# Direct laser written aperiodic photonic volume elements for complex light
shaping with high efficiency: inverse design and fabrication

Nicolas Barré  Institute of Biomedical Physics, Medical University of
Innsbruck, Müllerstraße 44, 6020 Innsbruck, AT  Institute of Photonic
Technologies, Friedrich-Alexander-University Erlangen-Nürnberg, Konrad-Zuse-
Straße 3/5, Erlangen 91052, DE  Erlangen Graduate School in Advanced Optical
Technologies (SAOT), Friedrich-Alexander-University Erlangen-Nürnberg, Paul-
Gordan-Straße 6, 91052 Erlangen, DE  Corresponding authors:
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>Ravi Shivaraman  Department of Engineering
Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK  Simon Moser
Institute of Biomedical Physics, Medical University of Innsbruck, Müllerstraße
44, 6020 Innsbruck, AT  Patrick Salter  Department of Engineering Science,
University of Oxford, Parks Road, Oxford OX1 3PJ, UK  Michael Schmidt
Institute of Photonic Technologies, Friedrich-Alexander-University Erlangen-
Nürnberg, Konrad-Zuse-Straße 3/5, Erlangen 91052, DE  Erlangen Graduate School
in Advanced Optical Technologies (SAOT), Friedrich-Alexander-University
Erlangen-Nürnberg, Paul-Gordan-Straße 6, 91052 Erlangen, DE  Martin J. Booth
Department of Engineering Science, University of Oxford, Parks Road, Oxford
OX1 3PJ, UK  Erlangen Graduate School in Advanced Optical Technologies (SAOT),
Friedrich-Alexander-University Erlangen-Nürnberg, Paul-Gordan-Straße 6, 91052
Erlangen, DE  Alexander Jesacher  Institute of Biomedical Physics, Medical
University of Innsbruck, Müllerstraße 44, 6020 Innsbruck, AT  Erlangen
Graduate School in Advanced Optical Technologies (SAOT), Friedrich-Alexander-
University Erlangen-Nürnberg, Paul-Gordan-Straße 6, 91052 Erlangen, DE
Corresponding authors<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Light plays the central role in many applications. The key to unlocking its
versatility lies in shaping it into the most appropriate form for the task at
hand. Specifically tailored refractive index modifications, directly
manufactured inside glass using a short pulsed laser, enable an almost
arbitrary control of the light flow. However, the stringent requirements for
quantitative knowledge of these modifications, as well as for fabrication
precision, have so far prevented the fabrication of light-efficient aperiodic
photonic volume elements (APVEs).

Here we present a powerful approach to the design and manufacturing of light-
efficient APVEs. We optimize application-specific 3D arrangements of hundred
thousands of microscopic voxels and manufacture them using femtosecond direct
laser writing inside millimeter-sized glass volumes. We experimentally achieve
unprecedented diffraction efficiencies up to 80%, which is enabled by precise
voxel characterization and adaptive optics during fabrication.

We demonstrate APVEs with various functionalities, including a spatial mode
converter and combined intensity shaping and wavelength-multiplexing. Our
elements can be freely designed and are efficient, compact and robust. Our
approach is not limited to borosilicate glass, but is potentially extendable
to other substrates, including birefringent and nonlinear materials, giving a
preview of even broader functionalities including polarization modulation and
dynamic elements.

##  1 Introduction

Light has become an indispensable tool in our modern day societies. It plays
the central role in countless technological solutions, such as in the
information transport along glass fibers or in various display, imaging and
sensing applications [1]. The increasing significance of light as a tool
sparked a sharply rising demand of technologies that enable the temporal and
spatial sculpting of light in user-defined ways [2, 3, 4].

Of notable importance are so-called multiplexing applications, where one of
many pre-defined output fields can be created upon presenting a unique pre-
defined input field. Elements that can handle such tasks with high efficiency
would find numerous applications in the analysis and synthesis of light
fields. Examples range from conventional spectroscopy to complex tasks around
information processing and transport. Mode-division multiplexing for example
is a potential solution for avoiding the threat of reaching an upper limit in
communication speed (“capacity crunch”) [5, 6, 7, 8]. Meanwhile optical
computing is currently experiencing a revival thanks to novel ways of
manufacturing computer-designed optical networks which enable sophisticated
data processing at the speed of light [9, 10, 11, 12, 13].

Optically recorded volume holograms, e.g. from photorefractive materials [14],
have traditionally been used for multiplexing applications [15, 16]. However,
creating such holograms requires the realization of matching physical
interference patterns at the recording step, and the approach is also limited
to photosensitive substrates, which severely limits its practical
applicability.

Aperiodic Photonic Volume Elements (APVE) can circumvent these limitations:
they are designed on the computer and open to a multitude of different
manufacturing techniques. For instance, they can be directly written into a
substrate by locally changing its 3D refractive index (RI) distribution in a
voxel by voxel fashion. However, a major challenge in the production of APVEs
is to meet the high accuracy requirements for manufacturing since producing an
efficient APVE puts high demands on the ability to modify the 3D RI
distribution with high precision. Since the output light field arises from the
interference of thousands of scattered waves, even small errors made on the
voxel-scale quickly accumulate and severely degrade the output field quality.
For this reason, past demonstrations of manufacturing APVEs in glass have been
limited to proof of concept studies with rather low diffraction efficiencies
[17] or restricted to 2D configurations [18]. Recently, 2-photon
polymerization additive manufacturing of multi-layer holograms [9] and the
fabrication of 3D graded index materials [10] have been demonstrated.

Here we introduce a novel approach for realizing APVEs using direct laser
writing in transparent dielectric media such as glass. Our method allows for
obtaining high diffraction efficiencies of up to 80%, many times more than
what could previously be achieved [17]. This is made possible by employing
precise tomographic voxel characterization [19] in combination with a design
algorithm based on numerical beam propagation and the use of adaptive optics
to ensure space invariant voxel shapes throughout the full body of the APVE.
We experimentally demonstrate the design and manufacture of three different,
highly integrated APVEs that are optimized for intensity shaping, spatial mode
and wavelength multiplexing.

We believe that our results represent an important step towards the
realization of robust and highly integrated 3D light shapers for many
important application fields, such as information transport [8], optical
computing [9, 10, 11, 12, 13], the imaging through multimode fibers [20] and
nonlinear photonics [21]. They may further help to obtain fundamental insights
in the behaviour of quantum states of light upon scattering [22].

##  2 APVE concept

Figure 1 (a) illustrates the basic concept of our photonic volume element,
which is a laser processed 3D region inside a glass substrate, containing
hundreds of thousands of spatially separated voxels of modified refractive
index at pre-defined positions. Our approach is based on past demonstrations
of direct laser written aperiodic volume optics [17], but also inspired by
previous work on computer-generated 2D holograms based on waveguides [23].

A single voxel measures only 1.75$\times$7.5$\times$10 µm3 and is created by
moving the z-stage (the long horizontal axis) for 10 µm during exposure to the
laser focus. Fabrication of the whole device takes about 20 minutes. The
quantitative RI profile of a single voxel across the x-y-axes, fabricated in
Eagle glass, is shown in (b). This profile must be precisely known, as it
serves as the basis for the numerical design process. The RI profile
characterization is performed in a home-build tomographic microscope [19]. The
RI distribution of each voxel is assumed to be invariant along the
z-direction. The input and output fields of the APVE are coupled through its
smallest end facets. A widefield transmission image of a fabricated APVE is
shown in (c).

Figure 1: Light manipulation with an aperiodic photonic volume element. (a)
Sketch of a laser processed glass substrate containing many voxels of modified
RI. (b) Tomographically measured RI cross section of a single voxel. (c)
Widefield image taken from a fabricated device.

Our design algorithm operates on a Cartesian grid of size
$I\times\leavevmode\nobreak\ J\times\leavevmode\nobreak\ K$, following the
directions $(x,y,z)$ of the coordinate system sketched in (a), which defines
potential voxel positions. For our demonstrators, $I=55$, $J=14$ and $200\leq
K\leq 400$ depending on the device extension along the propagation direction
$z$. This corresponds to a maximum number of voxels varying between 154,000
and 308,000. The number of voxels in the transverse $x$\- and $y$-directions
are quite different due to the elongated nature of our reference voxel
represented in (b). Basically, the algorithm simulates the light flow through
the device using a beam propagation method [24] and decides upon the
activation of a voxel, depending on whether it would improve the output
wavefront or not. For our experimental demonstration, we explored only binary
designs made of identical voxels depicted in (b), but the inverse design
algorithm we implemented generalizes well to multilevel RI modifications.
Details about this algorithm are provided in the Appendix F.

The successful implementation of our APVE requires not only precise knowledge
about the voxel properties, but also the manufacturing abilities to produce
the same profile reliably at all targeted substrate depths. This can be
ensured using dynamic wavefront control, which compensates for spherical
aberrations arising when focusing the laser into the substrate [25]. Further
information about the manufacturing is also provided in Appendix E.

##  3 Results

In the following, we explore three types of APVEs, designed for specific
applications with different levels of complexity. The first one is a single-
mode intensity shaper with a length of only 2 mm and discussed in section 3.1,
the second one is a multi-colour (RGB) multiplexer of 3 mm length and
discussed in section 3.2. Finally, a Hermite–Gaussian 6-mode sorter of 4 mm
length is discussed in section 3.3.

###  3.1 Sculpting intensity distributions

Our APVEs can be optimized to shape user-defined intensity patterns with high
efficiency. To demonstrate this, we designed and fabricated an element
producing a microscopic “smiley” at its output facet (i.e. the last layer of
voxels) when read out with a Gaussian beam at zero degree readout angle
(vacuum wavelength $\lambda_{0}$ = 640 nm, waist $w_{0}$ = 40 µm). The
physical dimensions of the element are 0.1$\times$0.1$\times$2 mm3 and it
contains 55 $\times$ 14 $\times$ 200 voxels. As mentioned earlier, the unequal
grid sizes along the $x$\- and $y$-directions are due to the anisotropic voxel
shape, which is a consequence of the likewise anisotropic shape of the focus
creating the voxel. The device itself, i.e., the 3D region containing
fabricated voxels, is embedded inside a larger glass chip for easier handling.

Figure 2 provides results obtained with this APVE. It shows the designed
output intensity template for a Gaussian input of 40 µm waist as well as the
simulated and experimentally obtained results. The simulated result was
obtained by numerical propagation of the input beam through a computer model
of the optimized voxel arrangement. It thus marks an upper limit for the
quality achievable in the experiment. The experimental result was obtained by
imaging the output plane with a regular widefield microscope (see Appendix B
for additional information about the experimental APVE characterization).

Figure 2: Results from a “smiley” generator. Left: Designed output intensity;
Middle: simulated readout; right: experimental result. The total light
efficiency $\eta_{tot}$ of the experimental result is about 80%. The scale bar
measures 20 µm.

The diffraction efficiency of the APVE has been determined by calculating an
overlap integral of the target amplitude pattern $u_{t}$ with the simulated
and experimentally measured output amplitude patterns $u_{out}$ over the area
of the observable end facet $A$:

$\eta=\left|\int_{A}u_{t}\leavevmode\nobreak\ u_{out}^{*}\leavevmode\nobreak\
dA\right|^{2},$ (1)

where the powers of $u_{t}$ and $u_{out}$ are both normalized to 1. We
introduce another efficiency metric $\eta_{tot}$, which further takes into
account scattering losses into non-observable angles:

$\eta_{tot}=T\leavevmode\nobreak\ \eta,$ (2)

with transmission factor $T\leq 1$. Therefore, $\eta$ quantifies the power
fraction of the measurable _output_ light111the light arriving at the
(simulated or real) camera plane which is effectively turned into the target
distribution, while $\eta_{tot}$ is the respective conversion efficiency for
the _input_ beam. Losses due to Fresnel reflections at the input/output facets
of the APVE are neglected in this consideration.

The theoretical and experimental diffraction efficiencies $\eta_{tot}$ of the
smiley converter are 90% and 80%, respectively. We note that $T\approx$1 for
the smiley shaper in both simulation and experiment, i.e., the amount of
scattering losses are close to zero. Since this application aims at intensity-
only shaping, the phase distributions of $u_{t}$ and $u_{out}$ are irrelevant
and thus assumed to be flat in Eq. 1. We further derived an error metric
$\epsilon$, defined as the root mean square (RMS) difference between
normalized target and simulated/experimental output intensities. We find
$\epsilon_{\mathrm{sim}}=0.25,\leavevmode\nobreak\
\epsilon_{\mathrm{exp}}=0.35$.

Therefore, even though the intensity shaping shown here can be easily achieved
using 2D diffractive optical elements, we are nevertheless able to simply
demonstrate that our computational design approach can be faithfully
implemented through the experimental characterization and reproducible
manufacture of 3D voxel arrangements inside glass.

###  3.2 Wavelength multiplexing

We demonstrate the feasibility of colour multiplexing by designing and
fabricating a multi-colour “smiley” shaper, where different parts of the
smiley (eyes, mouth, head) appear only for specific readout wavelengths as
illustrated in Fig. 3. This APVE measures 0.1$\times$0.1$\times$3 mm3 and
contains 55 $\times$ 14 $\times$ 300 voxels. When read out at 640 nm with a
beam waist of $w_{0}$ = 40 µm, only the circular head appears at the output.
Similarly, the eyes appear for a readout wavelength of 543 nm ($w_{0}$ = 30
µm) and the mouth for 455 nm ($w_{0}$ = 20 µm). Results obtained from
simulated readouts and experiments using fiber-coupled lasers (see Appendix C
for details) are summarized in Fig. 3. The image table depicts false-colour
intensity distributions at the output facet of the APVE, which are
individually normalized to their respective peak intensities. The fourth image
column on the right shows a computer-generated overlap of the three readouts.

Figure 3: Wavelength multiplexing. Different parts of the smiley appear
depending on the readout wavelength. Top row: target intensity patterns used
for the APVE design. Middle row: results from a simulated readout. Bottom row:
experimental readouts.

The simulated and experimentally determined power conversion efficiencies
$\eta$ from the three Gaussian input beams to the respective facial structures
are summarized in table 1. The table values are calculated using Eq. 1, i.e.,
they show which percentage of the _output_ laser power is shaped into the
respective facial structures. The average simulated efficiency over all
colours is 67% for the simulation and 55% in the experiment. From the table
values it is noticeable that in particular the blue wavelength shows the
largest disparity between simulation and experiment, with an efficiency of 64%
in the simulation and 49% in the experiment. This may be attributed to small
remaining differences between the assumed and real voxel profiles at length
scales of a few hundred nanometres. Such small irregularities will especially
affect shorter wavelengths and are likely to be overlooked by our tomographic
voxel characterization method, which employs the same wavelength for
inspection and a smoothing total variation norm regularizer [19]. The
simulated and experimentally measured transmission values $T$ for all three
readout wavelengths are shown in table 2. We observe well matching numbers
between simulation and experiment for the red and green wavelengths, while the
blue transmission value is around 25% lower in the experiment than predicted
by the simulation.

$\lambda_{0}/\textrm{nm}$ | head | eyes | mouth
---|---|---|---
640 | 77 | 2 | 1
543 | 8 | 60 | 7
455 | 3 | 7 | 64

$\lambda_{0}/\textrm{nm}$ | head | eyes | mouth
---|---|---|---
640 | 65 | 7 | 6
543 | 5 | 50 | 6
455 | 1 | 4 | 49

Table 1: Table of simulated (left) and experimentally (right) obtained conversion efficiencies $\eta$. For each wavelength, the numbers state the respective percentage of the output power forming head, eyes and mouth. | 640 nm | 543 nm | 455 nm
---|---|---|---
| (head) | (eyes) | (mouth)
T (sim) | 83 | 82 | 87
T (exp) | 84 | 82 | 66
Table 2: Table of simulated and experimentally measured transmission factors
$T$ for the colour smiley APVE.

The simulated (experimental) total efficiencies $\eta_{tot}$ for the head,
eyes and mouth features are therefore 64% (55%), 49 (41%) and 56% (32%).

We further investigated the wavelength-dependent APVE properties using a
fiber-coupled monochromator (Polychrome IV from TILL Photonics), whose output
wavelength was tuned from 420 nm to 680 nm in steps of 10 nm. In contrast to
the laser readouts where each beam had an individual, optimal waist value, the
beam waist of the monochromator light at the APVE input facet was about 30 µm
for all wavelengths (details are described in Appendix C). While this means
that the monochromator results are expected to be somewhat sub-optimal they
nevertheless provide valuable information about the wavelength dependence of
the APVE.

Figure 4: Power conversion efficiencies of the multi-colour APVE. The solid
curves indicate the measured percentage of the output power transformed into
the features "mouth", "eyes" and "head" depending on the readout wavelength.
The dashed lines correspond to simulated readouts.

At each wavelength we took images of the input beam and the APVE output (see
data in Appendix C), which allowed us to calculate wavelength dependent power
conversion efficiencies according to Eq. 1 for each spatial feature (mouth,
eyes, head). These efficiencies are plotted in Fig. 4. Of note, the peaks
appear at wavelengths which almost exactly match the targets at 455 nm, 543 nm
and 640 nm. The measured efficiencies $\eta$ at these wavelengths are 63%, 57%
and 63% and thus close to the simulation values obtained from the laser
readouts as shown in table 1 (left). Interestingly, the blue monochromator
image of the "mouth" shows a significantly higher efficiency than the
respective experimental blue laser readout (63% vs. 49%).

Figure 4 further contains data from simulated readouts (dashed lines), which
resemble the experimental data. To match the experimental conditions, these
simulations assumed a beam waist of 30 µm for all readout wavelengths, which
cause the peak efficiency values to be slightly lower than the values in table
1, which assume readouts taken at the designed waist values.

###  3.3 Angular multiplexing

The high angular selectivity of a photonic volume element allows for encoding
different outputs for varying input angles. This characteristic makes APVEs
highly interesting for tasks such as mode division multiplexing and sorting.
To investigate the feasibility of angular multiplexing for our approach, we
have designed a APVE measuring only 0.1$\times$0.1$\times$4 mm3 and containing
55 $\times$ 14 $\times$ 400 voxels, which produces different Hermite-Gaussian
(HG) modes depending on the angle of incidence (AOI) of a Gaussian readout
beam.

Such a device could be used as a mode division multiplexer to enhance data
transfer speed as sketched in Fig. 5: multiple signals are delivered by single
mode fibers in a triangular arrangement, which has been shown to facilitate
high Hermite-Gaussian (HG) mode conversion efficiencies [26]. The fiber ends
are in the focal plane of a convex lens, which collimates the light leaving
the fiber outputs and sends it into the APVE at different, fiber-specific
propagation angles. The APVE converts each signal into a particular HG mode
travelling in parallel to the optical axis, such that it can be efficiently
coupled into a single multimode fiber.

Figure 5: Principle of mode division multiplexing with our mode sorter.
Multiple signals are delivered via single mode fibers, arranged in a triangle.
A lens gives each input beam a specific AOI. The APVE transforms each input
beam into one of six different HG modes.

We designed a mode sorting APVE for readout with a Gaussian input of 640 nm
wavelength and 25 µm waist. Altering the input beam’s AOI by merely 1.4
degrees (defined in air) transforms the output field from one particular
Hermite-Gaussian (HG) mode to another. In total we encode six different HG
modes as shown in Fig. 5.

Figure 6 shows the output intensities and phases resulting from a simulated
readout of the mode sorting APVE. The matching experimental data are shown in
Fig. 7. For both, simulation and experiment, the numerical aperture (NA) at
the output needs to be reduced in order to block stray light produced by the
element. This reduces the overall light efficiency but enhances the quality of
the generated modes. In simulation the NA was restricted to 0.02, in
experiment to about 0.05.

Figure 6: Simulated results from the mode sorter. The images show intensities
(top row) and phases when reading out with a Gaussian beam ($w_{0}$ = 25 µm,
$\lambda_{0}$ = 640 nm) at six different AOIs. Each angle produces a different
HG mode at the output. The saturation of the phase images is weighted by the
intensity for enhanced clarity. Figure 7: Experimental results from the mode
sorter. The images show intensities (top row) and phases when reading out the
APVE with a Gaussian beam ($w_{0}$ = 25 µm, $\lambda_{0}$ = 640 nm) at six
different incidence angles.

As for the APVEs discussed previously, the overall quality of the mode sorter
is governed by two factors: (i) the purity of the generated modes and (ii) the
transmission of the APVE quantified by parameter $T$. Although the element
itself absorbs almost no light, there is some power loss caused by the
aperture stop restricting the imaging NA. We quantified point (i) by
calculating efficiency values in analogy to Eq. 1 and taking the different HG
modes as target fields:

$\eta_{i,j}=\left|\int_{A}\mathrm{HG}_{i,j}\leavevmode\nobreak\
u_{out}^{*}\leavevmode\nobreak\ dA\right|^{2},$ (3)

where $\mathrm{HG}_{i,j}$ and $u_{out}^{*}$ represent the target mode and
conjugate output field with their total powers normalized to 1. An efficiency
value of $\eta_{i,j}=1$ means that the field has been shaped into a perfect
$\mathrm{HG}_{i,j}$ mode, whereas $\eta_{i,j}=0$ means that the field is
orthogonal to this mode. Tables 3 and 4 contain efficiency values of the
simulated and experimental mode sorter. The off-diagonal terms represent
undesired crosstalks into neighbouring channels. The purest modes have
efficiency values of 90% in the simulation and 88% in the experiment. The
highest crosstalk values are 3% for the simulation and 5% for the experiment.

The mode-dependent transmission of the APVE (ii), including both losses due to
scattering and spatial filtering by the restricted imaging NA, are summarized
in table 5. These results are still far from what could be theoretically
obtained with a gradient index design [27], but they are surprisingly good for
a binary design. Introducing different types of voxels as new degrees of
freedom for the inverse design optimization could significantly improve the
capabilities of these mode multiplexers and approach the maximum performance
of gradient index designs, for which a mature manufacturing technology is not
yet available in glass.

input angle no. | HG00 | HG10 | HG20 | HG01 | HG11 | HG02
---|---|---|---|---|---|---
1 | 90.6 | $0.9$ | $0.6$ | $2.0$ | $0.1$ | $0.3$
2 | $0.2$ | 89.1 | $0.9$ | $0.3$ | $3.4$ | $0.1$
3 | $0.9$ | $0.5$ | 90.2 | $0.4$ | $0.2$ | $0.1$
4 | $0.9$ | $1.1$ | $0.2$ | 87.2 | $0.1$ | $0.4$
5 | $0.1$ | $1.0$ | $0.0$ | $0.2$ | 93.9 | $0.2$
6 | $1.1$ | $0.1$ | $0.1$ | $1.6$ | $0.1$ | 79.6
Table 3: Table of simulated efficiency values $\eta_{i,j}$ in percent. For each input angle, the numbers state the respective power fractions (in percent) of the transmitted output light that is shaped into the corresponding HG modes. input angle no. | HG00 | HG10 | HG20 | HG01 | HG11 | HG02
---|---|---|---|---|---|---
1 | 88.4 | $1.1$ | $2.6$ | $0.3$ | $5.0$ | $3.5$
2 | $0.3$ | 87.0 | $0.3$ | $1.8$ | $0.7$ | $2.5$
3 | $0.7$ | $0.5$ | 83.8 | $0.1$ | $0.1$ | $0.2$
4 | $0.6$ | $1.6$ | $0.1$ | 83.7 | $0.4$ | $4.4$
5 | $2.3$ | $1.4$ | $0.7$ | $0.6$ | 86.6 | $1.2$
6 | $1.0$ | $1.4$ | $1.2$ | $4.5$ | $1.3$ | 65.7
Table 4: Table of experimental efficiency values $\eta_{i,j}$ in percent. | HG00 | HG10 | HG20 | HG01 | HG11 | HG02
---|---|---|---|---|---|---
T (sim) | $50.0$ | $49.2$ | $49.5$ | $42.2$ | $57.9$ | $48.9$
T (exp) | $21.6$ | $29.6$ | $20.4$ | $38.8$ | $33.0$ | $32.3$
Table 5: Table of simulated and experimental transmission values $T$. For each
mode, the numbers state the output/input power ratio in percent.

##  4 Discussion

In this discussion section, we place our method in the context of related
technological approaches.

### Optically recorded volume holograms

An important difference between traditional volume holograms and our APVEs
exists in the physics of their creation. Traditional holograms are optically
recorded by transferring the properties of interference patterns into a
photosensitive material, such as a photopolymer [28, 29] or a photorefractive
crystal [30], which comes naturally with limitations. Firstly, each desired
function must be physically realized at the recording step, which becomes
unfeasible for advanced multiplexing applications. Secondly, one is bound to a
limited number of suitable photosensitive materials. Finally, the dynamic
range of optically recorded holograms is quickly consumed by only a few
multiplexed recordings, which has an adverse effect on the obtainable
diffraction efficiency [10].

### Meta-holograms

In contrast to the APVE concept presented here, meta-holograms rely on
nanostructured metallic or dielectric surfaces, on which “meta-atoms” interact
with light in pre-defined ways [31, 32, 33, 34]. While the concept has proven
to be especially powerful for wavelength and polarization-multiplexing tasks,
angular multiplexing, i.e., the generation of different output fields at
varying input angles, appears harder to achieve: only up to four multiplexed
fields could be generated upon a large angular variation of more then 60 deg.
[35]. Some folded multiplane designs using metasurfaces have been realized
recently in order to increase the multiplexing capabilities of these meta-
holograms [36]. Yet, the advantages of using metasurfaces in this context
seems limited since 3 reflections are required to achieve only a 3-mode
sorter, which is not significantly better than what more conventional cascaded
diffractive optical elements can provide [9].

### Silicon Photonics

The field of silicon photonics [37, 38], i.e., the manufacturing and use of
photonic integrated circuits (PICs) inside silicon, is attracting considerable
attention since more than a decade. The fabrication of photonic chips can rely
on existing infrastructure and lithographic techniques used in the electronic
semiconductor industry and enables structure sizes down to a few nanometres.
The field of silicon photonics is comparably mature, the demonstration of
wavelength multiplexing for instance dates back to the 1990s [39] and has been
further developed towards high integration by employing modern tools of
inverse design [40]. Spatial multimode operations have been developed later,
but are likewise possible in a highly integrated fashion [41]. Compared to
silicon photonics, which is bound to infrared wavelengths, our approach offers
a wavelength window reaching down to the UV, depending on the substrate, and
is potentially applicable to a larger variety of materials offering different
optical properties, including electro-optical and nonlinear ones. Furthermore,
glasses as substrate facilitate novel applications linking optical and
microfluidic technologies [42] and allow for easier light coupling compared to
silicon, where mode mismatch can cause important insertion losses [43].
Lastly, the manufacturing of 3D devices is more innate to direct laser writing
than it is to standard lithographic techniques [44] and puts also lower
demands on the required infrastructure (no clean room required).

### Writing voxels for data storage

From the manufacturing point of view, our approach is related to storing data
by writing of voxels inside polymers or glass [45, 46]. This application has
already reached an impressive maturity level, employing birefringent multi-
level voxels [47]. However, in contrast to our application, data storage
applications put significantly lower demands on the knowledge about the
quantitative 3D properties of a single voxel, as well as their reproducible
and spatially invariant fabrication, since they can be read out sequentially
and any slight deviation from the assumed voxel shape would be irrelevant
since only a few Bits are encoded in a single voxel. Conversely, hundreds of
thousands voxels act jointly on the readout light in our APVEs, such that even
small systematic errors at the single voxel level would have pronounced
adverse effects on the sculpted fields.

##  5 Conclusion and summary

We present a novel design and fabrication concept for robust and light
efficient aperiodic photonic volume elements. Our design algorithm uses a
mode-matching method and numerical beam propagation. The manufacturing is
based on directly writing an optimized 3D arrangement of voxels into the
volume of a transparent dielectric such as glass using a focused femtosecond
laser. Each voxel consists of a small volume (approx.
1.75$\times$7.5$\times$10 µm3) of slightly increased RI. Precise knowledge of
the properties of a single voxel, in conjunction with an advanced fabrication
technique, allowed us to physically realize APVEs of high efficiencies up to
80%.

We experimentally realize three different APVE designs demonstrating different
functionalities: An intensity shaper turning a Gaussian beam into a smiley-
shaped light distribution, a wavelength multiplexer producing different parts
of the smiley for three different readout wavelengths and a spatial mode
multiplexer, which converts a Gaussian input beam into particular HG modes
depending on the angle of incidence.

While our proof-of-concept studies already show unprecedented performance, our
APVE concept still has room for improvement in several respects, promising
significantly higher efficiency and more complex functionalities in the
future. For instance, our APVEs are made from a single voxel type, i.e., they
are binary devices. However, it is straightforward to extend the design
concept to non-binary devices. At the manufacturing side the use of varying
laser powers, focus shapes, or multi-pass writing could be used to fabricate
multiple different RI profiles. Non-binary devices will be able to store more
information and exhibit higher efficiencies. In addition, our concept is
extensible to different types of substrates, including birefringent, electro-
optic and nonlinear substrates, potentially offering the possibility of
polarization shaping, dynamic or even erasable ferroelectric APVEs [48].

## Acknowledgments

We thank Lisa Ackermann and Clemens Roider for valuable contributions in the
form of scientific discussions as well as for post processing the APVE
samples.

## Funding

The work is funded by the FWF (I3984-N36), the EPSRC UK (R004803/01) and the
DFG (409765270).

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##  Appendix A Appendix

##  Appendix B APVE characterization

Figure 8: Optical setup for characterizing APVEs. A motorized tip/tilt mirror
is imaged onto the input facet of the APVE, which rests on a manual x/y
microstage. The APVE can be simultaneously exposed to three laser beams
(red/green/blue) or optionally to light form the monochromator (not shown).
The APVE output facet is imaged from below, where the imaging NA can be
controlled by an aperture stop. The setup further features a Mach-Zehnder
interferometer for characterizing the mode sorter APVE.

The APVEs are characterized with the home-built imaging system shown in Fig.
8. The setup allows one to focus fiber coupled laser beams of three different
wavelengths (Toptica ibeam smart @640 nm, Lasos LGK7786P @543 nm, Thorlabs
laser diode PL450B @455 nm) onto the input facet of a APVE, which rests on a
manual 3D micro stage. The waists of the laser foci can be controlled by
choosing appropriate focal lengths for the lenses 4, 5 and 6. A motorized
tip/tilt mirror is imaged onto the APVE input facet using a Keplerian
telescope. This allows one to precisely control the laser’s AOI. The output
facet is imaged using a microscope objective lens (Olympus UPlanFL, 10x, 0.3
NA), whose exit pupil is imaged onto an iris with tunable aperture. This iris
was used to restrict the imaging NA to 0.05 when characterizing the mode
sorter. Finally, an image of the end facet is formed at the sensor of a CCD
camera (Basler puA1600-60uA). Optionally, a Bertrand lens can be flipped in
for imaging the objective’s exit pupil, which is used to calibrate the laser
AOIs.

The setup further comprises a Mach-Zehnder interferometer for measuring
complex field distributions at the output of the mode sorter APVE using off-
axis interferometry.

### Interferometry

To characterize the quality of the mode sorter, it is required to measure the
complex light field at the output facet of the APVE. This is done using off-
axis interferometry using a Mach-Zehnder interferometer. The reference wave is
tilted, such that the interference fringes have period of about 3 pixels as
shown in Fig. 9 (a). The intensity image is Fast Fourier transformed (FFT),
followed by cropping a 60$\times$60 pixel region around one sideband as shown
in Fig. 9 (b). The complex field is then obtained by taking an inverse FFT of
the cropped sideband. However, at this stage it will most likely show only a
poor overlap with any HG mode. This is because of several factors: Firstly,
the field is usually not perfectly centered on the pixel grid and there could
be a small geometric rotation in the experimental images. Secondly, the
presence of a global phase tilt and defocus is likewise reducing the overlap
integral. Finally, the waist of the measured mode could be slightly different
compared to the simulation.

Figure 9: Interferometry to characterize the light field shaped by the mode
sorter. Left: Raw interferogram captured by the camera; Right: Its logarithmic
power spectrum. One sideband is cropped for further processing.

To solve these issues, we employed a Nelder Mead optimizer (Julia programming
language, toolbox: Optim.jl) to find those seven parameters which maximize the
overlap integral between light field obtained at 0∘ AOI and the matching
target mode $\mathrm{HG}_{1,1}$. These parameters are then used to correct all
experimentally recorded fields.

##  Appendix C Additional information on the colour multiplexer

Figure 10 shows output intensities of the colour smiley APVE, taken with the
monochromator light source at wavelengths varied from 420 nm to 680 nm in
steps of 10 nm. For APVE characterization, the light form the monochromator’s
arc lamp was focused into a light guide of 1 mm diameter. About 1 cm from
light guide’s output we placed a 100 µm pinhole to further increase the
spatial coherence. This pinhole was then imaged with a slight demagnification
(telescope with 200 mm and 150 mm lenses) onto the motorized tip/tilt mirror
shown in Fig. 8, such that the monochromator light at the input facet of the
APVEs took the form of a Gaussian with $w_{0}\approx 30$ µm.

Figure 10: Wavelength dependent output intensities of the colour smiley APVE.
The illumination light was provided by a tunable monochromator. The numbers
below the images denote the respective wavelength in nm.

##  Appendix D Additional information on the mode sorter

Figure 11 presents a table of 30$\times$30 intensity images, which appear at
the output facet of the mode sorter when the AOI of the input beam is altered
in increments of 0.11 degrees (in air). One can clearly identify six angular
regions where the output pattern resembles a particular HG mode. Figure 12
shows the measured transmission $T$ of the mode sorter as a function of the
laser AOI. One can clearly define six local maxima, whose respective peak
values match the numbers in table 5. Finally, Fig. 13 visualizes the measured
angular dependence of all six efficiency values $\eta_{i,j}$ as defined by Eq.
3.

Figure 11: Angular dependent readouts of the mode sorter APVE. The figure
shows 30$\times$30 images, each one showing the intensity at the APVE output
facet for a different incidence angle of the readout beam. The angular
increment from one image to the next is 0.11 degrees. The image showing the
$HG_{1,1}$ mode in the center corresponds to a readout angle of zero. Figure
12: Angular dependence of the transmission $T$ of the mode sorter. The figure
shows the measured power fraction of the input beam arriving the camera
sensor. Figure 13: Angular dependence of the experimental efficiency values
$\eta_{i,j}$ of the mode sorter.

##  Appendix E Manufacturing of APVEs

The APVEs were manufactured using the direct laser writing technique [49]. The
beam from an ultrafast laser (Light Conversion, Pharos SP 6W) emitting pulses
with 170 fs duration, at a repetition rate of 1 MHz and wavelength 515 nm was
expanded and directed toward a phase-only liquid crystal spatial light
modulator (SLM) (Hamamatsu X10468). The SLM was imaged in 4f configuration
onto the pupil plane of a microscope objective (Zeiss 20$\times$ 0.5NA). The
beam was thus focused into the glass sample (Corning EAGLE 2000) to fabricate
the APVE voxels. The sample was mounted on a precision 3D translation stage
(Aerotech ABL10100 (x-z), ANT-95V (y)) and moved through the laser focus at a
speed of 2 mm/s in the $z$ direction, as defined in Figure 1 (a). The PSO
function was used to gate the laser according to the design specification and
fabricate voxels at the desired position. The laser pulse energy was 37 nJ and
the polarisation was linear along the translation direction. We note that this
pulse energy is significantly lower than that which would typically be used to
write single mode optical waveguides in such a system. As the sample was moved
in the $y$ direction to build up different layers of the APVE, the phase
pattern displayed on the SLM was updated to compensate for depth-induced
spherical aberration arising from refraction at the sample surface [50].

##  Appendix F Design algorithm

In the following, we present an algorithm that allows to design voxel-based
APVEs for the conversion of $N$ mutually incoherent transverse input modes
$\\{u_{n}\\}_{1\leq n\leq N}$ to $N$ target modes $\\{v_{n}\\}_{1\leq n\leq
N}$, with a one-to-one mapping. In general, any pair of modes $(u_{n},v_{n})$
has its own wavelength $\lambda_{n}$. We consider the design of a finite
extent sample in a rectangular coordinate system ($O,x,y,z$), $z$ being the
direction of propagation. We denote $z_{in}$ the transverse plane where the
input modes $\\{u_{n}\\}$ are defined, and $z_{out}$ the destination plane
where the target modes $\\{v_{n}\\}$ are defined. The bulk refractive index of
the sample at wavelength $\lambda_{n}$ is denoted $n_{n}$. In terms of
fabrication capabilities, we assume that we have access to a diversity of
transverse voxel distributions that we are able to write in the glass sample
at any 3D location, and that the length of any voxel (i.e., the $z$ dimension)
can be freely chosen. One example of a voxel cross section is given in Figure
1 (b), but many more are possible by varying the writing parameters such as
the pulse energy or the aberration compensation term. Let us note $D$ the
number of different transverse voxel distributions
$\\{\Delta\mathrm{RI}_{d}\\}_{1\leq d\leq D}$ which we consider to use for the
inverse design. These transverse distributions are relative refractive index
changes compared to the bulk refractive index, and are possibly wavelength
dependent. By convention, we also define $\Delta\textrm{RI}_{0}$ to refer to a
null refractive index modification. In order to make the inverse design much
simpler, the sample volume is virtually split into sub-volumes $\varv_{ijk}$,
$1\leq i\leq I$, $1\leq j\leq J$, $1\leq k\leq K$, acting as placeholders for
the elementary voxel distributions $\\{\Delta\mathrm{RI}_{d}^{ijk}\\}$. These
sub-volumes extensions in 3 dimensions are denoted $\mathrm{D}x$,
$\mathrm{D}y$ and $\mathrm{D}z$. Thus, the possible lengths of the waveguides
used for the design can only be integer multiples of the elementary length
$\mathrm{D}z$. In our designs, this elementary length is typically
$\mathrm{D}z=10\,\textrm{\textmu m}$. For the simulation to be numerically
accurate, the sampling resolution in 3 dimensions $\mathrm{d}x$, $\mathrm{d}y$
and $\mathrm{d}z$ must be much better, typically
$\mathrm{d}x=\mathrm{d}y=0.25\,\textrm{\textmu m}$ and
$\mathrm{d}z=0.5\,\textrm{\textmu m}$.

#### Forward model

The propagation of the input modes $\\{u_{n}\\}$ from the plane $z_{in}$ to
the plane $z_{out}$ relies on the split-step Fourier beam propagation method
(BPM), with spatial resolutions $\mathrm{d}x$, $\mathrm{d}y$ and
$\mathrm{d}z$. The input modes cross the sub-volumes $\varv_{ijk}$ layer by
layer, for $k$ varying from 1 to $K$. Each sub-volume $\varv_{ijk}$ may
contain (or not) one of the possible transverse distributions
$\\{\Delta\textrm{RI}_{d}^{ijk}\\}$, depending on the choice that was made in
a previous iteration by the selection algorithm that we present next. During
this forward propagation, the transverse distributions $\\{u_{n}\\}$ are saved
before crossing each sub-volume $\varv_{ijk}$, and we note them
$\\{u^{ijk}_{n}\\}$ for future reference. We also note $\\{z_{k}\\}$ the
common transverse planes where these distributions are saved.

According to the BPM, for a given input mode $u_{n}$, crossing each layer of
thickness $\mathrm{d}z$ is modeled as a small propagation of length
$\mathrm{d}z$ through an homogeneous medium of refractive index $n_{n}$,
followed by the multiplication of a local phase $\varphi_{ijk}$ depending on
the activated region:

$\varphi_{ijk}(\lambda_{n})=\exp\left(i\frac{2\pi}{\lambda_{n}}\Delta\textrm{RI}^{ijk}_{d}(\lambda_{n})\mathrm{d}z\right),\,0\leq
d\leq D.$ (4)

#### Error metric

Our inverse design relies on a multi-objective minimization based on the
squared $l_{2}$ distance between the input and target modes:

$O_{min}=\min\left\\{\sum\limits_{n=1}^{N}\|u_{n}-v_{n}\|_{2}^{2}\right\\}=\min\left\\{\sum\limits_{n=1}^{N}{\int\limits_{\,\,z_{out}}|u_{n}-v_{n}|^{2}}\right\\}.$
(5)

We can expand the squared modulus as
$|u_{n}-v_{n}|^{2}=|u_{n}|^{2}+|v_{n}|^{2}-2\Re{\left(v_{n}^{*}u_{n}\right)}$,
which allows to turn the minimization problem (5) into the following
maximization problem in the case of a lossless design ($\|\|_{2}^{2}$ of
$u_{n}$ and $v_{n}$ is conserved):

$O_{max}=\max\limits_{\left\\{\Delta\mathrm{RI}_{d}^{ijk}\right\\}}\left\\{\sum\limits_{n=1}^{N}{\int\limits_{\,\,z_{out}}\Re\left(v_{n}^{*}u_{n}\right)}\right\\}.$
(6)

The expression (6) consists only of overlap integrals that are conserved
during propagation (or backpropagation) through a lossless optical system,
thus allowing for an iterative propagation algorithm (IPA) [51], where the
same criterion $O_{max}$ is iteratively maximized in each plane $z_{k}$ (in a
reverse fashion).

#### Voxel distribution selection algorithm

We assume that we know the transverse distributions $\\{v_{n}^{ijk+1}\\}$
corresponding to the local distributions of the target modes $\\{v_{n}\\}$ in
the plane $z_{k+1}$ just after the sub-volumes $\\{\varv_{ijk}\\}$, and the
transverse distributions $\\{u_{n}^{ijk}\\}$ of the input modes $\\{u_{n}\\}$
stored just before the same sub-volumes. The selection of the refractive index
distribution $\Delta\textrm{RI}_{d}^{ijk}$ relies on the maximization of the
local multi-objective criterion

$O_{max}^{ijk}=\max\limits_{d}\left\\{\sum\limits_{n=1}^{N}{\int\limits_{\,\,z_{k+1}}\Re\left((v_{n}^{ijk+1})^{*}u_{n}^{ijk+1}(d)\right)}\right\\},$
(7)

where $u_{n}^{ijk+1}(d)$ denotes the local input modes distribution that have
been propagated from plane $z_{k}$, where they were stored during the forward
pass, to plane $z_{k+1}$ through a local refractive index distribution
$\Delta\textrm{RI}_{d}^{ijk}$. In order to find $d$ maximizing
$O_{max}^{ijk}$, all the distributions $\\{\Delta\textrm{RI}_{d}^{ijk}\\}$
need to be tried for each sub-volume $\varv_{ijk}$ ($k$ fixed).

Once the $\\{\Delta\textrm{RI}_{d}^{ijk}\\}$ distributions have been selected
for a given $k$, the target modes $\\{v_{n}^{ijk+1}\\}$ are backpropagated to
the plane $z_{k}$, and the same procedure is repeated iteratively until the
selection of the $\\{\Delta\textrm{RI}_{d}^{ij1}\\}$.

#### Iterative propagation algorithm

We summarize the different steps of the optimization procedure below:

  1. 1.

Propagate the input modes $\\{u_{n}\\}$ from their definition plane
$z_{in}=z_{1}$ to the destination plane $z_{out}=z_{K+1}$ using BPM, taking
into account the current waveguide selection in each sub-volume $\varv_{ijk}$
($1\leq k\leq K$). At the beginning of the first iteration, all
$\\{\varv_{ijk}\\}$ contain a null refractive index distribution
($\Delta\textrm{RI}_{0}$). During propagation, store the local field
distributions $\\{u_{n}^{ijk}\\}$ at planes $\\{z_{k}\\}$.

  2. 2.

Compute the final overlap criterion (6) for monitoring the convergence of the
algorithm. If the convergence is satisfactory or stagnant, then exit the
optimization procedure and consider the current waveguide selection as the
optimal design.

  3. 3.

Iteratively backpropagate the target modes $\\{v_{n}\\}$ from plane
$z_{out}=z_{K+1}$ to plane $z_{in}=z_{1}$, while proceeding to a new voxel
selection in each sub-volume $\varv_{ijk}$ according to criterion (7). For a
given $k$, this criterion leads to the best voxel selection in each sub-
volume, after which the modes $\\{v_{n}\\}$ in plane $z_{k+1}$ can be
backpropagated to plane $z_{k}$, and this iteratively until $k=1$.

  4. 4.

Repeat the procedure from step 1.