# Generation of biaxially accelerating static Airy light-sheets with
3D-printed freeform micro-optics

Yanis Taege
Laboratory for Micro-optics
Department of Microsystems Engineering (IMTEK)
University of Freiburg, Freiburg, Germany
&Tim Samuel Winter
Laboratory for Micro-optics
Department of Microsystems Engineering (IMTEK)
University of Freiburg, Freiburg, Germany
&Sophia Laura Schulz
GRINTECH GmbH
Jena, Germany
&Bernhard Messerschmidt
GRINTECH GmbH
Jena, Germany
Çağlar Ataman
Microsystems for Biomedical Imaging Laboratory
Department of Microsystems Engineering (IMTEK)
University of Freiburg, Freiburg, Germany<EMAIL_ADDRESS>
###### Abstract

One-dimensional Airy beams allow the generation of thin light-sheets without
scanning, simplifying the complex optical arrangements of light-sheet
microscopes (LSM) with an extended field-of-view (FOV). However, their
uniaxial acceleration limits the maximum numerical aperture of the detection
objective in order to keep both the active and inactive axes within the depth-
of-field. This problem is particularly pronounced in miniaturized LSM
implementations, such as those for endomicroscopy or multi-photon neural
imaging in freely-moving animals using head-mounted miniscopes. We propose a
new method to generate a static Airy light-sheet with biaxial acceleration,
based on a novel phase profile. This light-sheet has the geometry of a
spherical shell whose radius of curvature can be designed to match the field
curvature of the micro-objective. We present an analytical model for the
analysis of the light-sheet parameters and verify it by numerical simulations
in the paraxial regime. We also discuss a micro-optical experimental
implementation combining gradient-index optics with a 3D-nano-printed, fully
refractive phase plate. The results confirm that we are able to match
detection curvatures with radii in the 1.5 to 2 mm range.

_Keywords_ light-sheet microscopy, Airy beam, accelerating beams, field
curvature, two-photon polymerization

##  1 Introduction

Light-sheet fluorescence microscopy (LSM) has become an essential imaging
modality in the life sciences, providing rapid volumetric imaging with minimal
photodamage [1]. Although this makes it an ideal candidate for in-vivo imaging
such as endomicroscopy[2, 3] and neural imaging in freely moving animals[4,
5], translation to miniaturized optical systems has been limited to date[6,
7]. A critical technology for incorporating LSM into these applications is
high performance micro-objectives that can match their macroscopic
counterparts in numerical aperture (NA) and field-of-view (FOV)[8]. Their
performance comes at the cost of field curvature[9, 10], which presents an
additional challenge for LSM: With the generally planar geometry of a Gaussian
beam, only a limited portion of the illuminated area is brought into perfect
focus in the image plane, limiting the effective FOV, imposing limits on the
detection NA, or leading to loss in image contrast. Although the incorporation
of scanning micromirrors could address this challenge to some extent[11], it
further increases the system complexity and limits the achievable degree of
miniaturization. Using Airy beams [12] has been proposed to match field
curvature due to its uniaxial acceleration when implemented as a static light-
sheet.[13]. An Airy light-sheet can be generated by imposing a one-dimensional
cubic phase profile on a Gaussian beam, which is then focused by a cylindrical
lens. Similar to other configurations using propagation-invariant beams[14,
15], it has been used to excite a larger FOV compared to a Gaussian sheet,
while maintaining the same sectioning ability[16, 17]. However, incorporating
the static Airy light-sheet only partially solves the curvature problem, as
the planar Gaussian profile along its inactive axis still introduces a FOV
reduction, limiting the performance required for micro-optical applications.

We have recently shown that this challenge can be addressed by an additional
phase modulation on top of the cubic profile needed for the static Airy light-
sheet.[18] This phase profile depends linearly on the active coordinate and
quadratically on the inactive coordinate, and modifies the Gaussian profile of
the light-sheet to achieve the same radius of curvature as the Airy beam. The
result is a static, biaxially accelerating light-sheet that is unrestricted in
its ability to section, thus increasing the effective FOV of the micro-
objective. In this paper, we provide a detailed analytical derivation of the
phase-plate parameterization using typical micro-objective specifications,
allowing the light-sheet geometry to be tailored for maximized in-focus
excitation. To verify the validity of our analysis, we perform simulations in
the paraxial regime using ray tracing and beam propagation methods. By
designing, implementing, and characterizing a micro-optical excitation system
that incorporates a 3D-printed millimeter-scale phase plate, we experimentally
demonstrate our ability to generate such a light-sheet in a fully miniaturized
fashion.

##  2 Theory

Figure 1 illustrates the optical architecture of a miniaturized light-sheet
microscope with a high NA micro-objective. The illumination arm consists of a
collimator (not shown) and a cylindrical lens to form a Gaussian light-sheet
within the FOV of the micro-objective which is imaging the excited
fluorescence onto a flat image sensor. The field curvature of the detection
objective $z_{d}$ can be characterized by a radius $r_{d}\gg\text{FOV}$ so
that the detection field follows

$z_{\text{d}}(x,y)\approx\frac{x^{2}+y^{2}}{2r_{d}}.$ (1)

As common in LSM we use the coordinate system of the detection objective for
the light-sheet, in which

  * –

$z$ is the micro-objective’s optical axis, corresponding to the cylindrical
lens’ active axis,

  * –

$x$ is the optical axis of the cylindrical lens corresponding to the beam
propagating direction, and

  * –

$y$ is the inactive coordinate of the cylindrical lens.

Figure 1 extends the concept to the generation of a static Airy light-sheet,
resulting from a one-dimensional cubic phase modulation of the collimated beam
incident on the cylindrical lens, which allows field curvature matching in
$x$. This already increases the effective FOV in this coordinate compared to
the Gaussian light-sheet, and improves the sectioning capability due to a
thinner light-sheet. A secondary phase modulation of the proposed phase
profile aims at bending the Airy light-sheet along its inactive axis, thus
generating a static and yet biaxially accelerating sheet (Fig. 1). In the next
two sections we will discuss the design of the active and inactive phase
profiles to achieve biaxial field curvature matching.

Figure 1:  Comparison of three different static light-sheet illumination beams
for a typical micro-objective for a given light-sheet length and detection
numerical aperture. The detected in-focus fluorescence, which is imaged by the
micro-objective with a curved detection field (gray cone, center) onto the
image plane, is plotted on the top. In the bottom row, the intensity profiles
of the illumination beams are plotted in cross-sections along the center in
the propagation coordinate (XZ-plane, to the back) and the focal plane (XY-
plane, to the right). Black dots trace the maximum intensity of the beams
along the detection axes, visualizing the mismatch between beam- and detection
field-geometry. a: Gaussian light-sheet. Due to its planar geometry the in-
focus fluorescence is significantly smaller than the available FOV. b: Static
Airy light-sheet. Due to the matched curvature along the propagation axis, the
effective FOV is increased. In addition, the Airy beam is thinner than the
Gaussian beam with the same FOV. c: Proposed biaxial Airy beam. By adjusting
the curvature also along the inactive direction, the fluorescence collection
over the entire illumination area will be in focus.  Figure 2: Configuration
to generate a 1D-Airy light-sheet to illustrate the mathematical derivation.
An input Gaussian beam with a beam waist of $w_{0}$ is modulated by a one-
dimensional cubic phase plate. The phase plate is placed in the back-focal
plane (BFP) of a cylindrical lens that focuses the beam in the $z$ coordinate.
The resulting beam profiles, illustrated by the insets in the XZ and YZ plane,
show an Airy profile in the $z$\- and a Gaussian profile in the
$y$-coordinate.

###  2.1 Acceleration along the propagation direction

An Airy light-sheet can be generated by modulating the collimated Gaussian
beam with a beam waist $w_{0}$ by a one-dimensional cubic phase
$\phi_{\text{Ai}}$ at the back-focal plane (BFP) of the cylindrical lens [13]
along its active direction $z$, such that the electric field follows

$E^{\text{BFP}}_{\text{Ai}}(y,z)=\exp\left(-\frac{z^{2}+y^{2}}{w_{0}^{2}}\right)\cdot\exp\left[i2\pi\alpha\left(\frac{z}{w_{0}}\right)^{3}\right]$
(2)

where $\alpha$ is the amplitude of cubic modulation. A schematic depiction of
this configuration is shown in Fig. 2, where the approximate intensity profile
around the focal plane can be obtained in the paraxial domain as [19, 20]

$\displaystyle
I_{\text{Ai}}(x,y,z)\approx\left|\text{Ai}\bigg{(}\frac{z}{z_{0}}-\frac{x^{2}}{2r_{\text{Ai}}z_{0}}+i\frac{ax}{kz_{0}^{2}}\bigg{)}\right|^{2}\cdot\exp\bigg{[}2a\left(\frac{z}{z_{0}}-\frac{x^{2}}{r_{\text{Ai}}z_{0}}\right)-\frac{2y^{2}}{w_{0}^{2}}\bigg{]}$
(3)

where $\text{Ai}(\cdot)$ denotes the Airy function, $x$ is the axial
coordinate, $z_{0}=f(6\pi\alpha)^{\frac{1}{3}}/(w_{0}k)$ is a characteristic
length, $a=\left(6\pi\alpha\right)^{-2/3}$ is an apodization factor and
$k=2\pi n/\lambda$ is the wave vector in a medium of refractive index $n$[21].
It has been demonstrated by numerous groups[19, 22, 23, 24, 25] that this
function is highly asymmetrical with prominent side lobes, and its main lobe
follows a parabola with a radius of curvature $r_{\text{Ai}}$ of

$r_{\text{Ai}}=\frac{6\lambda n^{2}f^{3}}{w_{0}^{3}}\alpha.$ (4)

Therefore, one must solve Eq. 4 for $\alpha$ with $r_{\text{Ai}}=r_{d}$ to
obtain an Airy light-sheet with an acceleration matching the field curvature
of the detection objective. Using Eq. 2 we obtain

$\displaystyle\phi_{\text{Ai}}(z)=\frac{2\pi r_{d}}{6\lambda n^{2}f^{3}}\cdot
z^{3}.$ (5)

To finalize the geometry of the light-sheet,the beam needs to be matched to
the FOV of the micro-objective, which we define as the area around the focal
point within which the intensity remains larger than the half of the maximum
value.[21] Since Eq. 5 does not depend on $w_{0}$, we can control the geometry
of the light-sheet in the propagation coordinate and in the inactive
coordinate independently of each other via the focal length and the beam
waist. Explicitly, the FOV along the active direction is obtained from

$\text{FOV}_{x}=\frac{w_{0}r_{d}\sqrt{2\ln 2}}{nf},$ (6)

and along the inactive direction from

$\text{FOV}_{y}=\sqrt{2\ln 2}w_{0}.$ (7)

To obtain a symmetric FOV, the condition $r_{d}=nf$ must be satisfied, which
means that the focal length in the imaging medium must be equal to the radius
of curvature.

###  2.2 Acceleration along the inactive direction

To induce a parabolic axial shift of the focal point along the inactive
coordinate $y$ as well, we modify Eq. 3 by convolving it with a delta
distribution $\delta(\cdot)$ with the required shift in the detection
coordinate. This operation can be expressed as

$\displaystyle I_{\text{biaxial}}(x,y,z)$
$\displaystyle=I_{\text{Ai}}(x,y,z)\circledast\delta\left(z-\frac{y^{2}}{2r_{d}}\right)$
(8)
$\displaystyle\approx\left|\text{Ai}\bigg{(}\frac{z}{z_{0}}-\frac{x^{2}+y^{2}}{2r_{d}z_{0}}+i\frac{ax}{kz_{0}^{2}}\bigg{)}\right|^{2}\cdot\exp\bigg{[}2a\left(\frac{z}{z_{0}}-\frac{2x^{2}+y^{2}}{2r_{d}z_{0}}\right)-\frac{2y^{2}}{w_{0}^{2}}\bigg{]}$
(9)

where $\circledast$ denotes the convolution operation. By taking the Fourier
transform of the electric field at the focal point $x=0$ along $z$, we can
again obtain the required electric field at the BFP, which can be approximated
by

$\displaystyle
E^{\text{BFP}}_{\text{biaxial}}(y,z)\approx\exp\left(-\frac{z^{2}+y^{2}}{w_{0}^{2}}\right)\cdot\exp\left[i2\pi\alpha\left(\frac{z}{w_{0}}\right)^{3}\right]\cdot\exp\left(-i\frac{\pi}{r_{d}\lambda
f}\cdot y^{2}z\right).$ (10)

Details of this derivation can be found in Appendix A. The arguments of the
first two exponential functions in Eq. 10 represent the conventional profile
to generate an Airy beam as per Eq. 2, while the last term is an additional,
phase-only term that we define as

$\phi_{\text{inactive}}(y,z)=-\frac{\pi}{r_{d}\lambda f}\cdot y^{2}z.$ (11)

In ray optical terms, this phase profile adds an additional launch angle from
the phase plate to the cylindrical lens in its active coordinate, the
magnitude of which varies as the square of the inactive coordinate. Because of
the introduction of this additional angle in the BFP, each one-dimensional
Airy ray will be shifted in $z$ at the focal plane of the cylindrical lens.

###  2.3 Combined phase profile

A combined phase plate with a sag height of $h_{\text{PP}}$, which generates
both the one-dimensional Airy beam along the propagation coordinate, as well
as the focus-shifted Gaussian beam along the inactive coordinate of the beam,
can be obtained by

$\displaystyle h_{\text{PP}}(y,z)$
$\displaystyle=\frac{\lambda\cdot(\phi_{\text{Ai}}+\phi_{\text{inactive}})}{2\pi(n_{m}-1)}$
(12)
$\displaystyle=\frac{r_{d}}{6n^{2}f^{3}(n_{m}-1)}z^{3}-\frac{1}{2r_{d}f(n_{m}-1)}y^{2}z$
(13)

where $n_{m}$ is the refractive index of the phase plate. The individual
profiles are plotted alongside the combined one in Fig. 3. An important point
about Eq. 13 is that, the Airy-related modulation scales linearly with
$r_{d}$, while the modulation related to the inactive profile is inversely
proportional to it. This imposes a limit on the range of $r_{d}$ that can be
realized: For lower values of $r_{d}$, the $\alpha$ parameter is smaller, and
the assumptions made in the derivation of the apodized Airy beam become
invalid.[19, 25] On the other hand, the modulation generated by the inactive
profile dominates the phase plate. This leads to larger launch angles, which
may violate the paraxial approximation of the optical element. At the upper
end of the limit, it is useful to evaluate whether field curvature remains a
problem as the beam becomes flatter and thicker, bypassing the initial
problem.

Figure 3: Schematic illustration of the combined phase plate profiles for a
diameter of 1mm. a: Cubic profile $\phi_{\text{Ai}}$ along the active
direction of the cylindrical lens to generate the 1D-Airy beam. b: Inactive
profile $\phi_{\text{inactive}}$ that adds a linear phase along the active
direction, whose slope depends quadratically on the inactive direction of the
cylindrical lens. This profile leads to a focal shift in the active coordinate
such that it follows the same acceleration as the one induced by the cubic
profile. c: Linear combination of the individual profiles that generates the
biaxially accelerating light-sheet as given in Eq. 13.

##  3 Methods

###  3.1 Simulations

To assess that curvature-matching in the paraxial regime, we performed ray
tracing simulations using a commercial software (OpticStudio 22.1, Zemax LLC).
The simulation model implemented the configuration shown in Fig. 2 with the
phase profile described in Section 2.3. We calculated the curvature of the
beam in the active direction by evaluating the caustic[26] of 51 rays
displaced in the active $z$-coordinate (XZ-section in Fig. 2). For the
inactive direction, we obtained the the displacement of the rays in the image
plane by tracing 51 rays that were initially displaced along the
$y$-direction. To further investigate the validity of Eq. 9, we propagated the
field defined in Eq. 10 using a two-dimensional FFT-based beam propagation
method (BPM) in the Fresnel regime [27]. Propagation was performed to match
the ray tracing simulations, which include the same beam waist, phase plate
profile, and focal length of the cylindrical lens. 3-dimensional stacks were
calculated within a range of $\pm$300\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
around the focal point with a step size of $\Delta
z=$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, and a grid size of $(n_{x}\times
n_{y})=(2^{12}\times 2^{12})$ pixels with a width of $\Delta x=\Delta
y=$0.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. We set the input parameters
with an excitation wavelength $\lambda=$850\text{\,}\mathrm{n}\mathrm{m}$$, an
input beam waist $w_{0}=$240\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, an
effective focal length of the cylindrical lens
$f_{z}=$1.4\text{\,}\mathrm{m}\mathrm{m}$$ and a refractive index of the phase
plate $n_{m}=1.509$[28] to match the experimental setup described below. Two
radii of curvature were chosen in a typical range for micro-objectives of
$1.5\text{\,}\mathrm{m}\mathrm{m}$ and $2\text{\,}\mathrm{m}\mathrm{m}$. [8,
9, 29]

###  3.2 Fabrication and characterization

Phase plates with a diameter of $0.78\text{\,}\mathrm{m}\mathrm{m}$ were
fabricated using a commercial two-photon polymerization based 3D printer
(PPGT+, Nanoscribe GmbH & Co KG) with a commercial resin (IPS, NanoScribe GmbH
& Co KG). The height profiles were pre-processed and subsequently fabricated
on ITO-coated glass substrates using an in-house developed slicing and
printing strategy previously described [30]. After development, the height
profiles of the phase plates were measured by white light interferometry (WLI;
NewView 9000, Zygo Corporation). A 0.3-NA objective (I100384, Zygo
Cooperation) was used to obtain the surface profile of the entire plate and
then fitted with standard Zernike polynomials to quantify the shape deviation.
Since the tilt of the printing field in this system can only be controlled to
within a few microns from print to print, the tilt Zernike coefficients were
not included in the deviation analysis. The surface roughness was estimated by
measuring a $$25\text{\,}\mathrm{\SIUnitSymbolMicro
m}$\times$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ area of the phase plate
with a 0.8-NA objective (I190053, Zygo Cooperation). A high-pass Gaussian
spline filter with a cutoff frequency of
$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ was applied and the resulting root
mean square error (RMSE) was calculated. [31]

###  3.3 Experimental setup and analysis of the beam profiles

A cylindrical gradient-index (GRIN) lens with an effective focal length of
$1.4\text{\,}\mathrm{m}\mathrm{m}$ (Grintech GmbH) was bonded to the back of
the substrate after fabrication so that the phase plate was located at its
back focal plane. The entire assembly was illuminated with a fiber-based
collimator (Grintech GmbH) using a SLED with a wavelength of
$850\text{\,}\mathrm{n}\mathrm{m}$ as the light source (Fig. 4). The beam
waist of the illumination beam after the collimator was
$w_{0}=$240\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. In order to obtain the
3D-dimensional beam profile in the vicinity of the focal plane of the
cylindrical lens, the beam was imaged onto a CCD (UI-1240SE-NIR, IDS GmbH)
using a 0.28-NA objective (Plan Apo 10x, Mitutoyo AC) and a relay lens, while
the entire system was translated along the propagation direction of the light-
sheet. To compare the analytical analysis, simulations and experiments, two-
dimensional cross sections of the beam intensity along its propagation
direction at ($y=0$) and along the inactive direction at the focal plane
($x=0$) were measured. The thickness $\Delta z$ and the $\text{FOV}_{x,y}$
were determined by finding the full-width at half-maximum of the beam
intensity in the respective direction. The radius of curvature in $x$ and $y$
was determined by finding the location of the maximum intensity in the
respective coordinate and fitting Eq. 1 to these data points using a least-
squares algorithm.[32]

Figure 4:  Experimental setup used to measure the biaxially accelerating beam
profiles following the configuration in Fig. 2. For a micro-optical
implementation, input beam shaping was performed by a fiber-based collimation
unit. The glass substrate on which the phase plate has been fabricated
features a cylindrical GRIN lens at its back which focuses the modulated beam
from its backfocal plane to the focal plane. Around this plane the light-sheet
is imaged by moving an imaging arm, consisting of a 0.2-NA detection
objective, a relay lens and a CCD, in the propagation direction of the beam.

##  4 Results

Fig. 5 shows the characterization of the surface profile obtained with the
WLI. The surface errors become more pronounced towards the edge, where lift-
off from the substrate is expected, but were found to be less than
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ over a diameter of
$780\text{\,}\mathrm{\SIUnitSymbolMicro m}$. With an RMS surface roughness
below $10\text{\,}\mathrm{n}\mathrm{m}$, the phase plate can be considered an
optical quality surface.

The ray tracing simulations yielded radii of curvature that matched the design
radii within numerical accuracy. Furthermore, the simulated beam profiles
shown in Fig. 6 agree with those generated by evaluating Eq. 9. Deviations can
be narrowed down to an asymmetric intensity distribution around the focus
below $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and predicted offsets in $z$
are consistent with both design and numerical predictions. The radii of
curvature obtained from measured data matched the design values in most cases:
For $r_{d}=$1.5\text{\,}\mathrm{m}\mathrm{m}$$, the radii in the propagation
coordinate and in the inactive coordinate were found to be $0.2\pm
0.1\text{\,}\mathrm{m}\mathrm{m}$ and $0.1\pm
0.1\text{\,}\mathrm{m}\mathrm{m}$ smaller than the design value, respectively.
For $r_{d}=$2.0\text{\,}\mathrm{m}\mathrm{m}$$, the radius of the active
coordinate is equal to the designed radius, while the inactive coordinate is
$0.2\pm 0.1\text{\,}\mathrm{m}\mathrm{m}$ smaller than the designed radius.
The beam profiles themselves show the typical Airy profile, although a reduced
side lobe intensity is observed with an underestimation of the FOV in
propagation direction of $20-$36\text{\,}\mathrm{\char 37\relax}$$ and
$5-$10\text{\,}\mathrm{\char 37\relax}$$ in the inactive coordinate.

Figure 5:  Characterization of the phase plate profile with $r_{d}=$2\text{\,}\mathrm{m}\mathrm{m}$$ using white-light-interferometry. The left image depicts the measured surface profile, and the right is the tilt-corrected deviation from the ideal surface i. The inset on the right provides the high-pass filtered, high-NA scan for roughness estimation.  Table 1:  Comparison of the radius of curvature determined by Eq. 9, ray-tracing and BPM simulations with the experimental measurements. The corresponding beam profiles of the latter three are also shown in Fig. 6. Since the differences between the analytical and ray-tracing results are within the limits of numerical accuracy, they are not shown explicitly.  design radius |  | active coordinate | inactive coordinate
---|---|---|---
$r_{d}=$1.5\text{\,}\mathrm{m}\mathrm{m}$$ | analytical | $1.5\text{\,}\mathrm{m}\mathrm{m}$ | $1.5\text{\,}\mathrm{m}\mathrm{m}$
simulation (ray-tracing) | $1.5\text{\,}\mathrm{m}\mathrm{m}$ | $1.5\text{\,}\mathrm{m}\mathrm{m}$
simulation (BPM) | $1.5\pm 0.1\text{\,}\mathrm{m}\mathrm{m}$ | $1.4\pm 0.1\text{\,}\mathrm{m}\mathrm{m}$
experiment | $1.3\pm 0.1\text{\,}\mathrm{m}\mathrm{m}$ | $1.4\pm 0.1\text{\,}\mathrm{m}\mathrm{m}$
$r_{d}=$2.0\text{\,}\mathrm{m}\mathrm{m}$$ | analytical | $2.0\text{\,}\mathrm{m}\mathrm{m}$ | $2.0\text{\,}\mathrm{m}\mathrm{m}$
simulation (ray-tracing) | $2.0\text{\,}\mathrm{m}\mathrm{m}$ | $2.0\text{\,}\mathrm{m}\mathrm{m}$
simulation (BPM) | $2.1\pm 0.1\text{\,}\mathrm{m}\mathrm{m}$ | $2.0\pm 0.1\text{\,}\mathrm{m}\mathrm{m}$
experiment | $2.0\pm 0.1\text{\,}\mathrm{m}\mathrm{m}$ | $1.8\pm 0.1\text{\,}\mathrm{m}\mathrm{m}$
Figure 6: Biaxially accelerating beam profiles obtained by the analytical
evaluation (Eq. 9), the BPM-simulations and the beam profiling experiments.
The central cross-section of the beam in propagation direction (XZ-plane) and
along the inactive coordinate (XY-plane) are plotted and show the typical
Airy-profile of the beam, though with an acceleration also along the latter
coordinate. The $z$-offsets of the intensity maxima are plotted in $x$ and $y$
of both one-dimensional cross-sections. To also show that the main lobe’s
intensity resembles a spherical cap, the $z$-offsets are plotted along both
dimensions simultaneously and color-coded with the respective intensity. The
evolution of the design radii are shown alongside each plot.

##  5 Discussion and conclusions

The excellent agreement between the results of our analytical treatment and
the ray traced simulations is a good indication of the validity of the design
approach within the paraxial regime. Limitations of this approximation may
arise in different scenarios: As discussed in Section 2.3, lower values of
$r_{d}$ will lead to a reduced modulation of the Airy profile. Therefore, we
expect the radius to be underestimated in the propagation coordinate, as
observed in the experimental data. In the inactive direction, we also expect
to see an underestimated radius, but for a different reason: The higher
additional deflection leads to a gradual violation of the paraxial
approximation, resulting in aberrations and thus lower radii. On the other
hand, we observed the Airy radius to be the same as the design radius for
larger values of $r_{d}$, while the radius along the inactive direction is
slightly smaller. We attribute this to a manufacturing effect as for smaller
values of $r_{d}$, the sag height decreases and therefore requires more
accurate manufacturing capabilities. Both effects can lead to the deviations
reflected in the experimental results. Nevertheless, we have been able to
adjust both radii to a degree that does not affect the improvement in
effective FOV, which was the underlying design goal.

In summary, we have derived a phase profile to generate an Airy light-sheet
that follows the curved image plane of a typical micro-objective, while
incorporating the advantages of using an Airy beam to provide a larger FOV and
better sectioning. Ray tracing and BPM simulations in the paraxial regime
validated the analytical analysis, with deviations between the two within
numerical accuracy. A micro-optical implementation with mm-scale 3D-printed
refractive phase plate was developed and combined with GRIN optics, and
experimentally demonstrated to achieve curvature matching in the design range.

## Acknowledgements

We express our gratitude to Anders Kragh Hansen (Technical University of
Denmark) for having the initial idea of incorporating a quadratic phase
profile for curvature, as well as to Anja Borre, Madhu Veettikazhy and Peter
Andersen for discussions on the phase distribution. This project has received
funding from the European Union’s Horizon 2020 research and innovation program
under grant agreement No 871212.

## Data availability

The simulation and evaluation code used to generate the results presented is
available upon request.

## Disclosures

Sophia Laura Schulz and Bernhard Messerschmidt are full-time employees of the
company GRINTECH GmbH.

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###### List of Figures

  1. 1 Comparison of different illumination beams for static LSM
  2. 2 Schematic of the light-sheet configuration
  3. 3 Combination of the phase profiles
  4. 4 Experimental setup
  5. 5 Characterization of the phase plate
  6. 6 Simulation and experimental results

###### List of Tables

  1. 1 Resulting radii of curvature
  2. 2 Detailed results

##  Appendix A Derivation of the phase profile

To obtain the required phase profile, we need an expression of the biaxial
electric field at the focal plane $x=0$, which we can approximate from the
apodized Airy beam convoluted by the delta distribution $\delta$ as in Eq. 9
such that

$E_{\text{biaxial}}(x=0,y,z)=\text{Ai}\left(\frac{z}{z_{0}}\right)\cdot\exp\left(\frac{az}{z_{0}}-\frac{y^{2}}{w_{0}^{2}}\right)\circledast\delta\left(z-\frac{y^{2}}{2r_{d}}\right).$
(14)

The electric field at the backfocal plane can be obtained by taking the
Fourier transform along the active direction of the lens
$\mathcal{F}_{z}\\{\cdot\\}$, i.e.

$\displaystyle E^{\text{BFP}}_{\text{biaxial}}(y,z)$
$\displaystyle=\mathcal{F}_{z}\left\\{\text{Ai}\left(\frac{z}{z_{0}}\right)\cdot\exp\left(\frac{az}{z_{0}}-\frac{y^{2}}{w_{0}^{2}}\right)\circledast\delta\left(z-\frac{y^{2}}{2r_{d}}\right)\right\\}$
(15)

Since the Gaussian component is only dependent on $y$, it is neither affected
by $\mathcal{F}_{z}$, nor by the delta distribution and can thus be separated.

$\displaystyle E^{\text{BFP}}_{\text{biaxial}}(y,z)$
$\displaystyle=\mathcal{F}_{z}\left\\{\text{Ai}\left(\frac{z}{z_{0}}\right)\cdot\exp\left(\frac{az}{z_{0}}\right)\circledast\delta\left(z-\frac{y^{2}}{2r_{d}}\right)\right\\}\cdot\exp\left(-\frac{y^{2}}{w_{0}^{2}}\right).$
(16)

Using the convolution theorem we can also simplify the convolution operation
to a product, such that $\mathcal{F}_{z}$ operates individually on the
apodized Airy function, whose identity is known from Eq. 2. The Fourier
transform of a linear delta distribution is an exponential function

$\displaystyle\mathcal{F}_{z}\left\\{\delta\left(z-\frac{y^{2}}{2r_{d}}\right)\right\\}\propto\exp\left(-i\frac{k_{z}y^{2}}{2r_{d}}\right)$
(17)

where $k_{z}\approx 2\pi z/(\lambda f)$ is the wavevector in the active
coordinate. Inserting those identities into Eq. 16 leaves

$\displaystyle E^{\text{BFP}}_{\text{biaxial}}(y,z)$
$\displaystyle=\mathcal{F}_{z}\left\\{\text{Ai}\left(\frac{z}{z_{0}}\right)\cdot\exp\left(\frac{az}{z_{0}}\right)\right\\}\cdot\mathcal{F}_{z}\left\\{\delta\left(z-\frac{y^{2}}{2r_{d}}\right)\right\\}\cdot\exp\left(-\frac{y^{2}}{w_{0}^{2}}\right)$
(18)
$\displaystyle\approx\exp\left(-\frac{z^{2}}{w_{0}^{2}}\right)\cdot\exp\left[i2\pi\alpha\left(\frac{z}{w_{0}}\right)^{3}\right]\cdot\exp\left(-i\frac{\pi}{r_{d}\lambda
f}\cdot y^{2}z\right)\cdot\exp\left(\frac{-y^{2}}{w_{0}^{2}}\right)$ (19)
$\displaystyle=\exp\left(-\frac{z^{2}+y^{2}}{w_{0}^{2}}\right)\cdot\exp\left[i2\pi\alpha\left(\frac{z}{w_{0}}\right)^{3}\right]\cdot\exp\left(-i\frac{\pi}{r_{d}\lambda
f}\cdot y^{2}z\right)$ (20)

which is a Gaussian beam (first exponential function), on which the cubic Airy
phase (second exponential argument), as well as the biaxial tilt (third
exponential argument) is imposed. $\blacksquare$

###  A.1 Approximations

We would like to underline the fact that several assumptions were made in the
derivation of Eq. 20 which can be summarized by the following points

  * –

The Rayleigh range of the input Gaussian beam is much larger than the
propagation distance from the backfocal to the focal plane of the lens, i.e.
$x_{r}\gg 2f$. This ensures that the light-sheet does not expand in $y$.

  * –

All approximations made are strictly in the paraxial regime, such that the
Abbe sine condition is fulfilled, i.e. $k_{z}/k\approx z/f$.

  * –

There are several approximations made for the expression of the Airy beam, of
which one is that the apodization $a$ must be sufficiently strong, i.e. $a\ll
1$. For our application, this condidition is reasonably fulfilled if
$\alpha>2$.

  * –

All calculations are strictly proportionalities (e.g. constants like the
impedance or scaling factors like $\sqrt{2\pi}$ are not mentioned explicitly).

Any violation of those assumptions may result in aberrations of the Airy beam,
which lead to a variation in length and thickness of the light-sheet, and in
mismatch of the field curvature as discussed in Section 5.

##  Appendix B Other inactive geometries

In general, the argument of the $\delta$ function in Eq. 20 can be modified to
resemble an arbitrary focal shift in $z$ depending on a function $g(y)$, i.e.

$s(z,y)=\delta(z-g(y)).$ (21)

An example for a benchtop application is the combination of a spatial light
modulator (SLM) with a cylindrical lens to generate a 1D-Airy light-sheet. If
the imaging arm is rolled with respect to the detection arm the light-sheet
will intersect the focal plane at an angle. By setting $g(y)$ to match this
tilt one can compensate this misalignment without moving any mechanical
components by employing a phase of

$\phi_{\text{inactive}}=k_{z}\cdot g(y)$ (22)

onto the SLM.

##  Appendix C Further detailed results

Table 2: Detailed results accompying those in Table 1. In addition to the radii of curvature, the $\text{FOV}_{x,y}$ in each direction is given, as well as the thickness of the light-sheet $\Delta z$. |  | active $x$ | active $y$ |
---|---|---|---|---
design radius |  |  $r_{x}$ ($\mathrm{m}\mathrm{m}$) |  $\text{FOV}_{x}$ ($\mathrm{\SIUnitSymbolMicro m}$) |  $r_{y}$ ($\mathrm{m}\mathrm{m}$) |  $\text{FOV}_{y}$ ($\mathrm{\SIUnitSymbolMicro m}$) |  $\Delta z$ ($\mathrm{\SIUnitSymbolMicro m}$)
$r_{d}=$1.5\text{\,}\mathrm{m}\mathrm{m}$$ | analytical | 1.5 | 354 | 1.5 | 299 | 4.0
ray-traced | 1.5 | - | 1.5 | - | -
BPM | $1.5\pm 0.1\text{\,}$ | 362 | $1.4\pm 0.1\text{\,}$ | 301 | 4.0
experiment | $1.3\pm 0.1\text{\,}$ | 216 | $1.4\pm 0.1\text{\,}$ | 291 | 3.7
$r_{d}=$2.0\text{\,}\mathrm{m}\mathrm{m}$$ | analytical | 2.0 | 462 | 2.0 | 301 | 4.4
ray-traced | 1.5 | - | 1.5 | - | -
BPM | $2.1\pm 0.1\text{\,}$ | 482 | $2.0\pm 0.1\text{\,}$ | 301 | 4.4
experiment | $2.0\pm 0.1\text{\,}$ | 361 | $1.8\pm 0.1\text{\,}$ | 270 | 3.9


holds with the implicit constant independent of $N$.

###### Proof.

Let us write $R=\sqrt{N}$. We have

$\displaystyle\int_{{\mathbb{T}}^{2}}\left|\sum_{{\textbf{n}}\in
S_{N}}a_{{\textbf{n}}}\,e({\textbf{n}}\cdot x)\right|^{4}d\sigma(x)$
$\displaystyle\lesssim\sum_{0\leq
j\leq\log_{2}R}2^{-j\beta}\sum\limits_{\begin{subarray}{c}{\textbf{n}}_{1},{\textbf{n}}_{2},{\textbf{n}}_{3},{\textbf{n}}_{4}\in
S_{N}\\\
|{\textbf{n}}_{1}+{\textbf{n}}_{2}-{\textbf{n}}_{3}-{\textbf{n}}_{4}|\simeq
2^{j}\end{subarray}}|a_{{\textbf{n}}_{1}}a_{{\textbf{n}}_{2}}a_{{\textbf{n}}_{3}}a_{{\textbf{n}}_{4}}|+\|a\|_{\ell^{2}}^{4}$
(13.2) $\displaystyle\lesssim\sum_{0\leq
j\leq\log_{2}R}2^{-j\beta}I_{j}\|a\|_{\ell^{2}}^{4},$

where

$I_{j}=\max_{({\textbf{n}}_{1},{\textbf{n}}_{2})\in S_{N}\times
S_{N}}|\\{({\textbf{n}}_{3},{\textbf{n}}_{4})\in S_{N}\times
S_{N}:\;|{\textbf{n}}_{1}+{\textbf{n}}_{2}-{\textbf{n}}_{3}-{\textbf{n}}_{4}|\simeq
2^{j}\\}.$

We distinguish two regimes. If $2^{j}>R^{2\gamma-1}$, the trivial bound
$I_{j}\lesssim_{\epsilon}N^{\epsilon}$ suffices. Assume now that $2^{j}\leq
R^{2\gamma-1}$. We cover $\mathbb{S}_{+}^{1}(N)$ with arcs $\tau$ of length
$\simeq(2^{j}R)^{1/2}$. The Cordoba–Fefferman geometric argument (see for
example Section 3.2 in [10]) shows that we can split the arcs $\tau$ into
$O(1)$ many collections, such that

$\operatorname{dist}(\tau_{1}+\tau_{2},\tau_{3}+\tau_{4})\gg 2^{j}$

for each $\tau_{1},\ldots,\tau_{4}$ in each collection, subject only to the
requirement that $\\{\tau_{1},\tau_{2}\\}\not=\\{\tau_{3},\tau_{4}\\}$. To see
that this is indeed the case, note that after rescaling this is equivalent to
the fact that the sums of two arcs of length $\delta=(2^{j}/R)^{1/2}$ on
${\mathbb{S}}_{+}^{1}(1)$ are separated by $\gg\delta^{2}$. Since each
$\tau_{i}$ contains at most $C(\gamma)$ lattice points, it follows that
$I_{j}=O(1)$. The contribution of these $j$ to (13.2) is thus acceptable.

∎

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