi,z)=$
$\displaystyle\exp[i(z/2k\kappa^{2}-ia)(\eta^{2}-\xi^{2})/2]\times$ (1)
$\displaystyle\exp[i(z/2k\kappa^{2}-ia)^{3}/3]\Theta_{n}(\eta)\Theta_{n}(i\xi),$

where the parabolic coordinates $(\eta,\leavevmode\nobreak\ \xi)$ are related
to the Cartesian coordinates by $(\eta^{2}/2-\xi^{2}/2,\leavevmode\nobreak\
\eta\xi)=(x/\kappa-(z/2\kappa^{2})^{2}+iaz/k\kappa^{2},\leavevmode\nobreak\
y/\kappa)$, $\kappa$ is a transverse scale parameter, $k$ is the wave number,
$z$ is the propagation distance and $a$ is parameter that controls the
exponential aperture of the beam at $z=0$. The functions $\Theta_{n}(\cdot)$
correspond to the solutions of the differential equation

$\displaystyle\left(-\frac{1}{2}\frac{d^{\text{2}}}{d\eta^{\text{2}}}+\frac{\eta^{4}}{4}\right)N(\eta)=E\>N(\eta),$
(2)

which corresponds to the one–dimensional Schrödinger equation with potential
$V(\eta)=\eta^{4}/4$ (known as quartic potential) and $m=\hbar=1$ [34].
Importantly, the eigen-solutions $\Theta_{n}$ ($n\in\mathbb{N}$) of Eq. 2 form
an orthogonal set of functions, whose parity is governed by $n$. Since these
eigen-solutions cannot be expressed in a closed form, a suitable numerical
method must be employed to obtain them [35]. In particular, we are interested
in square integrable eigen-solutions of Eq. 2. Fig. 1(a) shows the intensity
profiles of the scalar APWs for $n=\\{0,1,2,3\\}$. As can be seen, for $n=0$
the intensity profile of the beam contains only one main lobe of maximum
intensity and additional subsequent lobes of decaying intensity. In general,
for $n>0$ the intensity profile is formed by $n+1$ lobes of maximum intensity.

Fig. 1: (a) Intensity profiles of accelerating parabolic scalar waves of
orders $n=\\{0,1,2,3\\}$. (b) Schematic representation of the non-separable
superposition of two orthogonal scalar modes carrying orthogonal polarisations
to generate an accelerating vector wave ${\bf AVW}_{n,m}(\eta,\xi,z)$. (c)
Illustration of the free-space propagation trajectory of an AVW along the $z$
axis. Green and orange ellipses represent right- and left-handed circular
polarisation, respectively, and white lines represent linear polarisation.

Mathematically, the accelerating Vector Waves (AVWs) are constructed as a
superposition of two scalar APWs with orthogonal polarisations, such that at
the $z$–plane and for indices $n,m$ they are given by

$\displaystyle{\bf AVW}_{n,m}(\eta,\xi,z)=$
$\displaystyle\cos\alpha\>\phi_{n}(\eta,\xi,z)\hat{{\mathbf{e}}}_{1}(\theta,\varphi)+$
$\displaystyle\sin\alpha\exp(i\beta)\>\phi_{m}(\eta,\xi,z)\hat{{\mathbf{e}}}_{2}(\theta,\varphi),$
(3)

where the weighting factor $\alpha\in[0,\pi/2]$ allows the field to change
from scalar to vector, and the parameter $\beta=[0,\pi]$ controls the inter-
modal phase. The basis vectors

$\displaystyle\hat{{\mathbf{e}}}_{1}(\theta,\varphi)=\cos(\theta/2)\hat{{\mathbf{e}}}_{R}+\exp(i\varphi)\sin(\theta/2)\hat{{\mathbf{e}}}_{L},$
(4)
$\displaystyle\hat{{\mathbf{e}}}_{2}(\theta,\varphi)=\sin(\theta/2)\hat{{\mathbf{e}}}_{R}-\exp(i\varphi)\cos(\theta/2)\hat{{\mathbf{e}}}_{L},$
(5)

represent the general elliptical polarisation basis. Note that we can obtain
the left/right-handed circular polarisation basis by setting $\theta=\pi$ and
$\varphi=0$ and the horizontal/vertical basis with $\theta=\pi/2$ and
$\varphi=0$. Without loss of generality, here we will restrict our results to
the circular polarisation basis, only briefly mentioning some theoretical
examples of the horizontal/vertical basis. Fig. 1(b) illustrates conceptually
the above description for the specific case ${\bf AVW}_{0,1}(\eta,\xi,z)$ with
$\alpha=\pi/4$ and $\beta=0$ as polarisation distributions overlay onto their
corresponding intensity profiles. Left and middle panels show the two scalar
modes $\theta\>\phi_{n}(\eta,\xi,z)\hat{{\mathbf{e}}}_{L}$ and
$\phi_{m}(\eta,\xi,z)\hat{{\mathbf{e}}}_{R}$ with Right Circular polarisation
(RCP) and Left Circular Polarisation (LCP), respectively, represented by green
and orange ellipses for the first and second case respectively. Notice the
intensity patters of the scalar modes are different, as required to obtain
vector modes. In a similar way, the right panel presents the non-separable
superposition of both scalar modes (Eq. 3). The parabolic trajectory described
by AVWs can be seen schematically in Fig. 1(c) for ${\bf
AVW}_{2,3}(\eta,\xi,z)$ propagating along the $z$ axis. Mathematically, this
is expressed in the transverse shift $y_{s}=[z/(2k)]^{2}/\kappa^{3}$ [36],
which is independent of the indices $n$, $m$.

##  3 Experimental details

We implemented the AVW described above using a Digital Micromirror Device
(DMD) and following the technique that we proposed and fully characterised in
a previous article [37]. This device is polarisation-insensitive, very
flexible and versatile, allowing the generation of vector modes with arbitrary
spatial distributions, such as elliptical or parabolic [37, 31, 30, 38]. In
essence, a DMD is illuminated with two modes carrying orthogonal
polarisations, impinging at slightly different angles but exactly at the
geometric centre of the hologram displayed on the DMD. The hologram contains a
superposition of the two transmittance functions that generate the
constituting scalar modes of Eq. 3, each with an additional unique linear
spatial grating that redirects the mode along a specific angle, and whose
periods are carefully chosen to guarantee both generated beams co-propagate
along the same axis, where the desired vector beam is created. Transmittance
functions are calculated as the inverse Fourier transform of the desired modes
$\phi_{n}(\eta,\xi,z)$ [36, 12], thus we add a lens in a $2f$ configuration,
where $f$ is the focal length of the lens ($f=200$ mm in our case), and
measure at its back focal plane. Intensity patterns of the generated beams
were captured with a high resolution CCD camera (FL3-U3-120S3C-C with
resolution of $4000\times 3000$ pixels and a pixel size of 1.55 $\mu$m).
Polarisation reconstruction was achieved through Stokes polarimetry, using a
set of intensity measurements as detailed in [31]. Fig. 2(a) show s an example
of the experimentally measured Stokes parameters $S_{0}$, $S_{1}$, $S_{2}$ and
$S_{3}$ for the specific mode ${\bf AVW}_{1,2}(\eta,\xi,z=0)$. Reconstructed
intensity and polarisation distributions of a set of representative examples
of the experimentally generated ${\bf AVW}_{n,m}(\eta,\xi,z=0)$ modes using
the circular polarisation basis are presented in Fig. 2(b), both for the
theory (top row) and the experiment (bottom row). Notice the high similarity
between the latter, demonstrating the good performance of our generation
method.

Fig. 2: (a) Example of experimentally measured Stokes parameters $S_{0}$,
$S_{1}$, $S_{2}$ and $S_{3}$ for ${\bf AVW}_{1,2}(\eta,\xi,z=0)$. (b)
Theoretical and experimental reconstructed intensity and polarisation
distribution from the Stokes parameters for the modes ${\bf
AVW}_{0,2}(\eta,\xi,0)$, ${\bf AVW}_{1,2}(\eta,\xi,0)$, ${\bf
AVW}_{1,3}(\eta,\xi,0)$ and ${\bf AVW}_{2,3}(\eta,\xi,0)$.

##  4 Results and discussion

Fig. 3: Shift of the $y$-coordinate of the maximum intensity lobe position for
three AVWs upon free-space propagation. The continuous curve represents the
expected position predicted by theory, whereas the data points correspond to
experiment for the cases ${\bf AVW}_{0,2}(\eta,\xi,z)$ (squares), ${\bf
AVW}_{1,2}(\eta,\xi,z)$ (triangles) and ${\bf AVW}_{2,3}(\eta,\xi,z)$
(circles). Notice that all three cases accelerate in an identical way. Insets
show the transverse polarisation distribution overlapped with the intensity
profile of the ${\bf AVW}_{1,2}(\eta,\xi,z)$ at three different planes.

The vector modes described by equation 3 and shown in Fig. 2 propagate along
parabolic trajectories maintaining not only their intensity and polarisation
distribution but also a maximum coupling between both. We corroborated this by
tracking the transverse spatial coordinates $(x,y)$ of one of the lobes of
maximum intensity as function of their propagation distance $z$. We observed
that while the $x$ coordinate remains almost constant, the $y$ coordinate
shifts following a quadratic trend. Figure 3, in which the coordinate $y$ is
plotted against the propagation distance $z$, clearly shows such behaviour for
a representative set of AVWs given by ${\bf AVW}_{0,2}(\eta,\xi,z)$, ${\bf
AVW}_{1,2}(\eta,\xi,z)$ and ${\bf AVW}_{2,3}(\eta,\xi,z)$. Since all modes
shown where generated with the same initial parameters $k$ and $\kappa$, they
accelerated in an identical way. Insets show examples of the polarisation
distribution overlapped with the intensity distribution at three propagation
distances $z=0$ mm, $z=20$ mm and $z=60$ mm for the mode ${\bf
AVW}_{1,2}(\eta,\xi,z)$.

As mentioned earlier, AVWs can be generated with arbitrary degrees of non-
separability, evolving from scalar to vector, via the parameter $\alpha$ (see
Eq. 3). More precisely, as $\alpha$ increases from 0 to $\pi/4$ the mode
changes monotonically from a pure scalar mode with right-handed circular
polarisation ($\alpha=0$) to a pure scalar mode with left-handed circular
polarisation ($\alpha=\pi/2$), passing through a pure vector mode
($\alpha=\pi/4$). Intermediate values of $\alpha$ produce vector modes with
intermediate degrees of non-separability, which can be measured through the
concurrence or Vector Quality Factor (VQF), which is a measure borrowed from
quantum mechanics that allows to quantify the degree of coupling between the
spatial and polarisation degrees of freedom [39, 40, 41]. Experimentally, the
VQF can be quantified directly from the Stokes parameters as [42, 43],

$VQF=\sqrt{1-\left(\frac{\mathbb{S}_{1}}{\mathbb{S}_{0}}\right)^{2}-\left(\frac{\mathbb{S}_{2}}{\mathbb{S}_{0}}\right)^{2}-\left(\frac{\mathbb{S}_{3}}{\mathbb{S}_{0}}\right)^{2}},$
(6)

where $\mathbb{S}_{i}$ ($i=0,1,2,3$) is a number that results from integrating
the Stokes parameters $S_{i}$ over the entire transverse profile, i. e.,
$\mathbb{S}_{i}=\iint_{-\infty}^{\infty}S_{i}dA$. Figure 4 shows a
representative example of the VQF as function of $\alpha$ for the specific
case ${\bf AVW}_{2,3}(\eta,\xi,z=0)$. As expected, the VQF increases from 0 to
1, as $\alpha$ increases from 0 to $\pi/4$ and then it decreases back to zero,
as $\alpha$ reaches the value $\pi/2$. Insets show the intensity profile
overlapped with the polarisation distribution for three key values, namely
$\alpha=0,\pi/4$ and $\pi/2$.

Fig. 4: Vector Quality Factor (VQF) as a function of the weighing coefficient
$\alpha\in[0,\pi/2]$ for the mode ${\bf AVW}_{2,3}(\eta,\xi,z=0)$. Insets show
the intensity and polarisation distributions for $\alpha=0,\pi/4$, and
$\pi/2$.

Finally, we analyse the polarisation states distribution of AVWs on the
Poincaré sphere. It turns out that for a particular AVW all of them are mapped
onto a great circle. Fig. 5(a) shows numerical simulations in the circular
(top row) and horizontal/vertical linear (bottom row) bases. It can be seen
that in the first case, the great circles intersect the North and South poles,
meaning all polarisation states in the AVW are oriented at exactly the same
polarisation angle and only differ in their ellipticity, containing all
polarisation states from circular to linear. Interestingly, a change in the
inter-modal phase $\beta$ originates a rotation of the great circle around the
$S_{3}$ axis (Fig. 5(a) top left panel), and a change in the weighting
coefficient $\alpha$ causes the polarisation states location from complete to
incomplete great circles, in such a way that the arc length is proportional to
it (Fig. 5(a) top right panel). Similarly, in the case of horizontal/vertical
linear polarisation basis (Fig. 5(a) bottom row), the great circles intersect
the cross points between the $S_{1}$ axis and the sphere, and rotate around
the $S_{1}$ axis when changing $\beta$. The effect of $\alpha$ is as in the
circular basis. For instance, for $\alpha=\pi/2$ and $\beta=0$, the
polarisation distribution is mapped to the equator of the Poincaré sphere. For
the case of circular polarisation basis, we corroborated this experimentally,
as shown in Fig. 5(b), that shows a remarkable similarity to its theoretical
counterpart.

Fig. 5: Geometric representation on the Poincaré sphere of the transverse
polarisation distribution of the vector mode ${\bf AVW}_{0,2}(\eta,\xi,z=0)$.
Left column shows modes with increasing values of the inter-modal phase,
namely $\beta=0$ (blue), $\beta=pi/4$ (purple), $\beta=pi/4$ (green) and
$\beta=3pi/4$ (yellow). Right column shows modes with different weighting
coefficients, namely $\alpha=\pi/12,\pi/6,\pi/3$ and $4\pi/12$, with the same
inter-modal phases as in the left column for the sake of better visibility.
(a) Theoretical results for the case of AVW in the circular (top row) and
linear (bottom row) polarisation basis. (b) Corresponding experimental results
for the case of the circular basis.

In summary, we have introduced theoretically and demonstrated experimentally a
new family of vector beams with the ability to accelerate along parabolic
trajectories upon free space propagation. Such accelerating beams differ quite
dramatically from common vector beams, which always propagate along straight
trajectories. These families of vector beams are constructed as a weighted
superposition of the spatial and polarisation degrees of freedom carrying an
inter-modal phase. To generate them, the spatial degree of freedom is encoded
in a set of orthogonal solutions of the one–dimensional Schrödinger equation
with a quartic potential, known as accelerating waves. An important feature of
such modes is their propagation-invariant spatial and polarisation structures,
as we corroborated experimentally. Further, the weighting coefficient allows
tuning from purely scalar to completely vectorial, passing through
intermediate states, which was also corroborated experimentally using the
well-known measure of concurrence from quantum mechanics adapted for vector
beams. Another important feature of these accelerating vector modes lies in
their transverse polarisation distribution, which is mapped onto great circles
on the Poincaré sphere. In particular, in the circular polarisation basis the
great circles intersect the North and South poles, and contain states of
polarisation from linear to circular, all with the same ellipticity angle. For
comparison, cylindrical vector modes are mapped either to an equatorial line
or to the whole Poincaré sphere (known as full-Poincaré modes). Noteworthy,
the inter-modal phase allows to rotate the circle of polarisation around the
Poincaré sphere, leaving the points on North and South poles fixed. Given
their interesting properties, we expect AVWs to find applications in fields
such as optical manipulation, laser material processing, among others.

Consejo Nacional de Ciencia y Tecnología (PN2016-3140); National Natural
Science Foundation of China (61975047).

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