# High-accuracy longitudinal position measurement using self-accelerating
light

Shashi Prabhakar<EMAIL_ADDRESS>Stephen Plachta  Marco Ornigotti
Robert Fickler  Photonics Laboratory, Physics Unit, Tampere University,
Tampere, FI-33720, Finland

###### Abstract

Radially self-accelerating light exhibits an intensity pattern that describes
a spiraling trajectory around the optical axis as the beam propagates. In this
article, we show in simulation and experiment how such beams can be used to
perform a high-accuracy distance measurement with respect to a reference using
simple off-axis intensity detection. We demonstrate that generating beams
whose intensity pattern simultaneously spirals with fast and slow rotation
components enables a distance measurement with high accuracy over a broad
range, using the high and low rotation frequency, respectively. In our
experiment, we achieve an accuracy of around 2 $\mu$m over a longitudinal
range of more than 2 mm using a single beam and only two quadrant detectors.
As our method relies on single-beam interference and only requires a static
generation and simple intensity measurements, it is intrinsically stable and
might find applications in high-speed measurements of longitudinal position.

##  I Introduction

Structuring the spatial shape of light fields has become a broad research
field spanning areas from the foundations of optics to optical communication,
materials processing, quantum optics, and microscopy, to name a few
rubinsztein2016roadmap . Amongst many interesting features structured light
may have, one in particular has attracted a lot of attention and might even be
seen as the starting point of the field, namely the azimuthal phase structure
connected to the orbital angular momentum (OAM) of light padgett2017orbital .
Light fields carrying such OAM have a transverse phase of the form
$\exp(-i\ell\varphi)$, where $\varphi$ is the azimuthal coordinate and $\ell$
defines the quanta of OAM each photon carries allen1992orbital . These
transverse scalar modes are commonly known as vortex modes, or donut beams as
they have a phase singularity and, thus, an intensity null along the optical
axis. Over the last decades, various techniques to imprint such twisted
structures have been established, e.g., spiral phase plates
beijersbergen1994helical ; holographic generation using spatial light
modulators heckenberg1992generation ; carpentier2008making ;
forbes2016creation ; cylindrical lenses beijersbergen1993astigmatic ; $q$\-
and $J$-plates marrucci2006optical ; larocque2016arbitrary ;
devlin2017arbitrary ; or direct generation of the light field inside the
cavity of a laser forbes2019structured .

One particular family of modes, whose higher orders have an azimuthal phase
ramp, is known as Bessel beams. Bessel beams are propagation invariant light
fields described by Bessel functions gori1987bessel . They have received
significant attention due to being diffraction-less zahid1989directionality ;
vetter2019realization  and self-healing chu2012analytical ; mcgloin2005bessel
. While zero-order Bessel beams have an intensity maximum along the optical
axis and do not carry OAM, higher orders have a twisted phase structure
leading to well-defined quanta of OAM per photon. In addition, Bessel beams
are comparatively easy to generate using a ring aperture, with or without an
azimuthally varying phase, at the back focal plane of a lens ($k$-space). The
light fields in the focal plane of the lens, which have undergone an optical
Fourier transformation, then resemble the theoretical Bessel beams; however,
they have a finite beam extent and are therefore called Bessel-Gauss beams. If
not only one ring but multiple rings of different radii and different orders
are put into the back focal plane of the lens, a superposition of multiple
Bessel beams is generated. Interestingly, the obtained superposition
structures show the peculiar feature of spiraling around the optical axis
along the propagation direction if the constituents forming the superposition
have different OAM values chavez1996nondiffracting ; schechner1996wave ;
abramochkin1997generation ; paakkonen1998rotating . This property of complex
superpositions of higher-order Bessel beams has been the focus of various
research efforts. Thorough theoretical and experimental studies of such
spiraling beams have been performed, which have been enabled by the progress
in experimental techniques of generating such beams with high precision and
flexibility tervo2001rotating ; kotlyar2007rotation ; vasilyeu2009generating ;
vetter2015optimization ; schulze2015accelerated .

Contrary to Airy beams, whose self-accelerating character is essentially given
by the fact that an observer in a reference frame solidal with an Airy beam
would experience a tangential fictitious force greenberger1980comment  that
results in their characteristic parabolic propagation profile, radially self-
accelerating beams (RSABs) are characterised by a centrifugal fictitious force
linked to their characteristic spiraling motion vetter2014generalized . As
such they are also distinct from another class of self-accelerating beams,
recently investigated in vetter2017real ; webster2017radially .

In this work, we demonstrate a novel application of spiraling beams as a means
to determine the longitudinal distance with high accuracy by only measuring
the intensity using a quadrant detector, i.e., in a limited number of off-axis
locations. First, we briefly introduce the theory behind spiraling beams and
describe how superpositions of three Bessel modes can lead to a complex
rotating pattern having both quickly and slowly rotating parts at the same
time. We then show in simulations and experiments that using such a spiraling
beam enables the accurate determination of distance over a long range when the
intensity using two quadrant detectors (or in minimally three off-axis
positions) is recorded. In the experimental implementation, we are able to
achieve an accuracy of around 2 $\mu$m over a range of 2 mm. The obtained
result is mainly limited by the aperture of our optical system, as well as the
resolution of the generating and detecting devices. Hence, the proposed and
demonstrated method of measuring a longitudinal distance using structured
light might find promising applications, similar to self-accelerating Airy
beams, which for example have been used to resolve depth in microscopy
applications recently jia2014isotropic ; he2019depth . Due to its simplicity
and high accuracy, our method nicely complements other available techniques
using light, e.g., time-of-flights measurements as used in LIDAR systems
jarvis1983laser , interferometric approaches kubota1987interferometer , or
schemes that rely on complex scattering of structured light fields
berg2020microsphere , to name a few.

##  II Theoretical background

###  II.1 Spiraling light fields

Measuring the longitudinal position, i.e., a certain distance with respect to
a fixed reference, using an intensity structure that changes over propagation
requires a light field with well-defined propagation dynamics. The recently
demonstrated radially self-accelerating, or spiraling, light fields, which
show a constant rotation of the intensity pattern along the propagation
direction, are a very convenient solution to use. While light with more
complex propagation dynamics would require sophisticated evaluation
procedures, rotating structures allow the determination of the longitudinal
position through simple measurements of the rotation angle. Importantly, such
light fields can be easily realized by superimposing (at least) two vortex
beams, each having a different OAM value $\ell$ as well as two different
longitudinal wave vectors $k_{z}$ defining the propagation dynamics
vetter2014generalized . The two-component solution, i.e., the so-called
helicon beams, can then be written as

$u(r,\phi,z)=A_{\ell_{1}}(r)\exp{[i(k_{1z}z+\ell_{1}\varphi)]}+A_{\ell_{2}}(r)\exp{[i(k_{2z}z+\ell_{2}\varphi)]},$
(1)

where the indices $1,2$ label the two constituents and $A_{\ell}(r)$ is a
radially-dependent envelope function. The resulting intensity,
$I=u(r,\varphi,z)u(r,\varphi,z)^{*}$, can then be obtained to be

$I(r,\varphi,z)\propto\cos^{2}{[\Delta kz+\Delta\ell\varphi]},$ (2)

where $\Delta k=(k_{1z}-k_{2z})/2$ is the difference between the wave vectors
of the two beams and $\Delta\ell=(\ell_{1}-\ell_{2})/2$ is the difference
between their OAM values. Notice, moreover, that we have only kept the part of
the intensity that is of importance, i.e., the one including the required
angular and $z$-dependence. For more details we refer the interested reader to
earlier works vetter2014generalized . From (2) we find that the angular
orientation of the intensity profile $\phi(z)$ changes along the beam
propagation according to the relation

$\phi(z)=\frac{z\Delta k}{\Delta\ell},$ (3)

from which the angular velocity can be calculated as

$\omega=\frac{\partial\phi(z)}{\partial z}=\frac{\Delta k}{\Delta\ell}.$ (4)

One such beam propagation is shown in Fig. 1, where the spiraling of the mode
along the propagation axis is depicted.

Figure 1: Normalised three-dimensional representation of the propagation of
the central lobe of a radially self-accelerating beam along the $z$ direction,
as defined in (1), consisting of the superposition of $\ell_{1}=0$ and
$\ell_{2}=1$ Bessel modes, which matches the form of the single-frequency beam
used in the experiment. The transverse directions $\\{k_{0}x,k_{0}y\\}$ are
normalised to the central $k$-vector of the beam, i.e., $k_{0}$, while the
longitudinal direction (i.e., the propagation direction $z$) has been
normalised to the rotation period $\Lambda=2\pi/\omega$, with $\omega$ being
the rotation speed defined by (4). The plot shows propagation up to the first
two periods. The colour scale in the picture represents different iso-
intensity surfaces, with brighter colours indicating regions of higher
intensity. As can be seen, the whole intensity distribution rigidly rotates
around the $z$ axis with rotation speed $2\pi/\Lambda$. This peculiar
propagation pattern is the result of interference between the various Bessel
beam components constituting the radially self-accelerating beam, as described
by (1).

In other words, if we measure the intensity over a certain angular region, the
intensity along the beam propagation follows a periodic $\cos^{2}$-function
and can be used to determine the longitudinal distance unambiguously within
half a period. In principle, a simple measurement of the intensity at an off-
axis transverse position therefore allows the determination of a distance with
arbitrary precision. In practical situations, however, errors induced by the
generation or measurement of the structure result in an uncertainty in the
determination of the intensity’s angular position, which leads to a limitation
of the longitudinal accuracy. One direct way to improve the measurement
accuracy despite these imperfections is to increase the rotation frequency of
the structure with respect to its propagation. The faster the rotation, the
better the accuracy in measuring the longitudinal distance $z$. Hence, one aim
in high-accuracy distance measurement using self-accelerating light fields is
to achieve the largest possible difference in the longitudinal wave vectors
$\Delta k$ allowed by the optical system. We also see that the difference in
the OAM values $\Delta\ell$ should be kept as small as possible, i.e., the
$\ell$-values of the two constituent beams should only differ by 1.

Obviously, improving the longitudinal accuracy by increasing the rotation
frequency comes at the cost of a reduced longitudinal range over which an
unambiguous determination of the angular position is possible by only
examining the intensity pattern. To circumvent this limitation, it is possible
to realize a more complex structure, which shows a rotating intensity pattern
that includes two (or more) well-defined rotation frequencies. Ideally, the
intensity structure should have one very high rotation frequency used to
obtain locally a high-accuracy measurement of the longitudinal position. The
intensity dynamics should further include a rotating structure with very low
frequency from which it is possible to determine the global distance and
discriminate between different fast-varying periods. Both frequencies need to
be adjusted such that each period of the high-accuracy measurement can be
distinguished from any other using the slow rotation pattern. In the
theoretical description, this idea can be implemented by adding a third term
to the equation earlier introduced (1), such that we obtain

$\displaystyle u(r,\phi,z)$ $\displaystyle=$ $\displaystyle
A_{\ell_{1}}(r)\exp{[i(k_{1z}z+\ell_{1}\varphi)]}$ (5)
$\displaystyle+A_{\ell_{2}}(r)\exp{[i(k_{2z}z+\ell_{2}\varphi)]}$
$\displaystyle+A_{\ell_{3}}(r)\exp{[i(k_{3z}z+\ell_{3}\varphi)]}.$

As can be seen, the electric field defined above contains three different
contributions, each characterised by its spatial frequency $k_{i}$ and OAM
$\ell_{i}$. If we now calculate the intensity distribution generated by such a
field, we will have, together with the contributions of the single terms in
(5) – i.e., terms proportional to $|A_{\ell_{k}}|^{2}$ – also all the possible
interference terms between the three beams composing the field above. We can
therefore write, neglecting the $z$\- independent terms, which amount only to
an overall normalisation factor (supplemental material of
vetter2014generalized ),

$\displaystyle I$ $\displaystyle\propto$ $\displaystyle\cos^{2}{[\Delta
k_{1,2}z+\Delta\ell_{1,2}\varphi]}+\cos^{2}{[\Delta
k_{1,3}z+\Delta\ell_{1,3}\varphi]}$ (6) $\displaystyle+\cos^{2}{[\Delta
k_{2,3}z+\Delta\ell_{2,3}\varphi]},$

where we labelled $\Delta k_{i,j}=(k_{iz}-k_{jz})/2$ and
$\Delta\ell_{i,j}=(\ell_{i}-\ell_{j})/2$ as the pairwise differences between
the wave vectors and OAM values of the three fields. By choosing
$\Delta\ell_{2,3}=0$, i.e., $\ell_{2}=\ell_{3}$, the angular dependence of the
propagation dynamics of the last term vanishes. Hence, we obtain the required
light field, whose structure rotates with only two rotation frequencies at the
same time.

###  II.2 Experimental implementation

In an experiment, it is convenient to realize these radially accelerating
beams in the framework of Bessel beams, i.e., $A_{\ell}(r)=J_{\ell}(rk_{r})$
ornigotti2018vector ; rop2012measuring . For Bessel beams the longitudinal
wavevector $k_{z}$ can straightforwardly controlled by adjusting its radial
counterpart $k_{r}$. Both quantities are related through

$k_{z}=\sqrt{k^{2}-k_{r}^{2}},$ (7)

where $k=2\pi/\lambda$ labels the wavenumber and $\lambda$ corresponds to the
wavelength of the utilized light field.

As the angular spectrum of a Bessel beam forms a ring in $k$-space, such beams
are relatively simple to generate in the laboratory. By modulating an incoming
light field to have a ring-shaped amplitude at one plane ($k$-space), we can
transform the light into a Bessel beam (real space) by implementing an optical
Fourier transform using a properly placed lens. The lens is placed one focal
distance $f$ away from the initial modulation plane, i.e., in the back focal
plane of the lens, such that at around another focal distance behind the lens
a Fourier transform leads to a Bessel beam with a well-defined longitudinal
wave vector $k_{z}$. In the modulation plane is a phase-only spatial light
modulator (SLM) whose screen displays a holographic pattern, generated by a
MATLAB script, that modulates the amplitude and phase of the beam into the
required ring shape rosales2017shape . Depending on the radius $r$ of the ring
in the modulation plane, the Bessel beam with longitudinal wave vector

$k_{z}=\frac{2\pi}{\lambda}\cos{\left(\frac{r}{f}\right)}$ (8)

will be obtained, which follows from simple geometric arguments
vasilyeu2009generating ; rop2012measuring .

A radially self-accelerating beam, such as the one described above, follows
from simply modulating the light field to have two (or more) rings with
different radii $r_{i}$ and individual OAM values $\ell_{i}$. The spiraling
around the optical axis can thus be tuned by changing the radii of the two
rings, leading to a wave vector difference

$\displaystyle\Delta k_{i,j}$ $\displaystyle=$
$\displaystyle(k_{iz}-k_{jz})/2$ (9) $\displaystyle=$
$\displaystyle\frac{\pi}{\lambda}\left[\cos{\left(\frac{r_{i}}{f}\right)}-\cos{\left(\frac{r_{j}}{f}\right)}\right].$

As discussed earlier, to achieve the best possible longitudinal accuracy, we
aim at generating a structure that contains both a very high and a very low
rotation frequency at the same time. To realise such a rotating structure, the
light field generated by the SLM needs to contain three Bessel beam
components, so that their mutual coupling gives rise to a fast-rotating and a
slow-rotating self-accelerating beam. The former, for example, results from
Bessel beam components 1 and 2, whose difference in radii should be as large
as the optical system allows, while the OAM value differs by only a single
quanta, i.e., $\Delta\ell=1$. As one of the rings, ring 2, necessarily has to
have a large radius $r_{2}$ leading to a large difference in longitudinal wave
vectors $\Delta k_{1,2}$, the interfering light field and, thus, the rotating
structure will be strongly confined at a small transverse region around the
optical axis.

The slow-rotating part, on the other hand, results from Bessel components 1
and 3, so the radii of the two rings should be very similar to obtain a small
difference in wave vectors $\Delta k_{1,3}$. However, the difference should
also be large enough that the resulting rotation frequency allows us to
discriminate between the repeating periods of the fast-oscillating signal. If
these two rings are chosen to be similar in radius but much smaller than ring
2, the rotating light field in the Fourier plane of the lens, i.e., the slowly
rotating structure, will cover not only the area in close proximity of the
optical axis but also the outer region. Note that examples of the ring-shaped
modulation patterns and the resulting propagation dynamics of the spiraling
beams can be found in Figures 3 and 5 in later sections. This difference in
the radial intensity compared to the fast-rotating structure enables us to
discriminate the two differently varying patterns by observing the intensity
in different radial regions. In the simplest case, these regions might be
defined by a single transverse location where the intensity is evaluated.
However, higher experimental accuracy can be obtained using two quadrant
detectors, one for each rotating structure, which evaluate the differences
between opposing quadrants to increase the signal-to-noise ratio. As such
detectors can work with tens of nanoseconds of rise time, the proposed method
might also find applications in high-speed longitudinal position measurements.

Another important aspect is the overall distance over which the spiraling
intensity can be observed. Here, the ultimate limit is given by the width of
the ring in the modulation plane that generates the Bessel beam. The narrower
the ring, the longer will the spiraling beam survive, thus allowing a
measurement for longer distances. On the contrary, the wider the ring width
is, the shorter will be the self-accelerating beam in the focal region, thus
allowing measurements over only a very short distance. Obviously, in
applications the ring width is determined by the resolution of the modulation
device, which is in our case the SLM (see below), as well as the minimum
amount of light required to detect the rotating intensity patterns. Using
high-resolution generation methods, self-accelerating beams over a distance of
70 mm have been demonstrated already vetter2019realization .

We further note that, in principle, one can also tune the rotation frequency
by adjusting the difference in the OAM values $\Delta l$ of the two
constituent beams, as can be seen in formula (3). However, by doing so, one
should keep in mind that large differences in OAM result in beams of more
complex angular structures, which might also require a detection system that
is able to resolve those structures. Moreover, if the two constituents differ
by many OAM quanta, they also show a significant difference in their OAM-
induced divergence of the beam padgett2015divergence . As this difference also
leads to a fast decrease in their spatial overlap, the region over which the
interference and, thus, the rotation can be observed is reduced. Hence,
improving the accuracy as well as the distance over which it can be measured
is preferably done by tuning the rotation frequency through adjustments to the
longitudinal wave vector difference $\Delta k$.

##  III Results

###  III.1 High-accuracy measurements

As a first task, we investigated the largest possible rotation frequency and,
thus, the highest possible accuracy. Before testing the theory in the
laboratory, we performed the so-called split-step propagation
methodpoon2017engineering  to simulate the entire setup. A sketch of the setup
can be seen in Fig. 2.

During the first set of simulations, the main aim was to verify the idea and
find the fastest rotation of the angular modes that was still within the
limitations of our experimental system. These limitations were mainly the
aperture of our optics (1-inch); the pixel resolution of our modulating
device, a phase-only SLM (Holoeye, Pluto-2.1-NIR-011 LCOS, 1920$\times$1080
pixels, 8 $\mu$m pixel size); and the pixels of the detection system, a camera
(ZWO ASI120MM Mini, 1280$\times$960 pixels, 3.75 $\mu$m pixel size).

Figure 2: Sketch of the experimental setup used in simulation and experiment.
A laser beam with a wavelength of 780 nm is enlarged to a wide radius using a
telescope (lenses $f_{1}$ and $f_{2}$). For maximum efficiency, the beam’s
polarization is controlled by a half-wave plate $\lambda/2$ such that it
aligns with the orientation of the diffraction grating on the SLM, which
modulates the diffracted light to have a shape with a central circle and
ring(s) and imparts angular momentum to the ring(s). The beam is filtered
through a 4f optical system (lenses $f_{3}$) that extracts the first
diffraction order, thereby removing the un-modulated light. Beyond the image
plane, the structured beam is focused by another lens $f_{4}$. The resulting
spiraling structure is imaged by a camera on a translation stage (TS).

In order to achieve the best matching between simulation and experiment, we
included in the simulation the spatial resolution of our modulation and
detection plane using the values of our laboratory system. In particular, we
set the pixel size of the simulation to 1.875 $\mu$m, which is half the size
of the physical pixels of our camera used in the experiment.

At first, we started by simulating a beam that only rotates with a single
frequency, i.e., a beam consisting of two Bessel components, as described in
equation (1). To achieve the largest difference in their longitudinal wave
vectors, we generated two Bessel beam components from two ring-shaped patterns
of very different radii in the Fourier domain. The larger ring size was mainly
limited by the screen size of the SLM as well as the optical apertures in our
system. The ring we utilized had an inner radius of $r_{1}$=3.5 mm, with a
ring width of 0.1 mm, and imprinted an OAM value of $\ell=$1 onto the beam. As
the length over which the rotating intensity pattern exists is determined by
the ring width, we chose the smallest possible width allowed by the resolution
of the modulation device, i.e., the pixel size of the SLM. Utilizing an SLM
with a pixel size of 8 $\mu$m, we chose a ring width of 100 $\mu$m in order to
have around 11-13 pixels at any given angular position, which allowed an
efficient generation using holographic methods. We chose the smaller ring to
be a circular area of radius $r_{0}=0.4$ mm with a flat phase, i.e., an OAM
value of $\ell=0$ (see Fig. 3a). The circular area was optimized to have a
similar intensity as the ring, taking the Gaussian shape of our input light
field, with a beam waist of 4.1 mm, into account. This optimization was done
because equal amplitudes of the two components result in a rotating pattern
with improved visibility vetter2015optimization ; ornigotti2018vector , which
in our task improves the accuracy.

Figure 3: Simulation for high-accuracy measurements. a) In the inset, the
modulation pattern is shown to generate the spiraling structure. Its
brightness corresponds to the amplitude of the light, and the color depicts
the phase of the modulation. For clarity only the modulation is shown, not the
holographic pattern required in the experiment. The green lines depict the
region utilized to emulate a quadrant detector. b) The simulated angular
intensity changes over propagation distance. Four exemplary intensity patterns
are shown as insets to visualize the spiraling behaviour. Comparing the
intensities found in 4 quadrants, shown in a), around the optical axis enables
the determination of the longitudinal position. The intensity difference
between quadrants 1,2 and 3,4 (2,3 and 1,4) leads to a sinusoidal curve shown
in black (red) with a period of 311.2 $\mu$m. Using both fast-varying curves
the longitudinal positions can be unambiguously determined over one half
period.

Finally, the ring-shaped intensity patterns were transformed into a spiraling
beam through an optical Fourier transform using a lens of 50 mm focal length.
The focal length was mainly limited by the pixel size of the camera, as
shorter focal lengths lead to beams of smaller extent in the focus. We note
that a magnification system might be used to circumvent this constraint if
shorter focal length lenses are required.

In order to determine the propagation distance from the change in angular
intensity, we simulated the intensity in a 300$\times$300-pixel region
centered on the optical axis during propagation. We then registered the
intensity within a circular region of radius 3.75 $\mu$m around the optical
axis. To emulate a quadrant detector we summed up the registered intensity of
each of the four quadrants and evaluated the difference between the
intensities of quadrants 1,2 and 3,4, i.e., the upper and lower halves (see
Fig. 3a). We obtained a sinusoidal curve with a period of 311.2 $\mu$m, as
shown in Fig. 3b, which matches the value of 311 $\mu$m expected from theory.
In addition, if one evaluates the difference in intensity between quadrants
2,3 and 1,4, i.e., the left and right halves, it is possible to shift the
steepest slope of the sinusoidal curve by a quarter of a period (an effective
phase shift of $\pi/2$). When using both signals, it is possible to achieve
the same longitudinal accuracy at the positions where the shallow slope of one
signal would cause a significant decrease in accuracy (see Fig. 3b).

After having verified the method in simulations, we turned to the experimental
implementation to determine the actual accuracy limits of our system due to
experimental imperfections and errors. In the experiment, sketched in Fig. 2,
we used a fiber-coupled single-frequency Toptica laser (DLpro) at a wavelength
of 780 nm. We enlarged the laser beam using a telescope system to a beam waist
radius of approximately 4.1 mm, such that it illuminated the whole screen of
the SLM. Via reflection off the SLM screen, we modulated the beam to have the
multiple-ring shaped intensity structure with the dimensions described above.
As the SLM is a liquid-crystal phase-only modulation device zhang2014LCOS , we
used holographic modulation techniques to perform a complex amplitude
modulation. This is done by displaying the diffractive holographic pattern
only at the ring-shaped regions where we wanted to obtain a light field
rosales2017shape . Through filtering only the first diffraction order using an
aperture in a Fourier plane of a 4f-system, we not only imprint the required
phase but also carve out the required intensity structure. At the image plane,
we obtained the required ring-shaped amplitude that leads to a spiraling beam
after a Fourier transformation, as described earlier. Analogous to the
simulations, we performed this Fourier transformation using another lens with
a focal distance of 50 mm.

To record the spiraling structure, we placed the camera on a high-accuracy
motorized translation stage, which was scanned over 1 mm in steps of 1 $\mu$m.
The recording and translation of the camera were automated by interfacing the
camera and translation stage with LabVIEW. At each position, we recorded 50
frames, from which we obtained the average and standard deviation of the
angular intensity along the optical axis. Analogous to the simulation, we
emulated a quadrant detector by registering the intensity difference between
different quadrants illuminated by the beam. As the beam waist in the focal
plane is very small, we only used 4 pixels of size 3.75 $\mu$m, each
corresponding to one quadrant.

Figure 4: Experimental high-accuracy distance measurements. a) The scatter
plot of intensity differences between quadrants 1,2 and 3,4 (2,3 and 1,4) are
shown in black (red). The intensities are obtained by averaging 50 images for
each position. The solid lines show the corresponding sinusoidal fits to the
measurements. b) The displacement accuracy analysis is obtained using the
fitted sine functions and the error propagation method from the standard
deviation of intensity fluctuations at each position. We find a best accuracy
of 1-2 $\mu$m at the points of steepest slope. For maintaining the minimum
error, one must hop between the steep slopes of the red and black curves,
which leads to obtaining the accuracy of 2.2$\pm$0.9 $\mu$m over the full
range (larger symbols).

The resulting variations in intensity when comparing quadrants 1,2 and 3,4 are
shown in Fig. 4a, demonstrating a period of about 332.6$\pm$0.3 $\mu$m. This
result matches the expected period from theory and simulation, with the small
discrepancy attributed to experimental imperfections such as finite resolution
and misalignment. Using the standard deviation of the intensity at each
position as the experimentally determined error and taking the known
sinusoidal curve into account, we determined the longitudinal accuracy or
displacement accuracy through error propagation. Hence, we define the
displacement accuracy as the minimal displacement for which the errors of two
data point do not overlap, i.e., the two data points that can be discriminated
with a 1-$\sigma$ confidence interval. We found that at the steep slope of the
curve, two different longitudinal positions that are 1 to 2 $\mu$m apart can
still be resolved with one standard deviation significance (see Fig. 4b). As
expected, the accuracy decreases dramatically around the extremal regions of
the sinusoidal curve. However, as mentioned earlier, evaluating the difference
in intensity between the left and right halves (quadrants 2,3 and 1,4) results
in a periodic signal that is shifted by a quarter of a period relative to the
signal of the upper and lower halves. We can therefore always refer to a fast-
varying signal regardless of position, so the strong reduction of accuracy due
to slow intensity variations at the extremal points of the sinusoidal curve is
circumvented. When switching between the two signals such that the quarter
period with the steepest slope is always used for a given longitudinal region,
we obtain an average accuracy of 2.2$\pm$0.9 $\mu$m over the whole scanning
range.

###  III.2 Long range measurements

In order to overcome the ambiguity between multiple fast-varying periods, we
then studied a beam that spirals with two components simultaneously: a slow
one and a fast one. The former can be used to determine the coarse position,
while the latter can be used to obtain the longitudinal distance with high
accuracy. Again, we first investigated the method in simulation before
implementing the scheme in the laboratory.

Figure 5: Simulation for distance measurements over a longer range. a) An
intensity structure spiraling simultaneously with two frequencies is generated
using the modulation pattern shown in the inset. As in Fig. 3, only the
complex modulation is shown and the brightness depicts the amplitude and the
color depicts the phase. The simulated beam close the focal region shows the
small off-axis intensity, that is spiraling fast as before, as well as larger
regions of increased intensity, which is simultaneously spiraling slower
around the optical axis. The regions used to emulate the quadrant detectors
are depicted with green lines. b) The angular intensity variation found in the
inner and out regions around the optical axis over propagation distance.
Comparing the intensity differences found in four quadrants in the inner
region (1,2-3,4 in black and 2,3-1,4 in red) around the optical axis leads to
a fast-varying pattern used as a local high-accuracy measurement analog as
before. Comparing the intensity difference in the outer region (ring-quad,
1,2-3,4), however, leads to a slow-varying function (blue line) with a
periodicity of 12.63 mm, which will give information about the global position
such that the high-accuracy distance measurement can be extended over multiple
periods, i.e., a longer region. Small oscillations in the slow-varying curve
are due to an inevitable cross-talk from the strong fast-oscillating signal.

As discussed in the theory section, such a dual-frequency beam can be
generated by adding an additional ring to the modulation pattern. In our
realization, we used an additional ring that was slightly bigger than the
inner circular area, with an inner ring radius of $r_{1}$ = 0.6 mm and a width
of 100 $\mu$m. The modulation pattern to generate such a beam can be found in
Fig. 5a. As before, the ring width was limited by the pixel size of the
modulating device used in the subsequent experiment. The additional ring
results in a beam having a second, much smaller rotation frequency in the
intensity, as described by equation (5). To prevent a third rotation frequency
from appearing in the intensity pattern, we imprinted the additional ring with
an OAM value of $\ell=1$, such that the pairwise interference only appears
between the circular central area and each of the rings. Again, the width of
the additional ring was chosen such that all three beam components were
similar in amplitude, thus obtaining a high-visibility structure
vetter2015optimization . The rest of the simulation remained the same as
before.

Figure 6: High-accuracy distance measurements over a long range. a) The
connected-scatter plot of intensity differences between the inner quadrants
(red and black) and the ring quadrant (blue) as shown in Fig. 5a. At every
position, 25 images were recorded. b) Displacement accuracy analysis for the
slow-varying curve (blue), again obtained using the fitted function from a),
the standard deviation of the intensity fluctuations at each position, and
error propagation. To distinguish the different steep slopes of the fast-
oscillating signals, an accuracy of around 80 $\mu$m is required (dashed
line), which we achieved over a region of about 2 mm (orange shaded region).

Because the components of the light field that rotate slower are stemming from
the components on the inner parts of the generation hologram, the resulting
interference is distributed over a larger region around the optical axis (see
Fig. 5a and insets in b). In other words, the angular intensity found in
regions of larger radii is strongly determined by the slow-varying pattern.
This spatial separation enables a simultaneous measurement of both variations:
the fast-spiraling part close to the optical axis and the slow-spiraling part
further away from the beam center. As before, we used the small circular
region with a radius of 3.75 $\mu$m around the optical axis to determine the
fast variation. The slow-varying signal was obtained by recording the
intensity in a circular, ring-shaped region with an inner radius of 7.5 $\mu$m
and a width of 22.5 $\mu$m. Upon propagation, the intensity follows a fast as
well as slow-varying sinusoidal curve when the differences between the
quadrants are evaluated (see Fig. 5b). As expected, the fast oscillation
period is again 311.2 $\mu$m, while the slow-varying signal has a periodicity
of around 12.63 mm. While the fast-oscillating period again matches nicely
with the theoretical value of 311 $\mu$m, we find a bigger discrepancy of the
slow-oscillating signal to the theoretically expected periodicity of 10.2 mm.
We attribute the latter to the fact that we only obtained only one full
fringe, which also shows an additional modulation. The additional slow-varying
signal now allows a discrimination between the different fast-oscillating
periods and, thus, a global determination of the longitudinal position. In
other words, after a first gauging of slope of the slow-varying curve with
respect to a coarse distance measure, the fast-oscillating signal can then be
used to determine the distance with high accuracy. However, a closer
inspection of the slow-varying curve shows that it is also slightly modulated
by the fast-oscillating signal, as the two differently spiraling fields are
not complete separable. While this leads to an increase of the slope in some
regions, it also flattens the curve whenever the fast-varying modulation
counteracts the change of the slow-oscillating curve. Obviously, in these
regions the accuracy of the slow-varying signal will be reduced. Thus, an
important experimental question is to determine if the detrimental effect of
this additional modulation is small enough to allow a discrimination between
the different fringes using our quadrant measurement technique: in other
words, over what range is the displacement accuracy of the slow-varying signal
less than $1/4$ of the period of the fast-varying signal?

Apart from changing the modulation pattern at the SLM, the rest of the
experimental setup remained the same as before. However, this time we scanned
over a range of 10 mm with a step size of 10 $\mu$m and recorded 25 image
frames of the spiraling structure at each position. In the data analysis, we
again used the four pixels around the optical axis (2$\times$2 pixel array) as
a quadrant detector to obtain the fast-oscillating signal. Additionally, to
measure the slow-varying signal, we measured the intensity differences between
the upper and lower quadrants (1,2 and 3,4) in a disk-shaped region with a
radius of 30 $\mu$m around the optical axis, excluding the four inner pixels
used to determine the fast-oscillating signal. As can be seen in Fig. 6a, we
find a slow-varying curve with a periodicity of 11.29$\pm$0.03 mm and fast-
oscillating fringes with periods of 312.9$\pm$0.4 $\mu$m, as expected. We also
obtain, analogous to the simulations, the additional high-frequency modulation
of the slow-varying signal. To evaluate the displacement accuracy, we first
determine the function that describes the slow-varying curve, for which we use
a sum of two sinusoidal functions whose amplitude and periodicity we obtain
from fits. Using this function; experimentally obtained errors given by the
standard deviation of the measured intensities; and error propagation, we find
that we obtain the required accuracy of less than than 80 $\mu$m ($\sim 1/4$
of the period of the fast curve) over a range of more than 7 fast-oscillating
fringes, or around 2 mm. Most displacement accuracies for the slow-varying
curve in this region are as low as 10-20 $\mu$m (see Fig. 6b). This accuracy
is enough to distinguish the different fast-oscillating fringes and the
corresponding regions having a steep slope, thereby demonstrating an accuracy
of around 2 $\mu$m over the full region, i.e., three orders of magnitude or 2
mm. We note that in principle our method does not require an initial
calibration, as the longitudinal-varying angular intensities can be
theoretically obtained from the beam dimensions as well as the focusing lens.
However, in experiments the system might be initially characterized once, such
that a camera (or quadrant detector) placed anywhere within the possible
measurement range can be used to determine an absolute position without
rescanning the entire translation region.

##  IV Conclusion

We have demonstrated that radially self-accelerating beams can be used to
determine the distance with respect to a reference with an accuracy of 2
$\mu$m over three orders of magnitude. The main benefit of our technique is
its simplicity, as only a single beam having the appropriate structure and two
quadrant detectors are required. Apart from this strong benefit, there is also
an important precaution worth mentioning. The technique requires a very
accurate alignment of the beam’s optical axis with the center of the detector
such that the recorded intensity does not move transversely while the detector
is translated. Even a single pixel of transverse displacement can cause large
discontinuous jumps in the measured intensity differences. However, once
alignment has been ensured, the measurement is stable and only requires
minimal post-processing (after having gauged the system), such that very high
read-out speeds on the order of nano-seconds might be feasible. The obtained
accuracy of the distance measurements can be further improved by using
stronger focusing optics and custom-fabricated optics for beam generation, as
well as an additional imaging system to lift the limitation of the finite
resolution of the detection system. In addition, it might be interesting to
consider more complex spiraling structures, such as an accelerated rotation
schulze2015accelerated , which should further increase the accuracy over small
regions at the cost of the accuracy elsewhere. Finally, we hope to stimulate
further research into applications that benefit from the propagation-dependent
intensity variations of radially self-accelerating beams.

## Acknowledgments

SP, SP, MO, and RF acknowledge the support from Academy of Finland through the
Competitive Funding to Strengthen University Research Profiles (decision
301820) and the Photonics Research and Innovation Flagship (PREIN - decision
320165). RF also acknowledges support from Academy of Finland through the
Academy Research Fellowship (decision 332399).

## Disclosures

The authors declare that there are no conflicts of interest related to this
article.

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