1
Complex-amplitude metasurface-based orbital angular momentum
holography in momentum space
Haoran Ren1, *, †, Xinyuan Fang2, †, Jaehyuck Jang3, †, Johannes Bürger 1, Junsuk Rho3, 4, * and
Stefan A. Maier1, 5, *
1 Chair in Hybrid Nanosystems, Nanoinstitute Munich, Faculty of Physics, Ludwig-Maximilians-
Universität München, München, 80539, Germany
2 Centre for Artificial-Intelligence Nanophotonics, School of Optical-Electrical and Computer
Engineering, University of Shanghai for Science and Technology, Shanghai, 200093, China
3 Department of Chemical Engineering, Pohang University of Science and Technology
(POSTECH), Pohang, 37673, Republic of Korea
4 Department of Mechanical Engineering, Pohang University of Science and Technology
(POSTECH), Pohang, 37673, Republic of Korea
5 Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom
Email: Haoran.Ren@physik.uni-muenchen.de; jsrho@postech.ac.kr; Stefan.Maier@physik.uni-
muenchen.de
† These authors contributed equally to this work
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Abstract
Digital optical holograms can achieve nanometer-scale resolution thanks to recent advances in
metasurface technologies. This has raised hopes for applications in data encryption, data storage,
information processing and displays. However, the hologram bandwidth has remained too low for
any practical use. To overcome this limitation, information can be stored in the orbital angular
momentum (OAM) of light, as this degree of freedom has an unbounded set of orthogonal helical
modes that could function as information channels. Thus far, OAM holography has been achieved
using phase-only metasurfaces, which however are marred by channels crosstalk. As a result,
multiplex information from only 4 channels has been demonstrated. Here we demonstrate an OAM
holography technology that is capable of multiplexing up to 200 independent OAM channels. This
is achieved by designing a complex-amplitude metasurface in momentum-space capable of
complete and independent amplitude and phase manipulation. Information is then extracted by
Fourier transform using different OAM modes of light, allowing lensless reconstruction and
holographic videos being displayed. Our metasurface can be 3D printed in a polymer matrix on
SiO2 for large-area fabrication.
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Optical holography is a promising technology for realistic 3D displays1, 2, optical encryption3, 4,
data storage5, and artificial intelligence6. Recent advances in metasurface-based flat optics7-9 have
opened up the possibility of using ultrathin metasurface for hologram digitalization10-16, towards
high-capacity holographic technology. Development of high-bandwidth metasurface holograms is
essential for optically addressable holographic video displays with potentially ultrafast switching
of image frames. Optical multiplexing uses physical properties of light to carry independent
holographic image channels, thus improving the bandwidth of a metasurface hologram. In this
context, spatially interleaved metasurfaces consisting of multiple sets of polarization- and
wavelength-sensitive meta-atoms have been designed for holographic multiplexing17-20, albeit with
strong crosstalk and limited bandwidth. Alternatively, the anisotropic property of a single meta-
atom has been tailored for polarization- and angle-sensitive optical responses21-23, although the
availability of distinct optical modes for multiplexing is rather limited and bandwidth remains low.
Orbital angular momentum (OAM) manifests itself as a twisted wavefront of light, and has
emerged as a novel way to boost information capacity due to its physically unbounded set of
orthogonal helical modes24-28. Through appropriate spatial-frequency sampling of a digital
hologram in momentum space, the OAM property of an incident light can be preserved for
selectively addressing OAM-dependent holographic images4, 29. However, such phase-only
metasurface holograms prohibit an exact convolution between a complex-amplitude image
channel and an OAM helical wavefront. As a result, the linear superposition principle is broken
and image multiplexing suffers from strong crosstalk. Thus, this approach restricts the maximum
multiplexing channels to four29. Even though complex-amplitude metasurfaces have recently been
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developed to perform Fresnel holography based on wave propagation theory10, 30-32, Fresnel
holograms fail to implement OAM holography as they are not designed for momentum-space
image reconstruction. In addition, metasurface fabrication is costly, leading to planar metasurface
holograms designs with limited degrees of freedom, which require a bulky microscope for image
reconstruction.
Here we show the design and 3D laser manufacturing of a large-scale complex-amplitude
metasurface for ultrahigh-dimensional OAM-multiplexing holography in momentum space. Such
high-bandwidth metasurface holograms allow us to demonstrate lensless reconstruction of up to
200 of orthogonal image channels encoded within an OAM signature, leading to optically-
addressable holographic video displays without involving any spatial scanning approach33-35.
Without loss of generality, we select OAM modes with helical mode indices () ranging from -50
to 50 (Fig. 1a) to be sequentially incident on a large-scale complex-amplitude OAM-multiplexing
metasurface hologram (COMH) (Fig. 1b) for addressing OAM-dependent orthogonal image
frames (Fig. 1c), with two distinct holographic videos being simultaneously reconstructed in
momentum space. The independent reconstruction of holographic videos in two different planes
suggests that our approach can be applied for 3D holography. Unlike the conventional
metasurfaces with restricted degrees of freedom in a 2D plane, we introduce the design and laser-
based printing of a 3D metasurface, in which the height ( ) and in-plane rotation ( ) of a
birefringent polymer nanopillar are employed to independently control the amplitude and phase
responses of transmitted light, respectively (Fig. 1b).
Design principle
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Mathematically, OAM-multiplexing holography can be represented as superposition of complex-
amplitude fields of different image channels encoded with distinctive OAM modes in the hologram
plane:
, wherein
and
stand for the amplitude and phase
information of each image channel, respectively;
ϵ ℤ and represent the helical mode index
and azimuthal angle, respectively, and M denotes the total number of multiplexing channels. Since
the complex-amplitude hologram is Fourier-based (Supplementary Note 1), its reconstructed
optical fields in momentum space can be represented as
, where denotes the Fourier transform (FT) operator,
expressing multiplexing results as the superposition of a convolution of the amplitude (
), phase
(
), and encoded OAM (
) information of each image channel. It is obvious to see that the
amplitude of each image channel can be individually controlled, allowing the adjustment of image
frame intensity essential for high-quality holographic video displays. On the other hand, a phase-
only OAM-multiplexing hologram can be described as an argument result:
, wherein
denotes an iteratively-retrieved phase-only hologram for each
image channel. The reconstructed optical fields in momentum space can thus be represented as
, indicating that the amplitude information of each image
channel is completely lost, which degrades the reconstruction quality by making the image channel
intensity non-uniform (fig. S1). Moreover, neglecting the amplitude information results in strong
crosstalk34, due to the non-exact reproduction of the convolution relationship as compared to the
complex-amplitude multiplexing results. Specifically, we present a general multiplexing case by
superposing two blazed phase gratings in the hologram plane. The resulting multiplexing fields in
