# Reference-less complex wavefields characterization with a high-resolution
wavefront sensor

Tengfei Wu Université de Paris, SPPIN – Saints-Pères Paris Institute for
Neurosciences, CNRS, 75006 Paris, France  Sorbonne Université, CNRS, INSERM,
Institut de la Vision, 17 Rue Moreau, 75012 Paris, France  Université de
Paris, 75006 Paris, France  Pascal Berto Sorbonne Université, CNRS, INSERM,
Institut de la Vision, 17 Rue Moreau, 75012 Paris, France  Université de
Paris, 75006 Paris, France  Marc Guillon Université de Paris, SPPIN – Saints-
Pères Paris Institute for Neurosciences, CNRS, 75006 Paris, France  Université
de Paris, 75006 Paris, France  Institut Universitaire de France (IUF), Paris,
France<EMAIL_ADDRESS>
Wavefront sensing is a widely-used non-interferometric, single-shot, and
quantitative technique providing the spatial-phase of a beam. The phase is
obtained by integrating the measured wavefront gradient. Complex and random
wavefields intrinsically contain a high density of singular phase structures
(optical vortices) associated with non-conservative gradients making this
integration step especially delicate. Here, using a high-resolution wavefront
sensor, we demonstrate experimentally a systematic approach for achieving the
complete and quantitative reconstruction of complex wavefronts. Based on the
Stokes’ theorem, we propose an image segmentation algorithm to provide an
accurate determination of the charge and location of optical vortices. This
technique is expected to benefit to several fields requiring complex media
characterization.

Quantitative phase mesurements of complex wavefields is at the basis of many
applications including bio-imaging [1, 2, 3], fibroscopy [4, 5], diffractive
tomography [6], astronomy [7], and more generally for imaging through
scattering media by phase conjugation [8, 9]. When a coherent reference beam
is available, complex wavefields may typically be measured using digital
holography [10, 11]. Alternatively, several techniques have been developed for
cases when no coherent reference beam is available, such as common-path
interferometers [12, 13, 14, 15, 16], point diffraction interferometry [17],
Shack-Hartman wavefront sensors (WFS) [18], curvature sensing [19], light-
field imaging [20], phase diversity approaches [21], or phase retrieval
algorithms [22, 23, 24]. All these techniques have their own advantages and
try to optimize the compromise between ease of use and measurement
reliability. Among the latter techniques, WFS generally consist in a so-called
“Hartmann mask” simply placed before a camera and have the specific advantages
to be a robust single-shot technique offering a great ease of use. They have
thus been the most used systems to measure distorted wavefronts arising from
random refractive index inhomogeneities due to the atmosphere in astronomy, or
due to tissues in bio-imaging. The main limitation associated to Shack-
Hartmann WFS has long been the limited resolution associated with the low
maximum density of micro-lens arrays. The advent of high-resolution WFS,
either based on a modified Hartmann mask [25] or on a thin diffuser [26, 27,
28, 29], has represented an important step towards making WFS a promising
alternative for measuring complex spatial phase profiles [30]. However,
despite their promising potential to measure complex WFs, high-resolution WFS
have mostly been used, so far, for measuring smooth WF distortions, such as
optical aberrations [1, 2] (typically projected onto Zernike polynomials [31])
or optical-path-length profiles of thin and transparent biological samples
[25, 27]. In contrast, complex and scattered wavefields comprise a high
spatial density of phase singularities, namely optical vortices [32], whose
separation distance close to nucleation/annihilation centers can be
arbitrarily small [33, 34, 35]. These singular spiral phase structures are
associated with non-conservative gradient fields that are especially
challenging for WFS which, unlike interferometric methods, do not provide a
direct measurement of the phase, but only measure phase gradients ${\bf g}$
(_i.e._ the transverse component of the local wavevector). The problem of
phase-spirals integration has appeared since the early ages of adaptive optics
in astronomy [36]. In this context, it has been shown that neglecting “branch-
cuts” significantly degrades adaptive-optics performances [7]. The
identification of spiral phase structures in phase gradient maps ${\bf g}$
relies on their Helmholtz decomposition (HD) [37, 38]. According to
Helmholtz’s theorem [39], the vector field ${\bf g}$ can be split into an
irrotational (curl-free) component and a solenoidal (divergence-free or
rotational) component [40, 37, 7, 41]. Most of current integration techniques
for WFS basically consist in computing $\nabla\cdot{\bf g}$, so implicitly
canceling the solenoidal component of the vector field [42]. The vector
potential associated with the solenoidal component (so-called “branch-point”
potential [40, 37]) exhibits peaks at vortex locations, so allowing their
localization and reconstruction [7]. However, although WFS have demonstrated
their ability to detect and characterize optical vortices [36, 43, 44],
complete wavefront reconstruction by vortex-gradients integration has given
rise to many works considering only simple experimental cases involving a
single vortex [45, 46, 47], or more vortices but in numerical simulations [36,
40, 7, 48, 45]. We believe that the high density of singular phase structures
in complex wavefields has probably been the main obstacle for measuring
experimentally complex wavefronts with a WFS. In WF shaping experiments,
neglecting a single optical vortex is equivalent to adding a complementary
spiral phase mask, which has been described as yielding a two-dimensional
Hilbert transform of the field [49]. For complex speckled wavefields, such a
single spiral transform induces a major change in patterns since resulting in
an inversion of intensity contrasts [50, 51]. Here, we propose a robust and
systematic numerical data-processing technique and demonstrate the
experimental quantitative reconstruction of complex wavefronts containing up
to $133$ optical vortices with a high-resolution WFS based on a $1.3~{}{\rm
MP}$ camera [27]. The camera sensor is thus similar to those typically used by
regular WFS based on micro-lens arrays. Wavefront reconstruction was achieved
using a segmentation algorithm that optimizes the identification of nearby
optical vortices. According to the Stokes’ theorem [40], segmentation
optimizes the computation areas to measure the charge of vortices, based on
the integration of the “branch-point” potential.

Fig. 1:  Experimental vortex WF sensing. A phase profile containing optical
vortices (a) is addressed onto a spatial light modulator (SLM) (b) illuminated
by a colimated laser beam and imaged onto a high-resolution wavefront sensor
(WFS) with a telescope ($L_{1}$, $L_{2}$). Considering only the irrotational
component of the gradient field detected by the WFS leads to an erroneous WF
lacking optical vortices (c), unlike full Helmholtz decomposition (HD) of the
gradient field (d).

To summarize, the problem to rebuild singular WFs is mostly threefold. First,
the phase-gradient map ${\bf g}$ measured by a WFS is a non-conservative
vector field defined over a multiply connected domain. Direct spatial
integration is thus not possible since the integral value depends on the line-
path. Second, optical vortices are associated with an infinite phase-gradient
${\bf g}$ at the singularity locations, which appears as critically
incompatible with WFS. Third, WFS may imprecisely measure strongly fluctuating
gradients around anisotropic vortices, leading to inaccurate vortex
characterization, in particular their spiral spectrum [52]. Such cases
especially appear in random wavefields, wherein pairs of vortices of opposit
charges are frequently close to one another [33]. This configuration results
in large phase gradients in between vortices and makes the computation of the
charges (the circulation of ${\bf g}$) especially delicate. The image
segmentation we propose more specifically addresses this latter challenge
[53].

By way of illustration, a phase pattern exhibiting optical vortices has been
designed (Fig. 1a) and addressed to a phase-only spatial light modulator (SLM)
(Hamamatsu, LCOS-X10468-01), illuminated by a spatially filtered, polarized,
and collimated laser beam (Fig. 1b). The SLM allows displaying patterns
exhibiting both smooth local WF distortions (such as lenses for the eyes and
the face contour for instance) as well as optical vortices of any topological
charge (left- and right-handed optical vortices at the tips of the curling
mustache). The phase-gradient map is detected with a custom-built high-
resolution WFS comprising $\simeq 2\times 10^{4}$ phase pixels and described
in Ref. [27]. The WFS is conjugated to the SLM with a Galilean telescope in a
$4-f$ configuration. The full Helmholtz decomposition (HD) of the gradient
vector-field detected by the WFS can be achieved according to:

${\bf g}=\nabla\varphi_{ir}+\nabla\times{\bf A}$ (1)

so splitting appart the irrotational contribution of the regular phase
$\varphi_{ir}$ and the solenoidal contribution of a vector potential ${\bf
A}$. The sought-for complete phase profile $\varphi$, whose gradient-field is
${\bf g}$, can then be written as $\varphi=\varphi_{ir}+\varphi_{s}$. Typical
direct numerical integration [42] yields the regular WF shown in Fig. 1c,
missing phase singularities because the non-conservative (solenoidal)
contribution to the WF-gradient has been ignored. The singular phase
contribution $\varphi_{s}$ (or “hidden phase” [40]), is defined over a
multiply-connected domain, and satisfies $\nabla\varphi_{s}=\nabla\times{\bf
A}$. Solving this latter equation then allows proper reconstruction of the WF
(Fig. 1d).

Fig. 2:  Principle of full WF reconstruction. The divergence and the curl of
the phase gradient map ${\bf g}$ (a) are computed to extract the irrotational
phase $\varphi_{ir}$ (b) and the solenoidal phase $\varphi_{s}$ (c). Double
space integration of $\nabla\cdot{\bf g}$ yields the irrotational phase (a
parabola in (b)). The curl of ${\bf g}$ yields $-\Delta{\bf A}$ where ${\bf
A}$ is the potential vector of the Helmholtz decomposition. Image segmentation
and weighted centroid computation of the peaks in $-\Delta{\bf A}$ then allows
reconstructing a Dirac-like distribution whose convolution by a spiral phase
profile yields $\varphi_{s}$. The complete phase $\varphi$ is finally rebuilt
by summing the two components $\varphi_{ir}$ and $\varphi_{s}$ (d).

We now detail the steps allowing us to achieve rigorous HD (See also Supp.
Mat.). First, taking the curl of Eq. (1), it appears that the potential vector
${\bf A}$ is a solution of the equation: $\nabla\times{\bf
g}=\nabla(\nabla\cdot{\bf A})-\Delta{\bf A}$, where "$\Delta=\nabla^{2}$"
denotes the Laplacian operator. Determining the potential vector thus requires
fixing a gauge. The Coulomb gauge $\nabla\cdot{\bf A}=0$ is chosen here for
obvious convenience [40, 37, 7]. Since ${\bf g}$ is a two-dimensional vector
field (in say $x$, $y$ plane), we may then write ${\bf A}=A_{z}{\bf e_{z}}$
without loss of generality. Second, introducing the circular vector
$\boldsymbol{\sigma_{-}}={\bf e_{x}}-i{\bf e_{y}}$, simple manipulation of Eq.
(1) yields: ${\bf
g}\cdot\boldsymbol{\sigma_{-}}=g_{x}+ig_{y}=(\partial_{x}+i\partial_{y})(\varphi_{ir}-iA_{z})$.
The HD can then be efficiently achieved numerically thanks to a single
computation step:

$\varphi_{ir}-iA_{z}={\cal F}^{-1}\left\\{\frac{{\cal F}\left[{\bf
g}\cdot\boldsymbol{\sigma_{-}}\right]}{i{\bf
k}\cdot\boldsymbol{\sigma_{-}}}\right\\}$ (2)

where ${\bf k}$ stands for the two-dimentional coordinate vector in the
reciprocal Fourier space. The regular phase component $\varphi_{ir}$ is thus
recovered the same way as previously proposed [42, 54] (Fig. 2b). Third, for
complete HD over a bounded domain, the contribution of a so-called additional
“harmonic” (or translation) term ${\bf h}$ must be considered when the flow
through the boundary is not zero [39]. This translation term, accounting for a
global tip/tilt of the WF, is both curl-free and divergence-free. If computing
derivation and integration through discrete Fourier transforms without care,
implicit periodic boundary conditions cancels out this term. Here we thus
retrieve the term ${\bf h}$ by symmetrizing the gradient field ${\bf g}$ prior
to gradient integration (Eq. (2)) as performed in Ref. [54], which
conveniently includes ${\bf h}$ in the curl-free component
($\nabla\varphi_{ir}$) of the HD. The divergence-free component requires
futher processing steps to obtain the singular phase pattern $\varphi_{s}$
from the potential-vector component $A_{z}$, as detailed hereafter.

Fig. 3:  Reconstruction of phase spirals of various topological charges $n$
from $-10$ to $+10$. The potential vector ${\bf A}$ of a spiral phase of
topological charge $1$ is, in theory, the Green function of Laplace equation.
In practice, $\Delta A_{z}$ exhibit peaks at the vortex location (a) whose
size depend on the topological charge of the corresponding vortex (in (a),
$\Delta A_{z}$ has been slightly filtered for the sake of readability). After
segmentation, the integral computation of this peak yields the expected value
$2\pi n$ (with a $3\%$ precision) (b). Phase profiles are then rebuilt (c).

A vortex is essentially characterized by the circulation of the vector flow
around the singularity, namely its topological charge (or winding number)
defined from the circulation of ${\bf g}$ around the vortex. Applying Stokes’
theorem to the definition of $n$ yields:

$n=\frac{1}{2\pi}\displaystyle{\oint_{\cal C}}{\bf g}\cdot{\rm
d}\boldsymbol{\ell}=-\frac{1}{2\pi}\displaystyle{\int_{\cal S}}\Delta
A_{z}{\rm d}S$ (3)

Reducing the contour length ${\cal C}$ (and so the enclosed surface ${\cal
S}$) to zero, it appears that $-A_{z}/(2\pi n)$ is the Green function of the
two-dimensional Laplace equation. In theory, $-\Delta A_{z}/(2\pi n)$ is thus
a Dirac distribution, making it easy to identify optical vortices [40, 37, 44]
(see Fig. 2c). In principle, the corresponding sought-for singular phase
component $\varphi_{s}$ could then be simply obtained by convolving $-\Delta
A_{z}/(2\pi n)$ with a single $+1$ optical vortex. However, in practice,
rebuilding $\varphi_{s}$ this way is not possible. The main difficulty is that
the experimental $-\Delta A_{z}/(2\pi n)$ map is not a perfect Dirac
distribution (or a single-pixeled non-zero data map) for three main reasons:
first, experimental data are affected by noise, second, ${\bf g}$ is filtered
by the optical transfer function of the WFS and third, optical vortices are
associated with a vanishing intensity that compromises accurate gradient
measurement in their vicinity. As detailed in Ref. [27], the optical transfer
function of a WFS is especially limited by the non-overlapping condition,
which imposes a maximum magnitude for the eigenvalues of the Jacobi matrix of
${\bf g}$ (_i.e._ the Hessian matrix of $\varphi$). As a first consequence,
the large curvatures of the phase component $\varphi_{r}$ (_i.e._ its second
derivatives) may be underestimated. As a second consequence, the diverging
magnitude of ${\bf g}$ (as $1/r$) prevents its proper estimation at close
distances $r$ from the optical vortex location, as well as the estimation of
the Hessian coefficients $\partial_{x}g_{y}$ and $\partial_{y}g_{x}$.
Therefore, the measurement of the vector potential $A_{z}$ is wrong in the
vicinity of the vortex center and the obtained peak is not single-pixeled
(Fig. 2c).

In experiments, the precise identification of the location and the charge of
optical vortices from the computed $\Delta A_{z}$-map thus demands an image
processing step. Notably, the circulation of ${\bf g}$ in Eq. (3) can yield an
accurate measure of the vortex charge provided that the contour ${\cal C}$
surrounds the vortex at a large enough distance, so that ${\bf g}$ is small
enough to be accurately measured by the WFS. Consequently, although the
estimate of $\Delta A_{z}$ is wrong in the vicinity of the vortex, the peak
integral achieved over a large enough surface ${\cal S}$ (enclosed by ${\cal
C}$) provides the proper charge, under the Stokes’ theorem (Eq. (3)). Vortices
may be characterized by a simple peak detection algorithm, after
regularization of the $\Delta A_{z}$-map with a low-pass filter [38]. However,
this solution is not robust since it requires setting manually an integration
radius. Such an adjustment of the integration radius is all the more
impractical when considering two typical cases: either high densities of
optical vortices potentially located at arbitrarily small distances (see Fig.
4d), or high-order vortices, associated with charge-dependent peak-sizes in
the $\Delta A_{z}$-maps (see Fig. 3a). In contrast, the solution we propose
consists in performing an image segmentation that automatically optimizes the
integration surface by adjusting it to the size and density of peaks. The main
processing steps are summarized in Fig. 2c (See also Supp. Mat.). The
segmentation of the $\Delta A_{z}$-map is achieved in two substeps. First,
$|\Delta A_{z}|$ is filtered in order to avoid oversegmentation, with a
Gaussian filter $G$ whose waist is chosen equal to the WFS resolution [27]
($\simeq 8$ camera pixels). Second, a watershed operation (matlab®, _Image
Processing Toolbox_) is applied to $-G\ast|\Delta A_{z}|$. Noteworthy, we
observed that segmentation of $-G\ast|\Delta A_{z}|$ rather than
$-|G\ast\Delta A_{z}|$ more efficiently cancels out pairs of vortices too
close to be resolved by the WFS. The resulting segments are thus structured
according to extrema of $\Delta A_{z}$, the latters corresponding to positive
and negative vortices locations. Integration and weighted-centroid computation
of the unfiltered $\Delta A_{z}$ image over each segment yields the charge and
the precise location of each vortex. To obtain $\varphi_{s}$, a Dirac-like
vortex map is rebuilt based on the result of the former step, and convolved by
a $+1$ spiral phase mask (see Fig. 2c). Finally, the complete phase
reconstruction $\varphi=\varphi_{ir}+\varphi_{s}$ is computed and wrapped
between $0$ and $2\pi$ (Fig. 2d). Even if the proposed image processing
algorithm includes a filtering operation, it is of a very different nature
from previously suggested regularization approaches [38], because the filter
size is set according to the WFS resolution (to avoid oversegmentation) and
not to the vortices to be characterized. Oversegmentation was observed to
degrade charge-measurement reliability and lengthen the processing time,
measured to be $0.54s$ on an Intel® Core i5-9400H CPU for a $1.3~{}{\rm MP}$
map at maximal vortex density.

Fig. 4:  WF reconstruction of a complex wavefield. The phase pattern addressed
to the SLM (a) is retrieved thanks to the WFS (b) after removal of the first
few low-order aberrations (tip/tilt, defocus and astigmatism). The phase
difference in (c) clearly demonstrates the efficient wavefront
reconstructions, especially the non-conservative part. Remaining smooth
distortions in $\varphi_{ir}$ at the boundaries are partly attributed to
inaccurate phase modulation by the SLM. The (filtered) $\Delta{\bf A_{z}}$ map
(d) allows identifying the $133$ vortices. The inset shows the optimized
segmentation of nearby vortices.

To demonstrate the flexibility of this approach to characterize and rebuild
optical vortices of any charge, we addressed phase spirals with charges
ranging from $-10$ to $+10$ (Fig. 3). Because of the diverging phase gradient
at the vortex center, the $\Delta A_{z}$ maps exhibit peaks whose widths
increase with the charge $n$ of the vortex (Fig. 3a). Nevertheless,
integration of $-\Delta A_{z}/(2\pi)$ over segmented regions yields $n$ within
a $3\%$ accuracy range (Fig. 3b). Except for specially designed optical vortex
beams carrying fractional charges [55], for beams propagating in free space,
the topological charge is typically an integer. Rounding the integral to the
closest integer value then allows an accurate reconstruction of the optical
vortices (Fig. 3c). Differences between the rebuilt phase profiles and the
perfect ones addressed to the SLM are due to the contribution of
$\varphi_{ir}$ arising from uniformity imperfections of the SLM.

Finally, we demonstrate the possibility to retrieve the phase of complex
random wavefields with our algorithm. Random wavefields contain a high density
of optical vortices of charge $+1$ and $-1$ [56, 32, 57]. These vortices
exhibit elliptical phase and intensity profiles along the azimuthal
coordinate. The non-uniform increase of the phase around the singular point
may then alter the ability to detect them if the phase-gradient magnitude is
locally too large. Furthermore, the separation distance between vortices may
be much smaller than the speckle grain size, especially when close to creation
or annihilation events of pairs of vortices [33, 58, 35]. Such a complex
wavefield was numerically generated by taking the Fourier transform of a
random phase map of finite aperture, and addressed to the SLM (Fig. 4a).
Despite the aforementioned specific difficulties, the WF could be efficiently
rebuilt (Fig. 4b). The accurate reconstruction can be visually appreciated by
considering $0$-$2\pi$ equiphase lines, easy to identify with a gray-level
colormap as abrupt white-to-black drops. The difference between the rebuilt
and the input phase profiles is shown in Fig. 4c demonstrating the almost
perfect reconstruction of the $133$ optical vortices. The six lowest order
Zernike aberration modes were removed (piston, tip/tilt, defocus,
astigmatisms) for better visualization. Again, differences mostly appear on
the $\varphi_{ir}$ contribution on the edges of the SLM, where the SLM
reliability degrades and where aberrations introduced by the relay telescope
are maximum. The dense experimental map of vortices distribution and charges
is shown in Fig 4d.

Relying on a high-resolution WFS, we could thus propose a systematic and
robust approach to rebuild complex wavefields containing a high density of
optical vortices. The proposed method first consists in performing a HD of the
local wavevector field ${\bf g}$ measured by the WFS. Importantly, the
circulation of ${\bf g}/(2\pi)$ over vortices, computed as the integral
$\nabla\times{\bf g}$ over large enough surface areas (under the Stokes’
theorem), yields the topological charge of vortices. The systematic
reconstruction of the optical vortex map relies on an image segmentation that
optimizes surface integration and thus, vortex identification. The robustness
of phase-spiral-reconstructions further relies on the quantization prior about
the detected topological charges. Our method is applicable to any high-
resolution WFS or system requiring integrating the local wavevector map. Full
reconstruction of WFs with a WFS represents an important step to make WFS
efficient reference-less spatial-phase detectors and to allow random
wavefields characterization, in particular with incoherent light sources.
These developments are of interest for applications such as adaptive optics,
diffractive tomography, as well as beam shaping behind scattering and complex
media.

## Funding Information

This work was partially funded by the french Agence Nationale pour la
Recherche (SpeckleSTED ANR-18-CE42-0008-01), by the technology transfer office
SATT/Erganeo (project 520 and project 600) and by Region île de France (DIM
ELICIT, 3-DiPSI).

## Acknowledgments

The authors thank Jacques Boutet de Monvel and Pierre Bon for careful reading
of the manuscript, and Benoit Forget for stimulating discussions.

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# Controlled transport based on multiorbital Aharonov-Bohm photonic caging

Gabriel Cáceres-Aravena1,2,†, Diego Guzmán-Silva1,2,†, Ignacio Salinas1,2, and
Rodrigo A. Vicencio1,2 1Departamento de Física, Facultad de Ciencias Físicas y
Matemáticas, Universidad de Chile, Chile
2Millenium Institute for Research in Optics - MIRO, Universidad de Chile,
Chile

(August 27, 2024)

###### Abstract

The induction of synthetic magnetic fields on lattice structures allows to
effectively control their localization and transport properties. In this work,
we generate effective $\pi$ magnetic fluxes on a multi-orbital diamond
lattice, where first ($S$) and second ($P$) order modes effectively interact.
We implement a $z$-scan method on femtosecond laser written photonic lattices
and experimentally observe Aharonov-Bohm caging for $S$ and $P$ modes, as a
consequence of a band transformation and the emergence of a spectrum composed
of three degenerated flat bands. As an application, we demonstrate a perfect
control of the dynamics, where we translate an input excitation across the
lattice in a completely linear and controlled way. Our model, based on a flat
band spectrum, allows us to choose the direction of transport depending on the
excitation site or input phase.

Understanding the properties of transport and localization in lattice systems
has been crucial for the implementation of present technologies based in
electronics. Novel phenomena has been discovered where, for example, the
transport can be drastically reduced and localization effects could emerge.
One important example is the well known Anderson localization Anderson (1958),
where all the eigenmodes collapse due to a destructive interference process
associated to a random distribution of the site energies and/or lattice
positions Schwartz  _et al._ (2007); Lahini  _et al._ (2008). Another
interesting and more controllable method appears with the addition of a
magnetic flux on a lattice, which adds an extra phase on hopping (coupling)
coefficients. For very specific values of this flux, a phenomenon of extreme
localization is observed, the so-called Aharonov-Bohm (AB) caging Aharonov and
Bohm (1959); Vidal  _et al._ (1998). In this case, in contrast to the Anderson
localization, all the eigenmodes become degenerated and form a set of flat
bands (FBs) Vicencio (2021). Therefore, an input excitation is decomposed in
FB compact states, the energy is caged, and the transport is abruptly reduced
into a couple of unit cells. Although the use of a magnetic flux was initially
thought for electronic lattices, it was possible to map its effect to non-
electronic systems by the use of artificial gauge fields Aidelsburger  _et
al._ (2018). This opened the possibility to study the AB effect in other
physical systems Longhi (2014); Fang  _et al._ (2012); Bloch  _et al._ (2012);
Brosco  _et al._ (2021); Perrin  _et al._ (2020); Möller and Cooper (2012);
Bermudez  _et al._ (2011), as well as to experimentally observe it in
superconducting wire networks Abilio  _et al._ (1999) and in semiconductors
tailored in an electron gas Naud  _et al._ (2001). Very recently, a synthetic
magnetic flux has been implemented in photonics by applying a detuning and
modulation of curved waveguides Mukherjee  _et al._ (2018), by the addition of
auxiliary waveguides to generate an effective negative coupling Kremer  _et
al._ (2020), or by the excitation of special beams having orbital angular
momentum Jörg  _et al._ (2020).

While the AB effect allows to trap and steer the propagation of light on a
given region, other ways of controlling the dynamics in optical lattices have
been suggested as well Garanovich  _et al._ (2012). When studying mobility
properties to, for example, modify the direction of propagation of a light
beam, most of the efforts have been focused on the addition of nonlinearity to
the system Garanovich  _et al._ (2012); Molina  _et al._ (2006); Naether  _et
al._ (2011); Mejía-Cortés  _et al._ (2013); Peschel  _et al._ (2002), which
also induces instabilities and resonances with linear modes. Interestingly,
magnetic fluxes in non-Hermitian lattices have also the property to alter the
direction of propagation affecting both reflexion and transmission, and
producing unidirectional reflection, absorption and lasing Jin  _et al._
(2016); Regensburger  _et al._ (2012); Feng  _et al._ (2013); Ramezani  _et
al._ (2014, 2016); Jin and Song (2018). However, the control of localized
wavepackets, including a magnetic flux in Hermitian systems, has not been
experimentally observed so far. A controlled flow of energy is important in
general physics and also when thinking on concrete applications as, for
example, logical operations and concatenated optical gates in photonics and
integrated technologies Pérez  _et al._ (2020); Corrielli  _et al._ (2021).

Figure 1: (a) A multi-orbital diamond lattice formed by $S$ and $P$ sites.
Full (dashed) arrows indicate a positive (negative) coupling constant. The
effective induced magnetic flux per plaquette is denoted as $\phi$. A dashed
rectangle defines the unit cell formed by six lattice sites. (b1) and (b2)
Spectrum for $\phi=0$ and $\phi=\pi$, respectively. FB profiles are shown as
insets. (c) Intensity evolution for different input conditions: (c1) central
site for $\phi=0$; (c2) central site and (c3) top/bottom site, both for
$\phi=\pi$. In (c) $V=0.1$ mm-1 and $z_{max}=50$ mm.

In this Letter, we propose a novel form to implement a synthetic magnetic flux
in lattices via the use of inter-orbital coupling Guzmán-Silva  _et al._
(2021) on a diamond lattice Longhi (2014). By considering the effective
interaction between fundamental ($S$) and second-order ($P$) modes, a positive
and negative coupling can be obtained depending on waveguide and mode
orientation. This allows to induce an effective magnetic flux $\phi=\pi$ per
plaquette (a closed ring in the lattice), allowing a direct observation of AB
caging. By using a femtosecond (fs) laser writing technique Szameit  _et al._
(2005), we fabricate a multi-orbital photonic lattice having a mixture of
straight waveguides which support one or two modes. We observe quite clearly
AB caging for $S$ and $P$ states. As a remarkable application, we
experimentally demonstrate a novel mechanism for directional transport of
linear localized excitations. We show a perfect steering of the energy across
the lattice in a very compact way and suggest a concatenated photonic scheme.
Our results are certainly a step forward on the study of discrete dynamics
Lederer  _et al._ (2008); Flach and Gorbach (2008), offering a new and linear
solution for the mobility of localized wavepackets in lattices.

Our model consists of a multi-orbital diamond-like chain, which includes sites
that support fundamental modes only and sites that support also a second mode,
as it is sketched in Fig. 1(a). As the mathematical model is equivalent to the
propagation of quantum particles on a tight-binding lattice, we use the
hydrogen-like nomenclature of S and P orbitals. We have very recently
demonstrated Guzmán-Silva  _et al._ (2021) that an effective interaction
between different orbitals can be achieved when a tuning process is
implemented, such that the optical propagation constants (or the energies in a
quantum version) match. This also means that the $S$ mode of tuned $P$ sites
is not anymore in resonance with the $S$ mode of $S$ sites and, therefore,
they simply do not interact. The unit cell of this lattice consists of six
sites as shown by a dashed rectangle in Fig. 1(a). The coupling interaction
between sites is determined by the coupling constants $\pm V$, which depends
on the spatial orientation of waveguides and their respective modes. We have
defined the positive part of the wave functions using red colors, while the
negative part is in gray. For any distance, the horizontal coupling in between
$S$ and $P$ waveguides, at the central row, is zero due to the orthogonality
of both wavefunctions and the $\pi$ phase structure of the $P$ mode Sutton
(1993). In fact, this makes possible to achieve positive and negative
couplings constants on a given lattice geometry and induce new phenomena on
rather trivial systems, without the need of implementing more complex
configurations Mukherjee  _et al._ (2018); Kremer  _et al._ (2020); Jörg  _et
al._ (2020).

A single negative coupling constant on a plaquette effectively induces a
synthetic flux of $\phi=\pm\pi$ Longhi (2014), which is the main requirement
for the observation of AB caging. We use a tight-binding approach to write
down the coupled equations governing the dynamics of this system SM , which in
a compact form can be written as

$-i\frac{\partial\psi_{n}}{\partial z}=\sum_{m}V_{n,m}\psi_{m}\ .$ (1)

Here, $\psi_{n}(z)$ describes all the lattice amplitudes, $z$ the propagation
distance (dynamical coordinate), and $V_{n,m}$ the nearest-neighbor coupling
elements. First of all, we look for the lattice spectrum by assuming a Bloch
ansatz of the form $\psi_{n}(z)=\psi_{0}\exp\\{ik_{x}n\\}$, with $\psi_{0}$ a
site amplitude and $k_{x}$ the horizontal quasi-momentum. We notice that for a
homogeneous lattice, composed of $S$ waveguides only, the effective flux is
simply $\phi=0$, the unit cell reduces to only three sites, and the spectrum
has just three bands Longhi (2014): $E=0,\ \pm 2\sqrt{2}\cos(k_{x}/2)$. We
plot this spectrum in Fig. 1(b1) and observe a FB with compact localized
states Vicencio (2021); Mukherjee and Thomson (2015) as the ones shown at the
inset. FB localization is typically achieved by destructive interference of
positive and negative amplitudes at specific connector sites Vicencio (2021);
Morales-Inostroza and Vicencio (2016) which, in this case, correspond to the
ones at the central row. Therefore, the excitation of central sites does not
resonate with FB states and excites dispersive bands only (more details in SM
), as the numerical propagation shows in Fig. 1(c1).

On the other hand, a multi-orbital lattice as the one shown in Fig. 1(a)
induces an effective flux of $\pi$ at closed plaquettes. This small
modification on coupling coefficients produces a fundamental change on the
linear properties: all the linear modes become compact and completely
localized. The synthetic magnetic field cancels the lattice dispersion and no
extended eigenmodes are allowed to exist in the system. The linear spectrum
has six bands with only three different eigenvalues (each band having a two-
fold degeneracy):

$E_{0}=0\ ,\ \ E_{+}=2V\ ,\ \ E_{-}=-2V\ .$

This flat spectrum is shown in Fig. 1(b2), including the six different FB
states as insets. We immediately notice that a single-site excitation will
excite a reduced number of FB states only. The AB caging is usually understood
Mukherjee  _et al._ (2018); Kremer  _et al._ (2020); Jörg  _et al._ (2020) as
a symmetric oscillation of the energy on a narrow lattice region, as a
consequence of exciting a couple of localized FB states corresponding to two
different energies. When injecting light at sites $S$ or $P$ at the central
row, only two FB states will be excited with energies $\pm 2V$, with a caging
period of $z_{c}=\pi/2V$ SM . We simulate this scenario in Fig. 1(c2) where a
central row site ($S$ or $P$) is excited. We observe how the energy is trapped
and goes back and forward, oscillating around the input position; i.e., the
energy is caged. A perfect destructive interference occurs at the next central
sites and no amplitude is transmitted to the rest of the lattice. As a
consequence the energy returns back to the input position and a caging cycle
starts again. Fig. 1(c3) shows a periodic dynamics also, when light is
injected at top/bottom $S$ sites, as a result of exciting three FBs SM .

Figure 2: (a) fs laser writting technique. (b1) and (b2) Bright-field
microscopy images for a homogeneous and a multi-orbital diamond lattice,
respectively. (c) Top view of the image characterization setup and $z$-scan
concept. (d) Intensity output image at $z=L$ for $\phi=0$. (e) and (f)
Intensity output images at $z=z_{l}$ and $\phi=\pi$, for a $S$ and a $P$ mode
excitation, respectively. A circle shows the input site in (d)–(f). (g) and
(h) Average intensity at the input waveguide versus propagation length
$z_{l}$, for a $S$ and a $P$ mode excitation, respectively. Error bars
correspond to the standard deviation.

We fabricate a set of photonic lattices (PL) by using a fs laser writing
technique Szameit  _et al._ (2005), which is sketched in Fig. 2(a). In this
technique, pulses of $\sim 240$ fs and $\sim 82$ mW are tightly focused inside
a borosilicate glass wafer of length $L=50$ mm, while a high precision XYZ
Aerotech stage translates the wafer at a velocity of $0.4$ mm/s. We start by
fabricating a homogeneous diamond $S$ lattice as shown in Fig. 2(b1), with
nominal distances $\\{d_{x},d_{y}\\}=\\{36.00,29.72\\}\ \mu$m. We
experimentally test the lattices by using an image setup Vicencio  _et al._
(2015); SM , which modulates in amplitude and phase a wide laser beam of $640$
nm. We image a specific input condition onto the PL input facet, as sketched
in Fig. 2(c), and measure the output profile using a CCD camera. For the
homogeneous lattice, we excite a single $S$ waveguide and observe a well
diffracted spatial profile, as shown in Fig. 2(d). We notice that the energy
covers all the lattice, as a result of exciting only the dispersive part of
the spectrum SM . We increase the writing power to $104$ mW for the $P$
waveguides (four in our experiment), which are located at the central row as
shown in Fig. 2(b2). The writing powers were chosen to reach a maximum
coupling in between $S$ and $P$ modes Guzmán-Silva  _et al._ (2021), which
occurs when their propagation constants are precisely tuned SM . We implement
a $z$-scan method to characterize the dynamics along the propagation
coordinate, and fabricate twelve multi-orbital lattices. All $S$ and $P$
waveguides at the central row have a full length $L$, while the rest of
waveguides at the top and bottom rows have a shorter propagation length
$z_{l}$, from $6$ to $50$ mm in steps of $4$ mm, as sketched in Fig. 2(c). In
this way, by exciting a single $S$ or $P$ waveguide at the central row, we are
able to trace the dynamics at different distances.

First of all, we excite a $S$ central site and observe quite clearly $S$ mode
caging in Fig. 2(e), for the distances indicated in the central column. We
observe that the light is able to couple to the top and bottom nearest-
neighbour sites, but it can not propagate further through the lattice due to
destructive interference at the next central sites. We measure the output
intensity at the input site for different $z_{l}$ and plot the data in Fig.
2(g), where we average the data for the three central $S$ waveguides of our
lattice. The light oscillates twice for $50$ mm, with a caging period of
$z_{c}\approx 26$ mm, what implies a coupling of $V\approx 0.06$ mm-1. The
numerical simulation shown in Fig. 1(c2) described an oscillation for $S$ and
$P$ modes. Therefore, we should observe a similar caging phenomena also by
using a $P$ mode as an input condition, which consists of two lobes with a
$\pi$-phase difference. The output profiles obtained after exciting the
lattice with a second mode at $P$ central waveguides are shown in Fig. 2(f).
We nicely and quite clearly observe AB caging for a $P$ mode. The light tries
to escape from the input site and couple to its neighbour $S$ waveguides, but
already having a phase difference due to the $P$ parity. This makes impossible
for the light to continue escaping because of the destructive interference
occurring at the next central $S$ sites. Therefore, an effective wall is
created due to the induced synthetic magnetic field, and the light is
reflected back into the input position, and the cycle starts again. In Fig.
2(h) we observe a two cycle oscillation in $50$ mm and similar caging period
for a second mode excitation (we averaged the data considering the four
central $P$ waveguides). In both cases, we observe that more than $90\%$ of
the energy is caged in a first cycle, with an intensity reduction after two
beating lengths. As a perfect tuning and perfect coupling symmetry conditions
are really hard to be achieved experimentally, we expect to have some
dispersion while increasing the propagation distance. Once the destructive
interference is not perfectly balanced due to any symmetry breaking mechanism,
some energy will be radiated through the lattice. Nevertheless, we observe in
Figs. 2(e)–(h) that most of the energy is very well confined in a narrow
spatial region, and that a multi-orbital caging is clearly demonstrated
Mukherjee  _et al._ (2018); Kremer  _et al._ (2020); Jörg  _et al._ (2020).

Figure 3: (a) Optical mask with an in-phase (left) and an out-of-phase (right)
input excitation. (b1) and (b2) Intensity output images at $z=z_{l}$ and
$\phi=\pi$, for an in-phase and an out-of-phase input excitation,
respectively. Circles show the input sites. (c1) Concatenation scheme: red and
gray colors correspond to $S$ and $P$ waveguides, respectively. (c2) and (c3)
Numerical propagation for in-phase and out-of-phase input conditions,
respectively.

We notice that in a homogenous array the transport is uniformly distributed in
both directions [see Fig. 2(d)], while using a multi-orbital mask an
unidirectional displacement could be observed, allowing us to decide if the
light propagates to the right or to the left of the system. We fabricate a
smaller multi-orbital lattice, which we call an optical mask, as the one
sketched in Fig. 3(a). Again, we implement a $z$-scan study by fabricating
twelve lattices, where the $S$ top and bottom sites of the central column have
a full length $L$, while the rest lattice sites have a shorter propagation
distance $z_{l}$, which goes from $4$ to $26$ mm in steps of $2$ mm. We expect
to observe a maximum translation of the energy at $z_{l}\sim 26$ mm (in our
mask, the effective transversal translation is similar to the one for the AB
caging, as we need to move from two $S$ sites into similar $S$ sites to the
right or to the left). The experiment consists on exciting simultaneously two
full length $S$ sites at the center of the lattice using an image setup SM .
An in-phase (++) excitation [Fig. 3(a)-left] produces a perfect cancellation
of amplitudes at the right $P$ central waveguide and, therefore, zero energy
flowing to the right of the mask. In contrast, the light will propagate to the
left, as the first coupling occurs with the central $S$ site only, which then
couples with the following (left) top and bottom $S$ sites. There, a perfect
full translation is achieved with a precise control and discrete energy
transportation from one unit cell into another. The energy in this case can
not continue moving to the next $P$ waveguide: the energy find an interference
wall and the light is reflected back again to the input position, in a
dynamical picture similar to the caging one described previously. The
experimental intensity images obtained in this experiment are shown in Fig.
3(b1), where we observe how the light is not allowed to propagate to the right
and only is able to move to the left. In fact, our experiment nicely shows how
the input condition has been precisely translated to the left in a distance
$z_{l}\approx 24$ mm. Of course, this input condition also excites a caging-
like transport due to the presence of only FBs in the spectrum, and the energy
oscillates in a very well defined region. For a longer propagation the energy
will go back and will replicate the in-phase input profile. The out-of-phase
(+-) input condition shown in Fig. 3(a)-right will show an opposite dynamics.
The destructive interference will cancel the amplitude at the left $S$ central
site and no energy will flow to the left. However, the $\pi$-phase difference
will cancel the effect of the negative coupling and the energy will flow
through the $P$ central site, but keeping the initial phase difference.
Therefore, when arriving to a $S$ central site to the right, the amplitude
will be just zero there. The experimental images showing the out-of-phase case
are presented in Fig. 3(b2).

As a concrete and very important application, we numerically show that by
concatenating several optical masks, as it is schematically described in Fig.
3(c1) [every mask shifts the $S$ and $P$ central waveguides periodically in
the propagation direction], we achieve a controlled displacement on a larger
lattice structure. A perfect translation of the energy (to the right or to the
left) is obtained, as the numerical simulations in Figs. 3(c2) and (c3)
indicate. In fact, by changing the input phase or input position, one can
reverse the direction of movement. For example, an in-phase profile could move
to the left in position $\\{B_{n},C_{n}\\}$ and to the right in positions
$\\{E_{n\pm 1},F_{n\pm 1}\\}$. This is a very remarkable result for lattice
systems Lederer  _et al._ (2008); Flach and Gorbach (2008), considering that
our proposal is based on linear properties only. We therefore suggest a simple
and clear method for perfect energy transport, with high controllability and
stability. The injection of in-phase or out-of-phase input conditions allows
to propagate the energy freely through the lattice; while, on the other hand,
the injection of $S$ or $P$ states at the central row induces AB caging.
Therefore, our lattice model gives a perfect linear solution for controlled
transport and localization.

In conclusion, we have proposed a new scheme for inducing synthetic magnetic
fields in lattice systems. This effect relies in the interaction between
fundamental $S$ and excited $P$ modes, observing a net flux of $\pi$ per
plaquette. We experimentally proved this system by fabricating multi-orbital
diamond photonic lattices and we observed Aharonov-Bohm caging for both modes,
which is a direct experimental proof for the induction of an effective
magnetic field. We demonstrated the possibility of controlling the direction
of transportation determined by the input condition, which could be useful,
for example, in the creation and manipulation of logic gates Pérez  _et al._
(2020); Corrielli  _et al._ (2021). Our proposal gives a simple on demand
solution for localization and transport in lattices. We believe that our
multi-orbital configuration has the flexibility to include more effects in a
following study, such as non-hermiticity Zhang and Jin (2020) or nonlinear
effects Chang  _et al._ (2021), in a very closed and direct form.

The authors acknowledge stimulating discussions with Magnus Johansson and
Bastian Real. This work was supported in part by Millennium Science Initiative
Program ICN17-012, and FONDECYT Grants 1191205 and 3190601.

† Both authors contributed equally to this work

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