# Axial superlocalization with vortex beams

D Koutný1, Z Hradil1, J Řeháček1 and L L Sánchez-Soto2,3 Department of Optics,
Palacký University, 17. listopadu 12, 77146 Olomouc, Czech Republic
Departamento de Óptica, Facultad de Física, Universidad Computense, 28040
Madrid, Spain  Max-Planck-Institut für die Physik des Lichts, Staudtstraße 2,
91258 Erlangen, Germany<EMAIL_ADDRESS>
###### Abstract

Improving axial resolution is of paramount importance for three-dimensional
optical imaging systems. Here, we investigate the ultimate precision in axial
localization using vortex beams. For Laguerre-Gauss beams, this limit can be
achieved with just an intensity scan. The same is not true for superpositions
of Laguerre-Gauss beams, in particular for those with intensity profiles that
rotate on defocusing. Microscopy methods based on rotating vortex beams may
thus benefit from replacing traditional intensity sensors with advanced mode-
sorting techniques.

  *

Keywords: quantum metrology, quantum Fisher information, axial resolution,
superresolution

##  1 Introduction

Discerning the nanometer‐scale details of living cells, tissues, and materials
is of utmost importance for many modern research efforts. A trail towards to
this holy grail was opened with the appearance of a set of methods, dubbed
with the generic term of superresolution microscopy [1, 2], capable of
bypassing the diffraction limit [3, 4, 5]: a barrier that was traditionally
thought to be impenetrable.

A number of these techniques can also reveal three-dimensional (3D) structural
details: relevant examples include stimulated-emission-depletion microscopy
[6], PSF engineering [8, 9, 10, 11, 12], photoactivated-localization
microscopy [7], and multiplane detection [13, 14, 15], to cite only but a few.
All of them rely on a very accurate localization of point sources; they differ
in how point objects are excited and how the corresponding emitted photons are
collected.

For 3D imaging, the emitter is fluorescently labeled and determining its axial
position is an indispensable ingredient. This problem has been throughly
examined and some impressive results have been demonstrated so far [16].
However, the fundamental depth precision achievable by any such engineering
method has been considered only lately [17, 18, 19]. The rationale behind is
to make a systematic use of the quantum Fisher information (QFI) [20] and the
associated quantum Cramér-Rao bound (QCRB) to get a measurement-independent
limit [21, 22]. This is much along the lines of the work of Tsang and
coworkers to quantify the transverse two-point resolution [23, 24, 25, 26,
27], which has led to the dispelling of the Rayleigh curse [28, 29, 30, 31].

In a recent work [32], the ultimate precision in axial localization using
Gaussian beams has been established. This limit can be attained with just one
intensity scan, as long as the detection plane is placed at one optimal
position.

In this paper, we generalize those results and derive quantum limits for axial
localization with Laguerre-Gauss (LG) beams, which carry quantized orbital
angular momentum [33]. Here, the beam waist acts as a realization of the light
emitted by a point source after e.g. mode conversion. Another relevant
situation is the reflection of the beam from a surface in surface topology
measurements, etc. By linearly superposing different LG modes, one can realize
beams with amplitude, phase, and intensity patterns that simply rotate, under
free space propagation, maintaining the transverse shape. These rotating
structures lie at the core of a variety of sensing techniques [34, 35, 36,
37].

We demonstrate that a meager part of the full (quantum) information is
available in intensity scans and only a small fraction of this can be
attributed to the rotation. This clearly confirms the potential of modal
expansions inspired by quantum information protocols [38], which allow for
reaching the QCRB and thus can be considered as the optimal measurement. Our
results make 3D superresolution imaging more feasible and potentially useful
for improving the resolution of optical microscopes.

##  2 Theoretical model

The problem we address here is to estimate the distance traveled by a vortex
beam from the beam waist to an arbitrary detection plane. We thus consider the
beam waist as an object whose axial distance is to be determined. In what
follows, we shall represent the fields using Dirac notation, for it makes it
straightforward to expand the theory to other types of light states.

We take the beam to be represented by the pure state $|\Psi(0)\rangle$, where
$z=0$ denotes the position of the object plane. The axial displacement is thus
characterized by a unitary operation

$|\Psi(z)\rangle=\rme^{\rmi G\,z}\;|\Psi(0)\rangle\,,$ (2.1)

where the Hermitian operator $G$ is the generator. To pinpoint the specific
form of $G$, it is appropriate to use the transverse-position representation
$\Psi(x,y;z)=\langle x,y|\Psi(z)\rangle$. It then follows directly from
equation (2.1) that

$\rmi\partial_{z}\Psi(x,y;z)=-G\,\Psi(x,y;z)\,.$ (2.2)

On the other hand, vortex beams are solutions of the paraxial Helmholtz
equation [39, 40]

$2\rmi k\partial_{z}\Psi(x,y,z)=\nabla_{T}^{2}\Psi(x,y,z)\,,$ (2.3)

where $k$ is the wavenumber and $\nabla^{2}_{T}=\partial_{xx}+\partial_{yy}$
is the transverse Laplacian. A direct comparison leads us to

$G\mapsto-\frac{1}{2k}\nabla^{2}_{T}\,.$ (2.4)

The detection plane is placed at $z$, wherein we perform an arbitrary
measurement. Given the formal analogies between spatial modes in wave optics
and pure states in quantum theory, and also the mathematical similarities in
describing evolution and detection of such objects, the amount of information
about the axial distance $z$ carried by the measured signal is quantified by
the QFI. For pure states, as it is our case, the QFI reduces to [21]

$\mathcal{Q}(z)=4\mathop{\mathrm{Var}}\nolimits(G)\,.$ (2.5)

Except for the factor 4, the QFI is the variance of the generator $G$ computed
in the initial state $|\Psi(0)\rangle$. According to the time-honored QCRB,
the variance of any unbiased estimator $\widehat{z}$ of the axial distance $z$
satisfies

$\mathop{\mathrm{Var}}\nolimits(\widehat{z})\geq\frac{1}{\mathcal{Q}(z)}\,,$
(2.6)

whose saturation provides the ultimate precision in axial distance estimation.

To proceed further, we take the structure of the transverse field to
correspond with LG modes

$\displaystyle\displaystyle\mathrm{LG}_{pl}(r,\phi,z)=\langle
r,\phi,z|p,l\rangle=\sqrt{\frac{2p!}{\pi(p+|l|)!}}\frac{1}{w(z)}\left[\frac{\sqrt{2}r}{w(z)}\right]^{|l|}$
$\displaystyle\times\displaystyle
L_{p}^{|l|}\left(\frac{2r^{2}}{w(z)^{2}}\right)\,e^{-\frac{r^{2}}{w(z)^{2}}}\;\exp\left(\rmi\left[\frac{kr^{2}}{2R(z)}-l\phi-\psi_{pl}(z)\right]\right)\,,$
(2.7)

where $(r,\phi,z)$ are cylindrical coordinates, $L_{p}^{|l|}(\cdot)$ is the
generalized Laguerre polynomial, $l\in\\{0,\pm 1,\pm 2,\dots\\}$ is the
azimuthal mode index and $p\in\\{0,1,2,\dots\\}$ is the radial index. The
parameters $R(z)$, $w(z)$, and $\psi_{pl}(z)$ are

$\displaystyle R(z)$ $\displaystyle=$ $\displaystyle
z\left[1+(z_{R}/z)^{2}\right],$ $\displaystyle w^{2}(z)$ $\displaystyle=$
$\displaystyle w^{2}_{0}\left[1+\left(z/z_{R}\right)^{2}\right],$ (2.8)
$\displaystyle\psi_{pl}(z)$ $\displaystyle=$
$\displaystyle(2p+|l|+1)\arctan(z/z_{R}),$

and represent the radius of curvature of the wave front, the beam radius, and
the Gouy phase [41], respectively, at an axial distance $z$ from the beam
waist located at $z=0$. Here, $z_{\mathrm{R}}=kw_{0}^{2}/2$ is the Rayleigh
length and $w_{0}$ the beam waist radius [42].

##  3 Quantum limit for axial localization with vortex beams

To facilitate the derivation of the QFI corresponding to axial displacements
of vortex beams it is advantageous to use an established correspondence
between eigenstates of a two-dimensional harmonic oscillator and paraxial
beams [43]. For the case of LG beams, one defines

$a_{\pm}=\frac{1}{\sqrt{2}}(a_{\xi}\mp ia_{\eta})\,,$ (3.1)

where $a_{\xi}$ and $a_{\eta}$ are dimensionless bosonic operators for each
independent amplitude of the oscillator and obey the standard commutation
relations $[a_{s},a^{\dagger}_{s^{\prime}}]=\delta_{ss^{\prime}}$
($s,s^{\prime}\in\\{\xi,\eta\\}$). Similar relations are obeyed by $a_{\pm}$.

The eigenstates of the harmonic oscillator $|n_{+},n_{-}\rangle$, generated by
the action of $a_{+}^{\dagger}$ and $a_{-}^{\dagger}$ on the vacuum, are
nothing but LG modes with azimuthal and radial indices given by

$l=n_{+}-n_{-}\,,\qquad p=\min(n_{+},n_{-})\,.$ (3.2)

Applying the usual definition of the momentum operator
$p_{s}={\textstyle\frac{1}{\sqrt{2}\rmi}}(a_{s}-a^{\dagger}_{s})$, we have

$\widetilde{\nabla}^{2}_{T}=p_{\xi}^{2}+p_{\eta}^{2}=(p_{\xi}+ip_{\eta})(p_{\xi}-ip_{\eta})=(a_{+}-a^{{\dagger}}_{-})(a^{{\dagger}}_{+}-a_{-})\,.$
(3.3)

If we recall (2.4), and take into account that, in the proper units,
$(x,y)\mapsto(\sqrt{2}\xi/w_{0},\sqrt{2}\eta/w_{0}\\})$, we get
$\nabla^{2}_{T}\mapsto 2\widetilde{\nabla}^{2}_{T}/w_{0}^{2}$. In this way,
the QFI of a pure LG mode $\ket{n_{+},n_{-}}$ reads:

$\mathcal{Q}(z)=\frac{1}{z_{\mathrm{R}}^{2}}(2n_{+}n_{-}+n_{+}+n_{-}+1)=\frac{1}{z_{\mathrm{R}}^{2}}[2p(p+|l|)+2p+|l|+1]\,.$
(3.4)

For the particular case of the Gaussian mode $\mathrm{LG}_{00}$ we get

$\mathcal{Q}(z)=\frac{1}{z_{\mathrm{R}}^{2}}\,,$ (3.5)

that is, the quantum bound (per single detection) is precisely the Rayleigh
range [44]. Note that the QFI is linear in $|l|$, which means that axial
localization can be improved by using LG beams with large OAM.

Other sets of Gaussian transverse modes can be characterized using the so-
called Hermite-Laguerre sphere [45, 46]. These modes are represented by a
point, of spherical coordinates $(\theta,\phi)$, on that sphere, and they are
generated by the rotated operators

$\displaystyle a_{1}(\theta,\phi)$ $\displaystyle=$ $\displaystyle
a_{+}e^{-i\phi/2}\cos\left(\frac{\theta}{2}\right)+a_{-}e^{i\phi/2}\sin\left(\frac{\theta}{2}\right),$
$\displaystyle a_{2}(\theta,\phi)$ $\displaystyle=$ $\displaystyle-
a_{+}e^{-i\phi/2}\sin\left(\frac{\theta}{2}\right)+a_{-}e^{i\phi/2}\cos\left(\frac{\theta}{2}\right)\,.$

In particular, $\theta=\pi/2$ gives rise to Hermite-Gauss (HG) modes, and
$\theta=0,\pi$ to LG modes analyzed above. Combining equations (3.3) and (3),
we get a direct generalization of (3.4); viz,

$\mathcal{Q}=\frac{1}{z_{\mathrm{R}}^{2}}[4+n_{1}+n_{2}(3+n_{2})+n_{1}(3+4n_{2})+(n_{1}-n_{1}^{2}+n_{2}+4n_{1}n_{2}-n_{2}^{2})\cos(2\theta)]\,,$
(3.7)

where $n_{1}$ and $n_{2}$ denote the eigenvalues of the corresponding number
operators. Notice that the QFI (3.7) is independent of $\phi$ and is optimized
by (3.4). This proves that LG modes are better than their HG counterparts, as
axial localization is concerned.

Figure 1: Sketch of the evolution of the superposition of two LG modes, with
$l=2$ and $l^{\prime}=0$. The interference pattern is subject to rotation as
well as divergence during propagation.

Apart from a trivial divergence, transversal intensity profiles of pure vortex
beams do not change on propagation. More complex intensity transformations can
be realized by superposing two or more vortex beams. In particular, rotating
structures are of interest in microscopy [34, 35, 36, 37]. Rotation of highly
symmetric spots is easily detected and the corresponding defocusing parameter
can be estimated from the measured rotation angle. This is roughly sketched in
figure 1.

To simplify the details as much as possible, we take the simple example of the
superposition of two LG modes with different azimuthal numbers $l\neq
l^{\prime}$, and $p=p^{\prime}=0$

$|\Psi_{ll^{\prime}}\rangle=\frac{1}{\sqrt{2}}(|\mathrm{LG}_{0l}\rangle+|\mathrm{LG}_{0l^{\prime}}\rangle)=\frac{1}{\sqrt{2}}(\ket{n_{+},0}+\ket{n^{\prime}_{+},0})\,.$
(3.8)

In this case, the QFI about the axial position of the source reads

$\mathcal{Q}=\frac{1}{z_{\mathrm{R}}^{2}}[4+2(|l|+|l^{\prime}|)+(|l|-|l^{\prime}|)^{2}]\,.$
(3.9)

Considering OAM as a resource for axial localization and using the maximum
available OAM for one of the components of the superposition,
$l=\pm|l_{\mathrm{max}}|$, the QFI is maximized when the second component is
in the fundamental mode $l^{\prime}=0$, whereupon
$\mathcal{Q}_{\max}=[4+2|l_{\max}|+|l_{\max}|^{2}]/z_{\mathrm{R}}^{2}$,and the
quantum bound becomes quadratic in $|l|$. At first sight, the better
performance of vortex superpositions (3.9) over pure vortex beams (3.4) is
somehow related to beam rotation. This might suggest a proxy for the measured
axial distance traveled from the waist to the detection plane. However, as we
show in the next section, things are not that simple.

##  4 Intensity detection

In single-parameter estimation, the QFI can always be accessed and the
corresponding QCRB saturated with a von Neumann measurement projecting the
measured signal on the eigenstates of the symmetric logarithmic derivative of
the density matrix [21]. In our context, the practical implementation of such
measurements requires a spatial mode demultiplexer/sorter that performs
simultaneous projection of the signal onto a complete orthonormal set of modes
[47, 48]. Mode separation is usually achieved with a sequence of spatial light
modulators implementing a suitable unitary transformation. Unavoidable
systematic errors and losses introduced by such complicated experimental
setups may ruin any theoretical advantage offered by optimal strategies.

In consequence, we consider the performance of the possibly inferior, but much
more robust intensity detection, because it is the simplest method at hand for
the experimentalist. As the information (3.4) and (3.7) about the axial
distance is carried by both intensity and phase of the measured beam, we might
aptly ask how much information is sacrificed by completely ignoring the phase.

Figure 2: Fisher information about axial distance (relative to the QFI) for
different superpositions
$(|\mathrm{LG}_{0l}\rangle+|\mathrm{LG}_{00}\rangle)/\sqrt{2}$ as a function
of the position $z$ (relative to the beam waist) of the detection plane.

As usual, due to the noise, the detection can be considered as a random
process. In consequence, the (normalized) beam intensity,

$p(r,\phi|z)=|\Psi(r,\phi;z)|^{2}\,,$ (4.1)

can be seen as the probability density of a detection conditional on the axial
distance $z$. We take the detection as dominated by shot noise, which obeys a
Poisson distribution [49]: although this neglects nonclassical effects, it is
still a suitable model for realistic microscopy.

The classical Fisher information about $z$, per single detection, thus reads
[50]

$\mathcal{F}=\int_{0}^{\infty}\int_{0}^{2\pi}\frac{\left[\partial_{z}p(r,\phi|z)\right]^{2}}{p(r,\phi|z)}\,rdrd\phi\,,$
(4.2)

and it is a suitable tool to quantify the information content about axial
displacements accessible from the detected transversal intensity profile. For
simplicity, we take the pixel size negligibly small, so that any sampling
effect can be ignored. We also define the radial and azimuthal Fisher
informations

$\displaystyle\mathcal{F}_{r}$ $\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{\infty}\frac{[\partial_{z}\int_{0}^{2\pi}p(r,\phi|z)\,d\phi]^{2}}{\int_{0}^{2\pi}p(r,\phi|z)d\phi}\,rdr\,,$
$\displaystyle\mathcal{F}_{\phi}$ $\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{2\pi}\frac{[\partial_{z}\int_{0}^{\infty}p(r,\phi|z)\,rdr]^{2}}{\int_{0}^{\infty}p(r,\phi|z)\;rdr}d\phi\,,$

respectively. They quantify the information of the radial and the azimuthal
intensity components due to the $z$-dependence. For example, beam rotation
only contributes to $\mathcal{F}_{\phi}$, whereas beam divergence contributes
to $\mathcal{F}_{r}$.

For pure $|\mathrm{LG}_{pl}\rangle$ modes,
$\Psi(r,\phi;z)=\mathrm{LG}_{pl}(r,\phi,z)$, the classical Fisher information
can be obtained in a closed form. First notice the integrand of (4.2) is
independent of $\phi$. Carrying out $z$-derivatives of Laguerre polynomials
using the relation $\partial_{z}L_{p}^{l}(z)=-L_{p-1}^{l+1}(z)$, and changing
the integration variable $r$ yields

$\mathcal{F}=\frac{4p!}{(|l|+p)!}\left[\frac{\partial_{z}w(z)}{w(z)}\right]^{2}\int\limits_{0}^{\infty}e^{-t}t^{|l|}\left[2t\,L_{p-1}^{|l|+1}(t)+(t-|l|-1)L_{p}^{|l|}(t)\right]^{2}dt\,.$
(4.4)

We are thus left with evaluating six different integrals involving products of
Laguerre polynomials. Compact expressions for each case can be obtained from
the generalized orthogonality relation [51]

$\int_{0}^{\infty}\rme^{-t}t^{\mu}L_{p}^{l}(t)L_{p^{\prime}}^{l^{\prime}}(t)\rmd
t=(-1)^{p+p^{\prime}}\Gamma(\mu+1)\sum_{k=0}^{\min(p,p^{\prime})}{\mu-l\choose
p-k}{\mu-l^{\prime}\choose p^{\prime}-k}{\mu+k\choose k}\,.$ (4.5)

Straightforward simplifications leave us with the final result

$\mathcal{F}=4[2p(p+|l|)+2p+|l|+1]\left[\frac{\partial_{z}w(z)}{w(z)}\right]^{2}=\frac{2p(p+|l|)+2p+|l|+1}{R(z)^{2}/4}\,.$
(4.6)

For the planes of maximal wavefront curvature $z_{\mathrm{opt}}=\pm
z_{{\mathrm{R}}}$, where $R(z_{\mathrm{R}})^{2}=4z_{\mathrm{R}}^{2}$, the
quantum limit is saturated with intensity sensitive detection

$\mathcal{F}_{\mathrm{opt}}=\mathcal{Q}\,.$ (4.7)

Thus, for any pure LG beam and any beam waist location, two detection planes
can be found, where complete information about the axial distance can be
extracted with intensity-only detection. Hence, full potential of high-order
vortex beams for axial metrology can be exploited with direct detection
techniques. This generalizes the results obtained for Gaussian beams[32].

However, numerical analysis suggests that this result does not hold for
superpositions of LG modes: a single transversal intensity scan is no longer
optimal for any detector position $z$. In figure 2 we plot the classical
Fisher information (normalized to the optimal QFI) for different vortex
superpositions. First, notice that the optimal detector position is no longer
at $z=\pm z_{\mathrm{R}}$, as it was for pure LG beams. Second, the larger the
angular momentum $l$ carried by the signal, the smaller portion of the total
QFI can be extracted with intensity measurements.

Figure 3: Red dots represent Fisher information about axial distance $z$ for
optimally placed intensity detections for the superpositions
$(|\mathrm{LG}_{0l}\rangle+|\mathrm{LG}_{00}\rangle)/\sqrt{2}$. Blue squares
are the same Fisher information, but normalized to the optimal QFI. Figure 4:
Radial (green), azimuthal (brown) and total (red) Fisher information
$\mathcal{F}$ about axial displacement as a function of detector axial
position $z$ (in units of $z_{\mathrm{R}}$) for the vortex superposition
$(|\mathrm{LG}_{02}\rangle+\mathrm{LG}_{00})/\sqrt{2}$. The sum of radial and
azimuthal components is also shown (blue). All quantities are normalized to
the corresponding optimal QFI. Notice that for small (large) propagation
distances, most FI comes from azimuthal (radial) intensity distribution. Some
information is stored in correlations between those two marginal
distributions, as can be seen from the gap between the blue and the red
curves.

This is further illustrated in figure 3. Locating intensity detectors at the
optimal detection planes for every value of $l$, the classical Fisher
information grows sublinearly with $l$, so that progressively smaller portion
of quadratic QFI is available from intensity data and the gap between the
quantum bound and the performance of the best estimator from intensity data
worsens with $l$.

For example, for $l=2$, only about $17\%$ of the QFI is available from
intensity scan at the optimal detector position, as we can see in figure 4. Of
this, only a small fraction is due to changes in the azimuthal intensity
profile, as arises, e.g., in rotations, except for very small propagation
distances, where azimuthal profile accounts for up to $10\%$ of QFI. This
ratio quickly approaches zero with increasing $l$.

The information content of rotating beam intensity distributions is
surprisingly low. As bad as it sounds, this is not a negative result, but
encouraging news for quantum metrology. It highlights inadequacy of simple
detection techniques in this particular metrological scenario, where intensity
detection fails to reveal all potentially accessible information about the
parameter of interest. The hidden Information has to be accessed with advanced
detection techniques. Here a generic tool is projecting/sorting the signal
into optimal set of spatial modes derivable from the quantum detection theory.
Hence, we uncover a huge potential of axial superlocalization based on
spatial-mode projections applied to higher order vortex beam superpositions
[44].

##  5 Concluding remarks

We have established the ultimate quantum limits for axial localization using
vortex beams. For pure LG beams, this limit is attained with an intensity scan
with the detector located at one of two optimal planes. For superpositions of
LG beams, in particular of those with intensity profiles rotating on
defocusing, this is no longer true. This means that microscopy methods based
on rotating vortex beams may benefit from replacing traditional intensity
scans with advanced mode-sorting techniques.

We thank Robert W. Boyd and Aaron Z. Goldberg for helpful discussions. We
acknowledge financial support of the project ApresSF, under the QuantERA
programme, which has received funding from the European Union’s Horizon 2020
research and innovation programme, the program H2020 (project StormyTune),
Grant Agency of the Czech Republic (Grant No. 18-04291S), the Palacký
University (GrantNo. IGA_PrF_2020_004), and the Spanish MINECO (Grant
PGC2018-099183-B-I00).

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