Bright solid-state sources for single photons with orbital angular momentum

Tempering Multichannel Photon Emission from Emitter-Coupled Holographic Metasurfaces

On-chip manipulation of photon emission from quantum emitters (QEs) is crucial for quantum nanophotonics and advanced optical applications. At the same time, the general design strategy is still elusive, especially for fully exploring the degrees of freedom of multiple channels. Here, the vectorial scattering holography (VSH) approach developed recently for flexibly designing QE-coupled metasurfaces is extended to tempering the strength of QE emission into a particular channel. The VSH power is demonstrated by designing, fabricating, and optically characterizing on-chip QE sources emitted into six differently oriented propagation channels, each representing the entangled state of orthogonal circular polarizations with different topological charges and characterized with a specific relative strength. We postulate that the demonstration of tempered multichannel photon emission from QE-coupled metasurfaces significantly broadens the possibilities provided by the holographic metasurface platform, especially those relevant for high-dimensional quantum information processing.

Scalable Bright and Pure Single Photon Sources by Droplet Epitaxy on InP Nanowire Arrays

High-quality quantum light sources are crucial components for the implementation of practical and reliable quantum technologies. The persistent challenge, however, is the lack of scalable and deterministic single photon sources that can be synthesized reproducibly. Here, we present a combination of droplet epitaxy with selective area epitaxy to realize the deterministic growth of single quantum dots in nanowire arrays. By optimization of the single quantum dot growth and the nanowire cavity design, single emissions are effectively coupled with the dominant mode of the nanowires to realize Purcell enhancement. The resonance-enhanced quantum emitter system boasts a brightness of millions of counts per second with nanowatt excitation power, a short radiation lifetime of 350 ± 5 ps, and a high single-photon purity with g(2)(0) value of 0.05 with continuous wave above-band excitation. Finite-difference time-domain (FDTD) simulation results show that the emissions of single quantum dots are coupled into the TM01 mode of the nanowires, giving a Purcell factor ≈ 3. Our technology can be used for creating on-chip scalable single photon sources for future quantum technology applications including quantum networks, quantum computation, and quantum imaging.

Ultracompact Single-Photon Sources of Linearly Polarized Vortex Beams

Ultracompact chip-integrated single-photon sources of collimated beams with polarization-encoded states are crucial for integrated quantum technologies. However, most of currently available single-photon sources rely on external bulky optical components to shape the polarization and phase front of emitted photon beams. Efficient integration of quantum emitters with beam shaping and polarization encoding functionalities remains so far elusive. Here, ultracompact single-photon sources of linearly polarized vortex beams based on chip-integrated quantum emitter-coupled metasurfaces are presented, which are meticulously designed by fully exploiting the potential of nanobrick-arrayed metasurfaces. The authors first demonstrate on-chip single-photon generation of high-purity linearly polarized vortex beams with prescribed topological charges of 0, - 1, and +1. The multiplexing of single-photon emission channels with orthogonal linear polarizations carrying different topological charges are further realized and their entanglement is demonstarated. The work illustrates the potential and feasibility of ultracompact quantum emitter-coupled metasurfaces as a new quantum optics platform for realizing chip-integrated high-dimensional single-photon sources.

Metafiber transforming arbitrarily structured light

Structured light has proven useful for numerous photonic applications. However, the current use of structured light in optical fiber science and technology is severely limited by mode mixing or by the lack of optical elements that can be integrated onto fiber end-faces for wavefront engineering, and hence generation of structured light is still handled outside the fiber via bulky optics in free space. We report a metafiber platform capable of creating arbitrarily structured light on the hybrid-order Poincaré sphere. Polymeric metasurfaces, with unleashed height degree of freedom and a greatly expanded 3D meta-atom library, were 3D laser nanoprinted and interfaced with polarization-maintaining single-mode fibers. Multiple metasurfaces were interfaced on the fiber end-faces, transforming the fiber output into different structured-light fields, including cylindrical vector beams, circularly polarized vortex beams, and arbitrary vector field. Our work provides a paradigm for advancing optical fiber science and technology towards fiber-integrated light shaping, which may find important applications in fiber communications, fiber lasers and sensors, endoscopic imaging, fiber lithography, and lab-on-fiber technology.

High-Purity Generation and Switching of Twisted Single Photons

Quantum technologies, if scaled into a high-dimensional Hilbert space, can dramatically enhance connection capabilities with supporting higher bit rates and ultrasecure information transfer. Twisted single photons, carrying orbital angular momentum (OAM) as an unbounded dimension, could address the growing demand for high-dimensional quantum information encoding and transmission. By hybrid integration of two-dimensional semiconductor WSe_{2} with a spin-orbit-coupled microring resonator, we demonstrate an integrated tunable twisted single photon source with the ability to precisely define and switch between highly pure spin-OAM states. Our results feature a single photon purity of g^{(2)}(0)∼0.13 with a cavity-enhanced quantum yield of 76% and a high OAM mode purity up to 96.9%. Moreover, the demonstrated quantum-chiral control can also enable new quantum functionality such as single photon routing for efficient quantum information processing on chip.

Multichannel Quantum Emission with On-Chip Emitter-Coupled Holographic Metasurfaces

Multichannel quantum emission is in high demand for advanced quantum photonic applications such as quantum communications, quantum computing, and quantum cryptography. However, to date, the most common way for shaping photon emission from quantum emitters (QEs) is to utilize free-standing (external) bulky optical components. Here, we develop the multichannel holography approach for flexibly designing on-chip QE-coupled metasurfaces that make use of nonradiatively QE-excited surface plasmon polaritons for generating far-field quantum emission, which propagates in designed directions carrying specific spin and orbital angular momenta (SAM and OAM, respectively). We further design, fabricate, and characterize on-chip quantum light sources of multichannel quantum emission encoded with different SAMs and OAMs. The holography-based inverse design approach developed and demonstrated on-chip quantum light sources with multiple degrees of freedoms, thereby enabling a powerful platform for quantum nanophotonics, especially relevant for advanced quantum photonic applications, e.g., high-dimensional quantum information processing.

Nanoprinted Diffractive Layer Integrated Vertical-Cavity Surface-Emitting Vortex Lasers with Scalable Topological Charge

Vertical-cavity surface-emitting lasers (VCSELs) represent an attractive light source to integrate with OAM structures to realize chip-scale vortex lasers. Although pioneering endeavors of VCSEL-based vortex lasers have been reported, they cannot achieve large topological charges (less than l = 5) due to the insufficient space-bandwidth product (SBP) caused by the inherent limited device size. Here, by integrating a nanoprinted OAM phase structure on the VCSELs, we demonstrate a vortex microlaser with a low threshold and simple structure. A monolithic microlaser array with addressable control of vortex beams with different topological charges (l = 1 to l = 5) was achieved. Nanoprinting offers high degrees of freedom for the manipulation of spatial structures. To address the challenge of insufficient SBP, two-layer cascaded spiral phase plates were designed. Thereby, a vortex beam with l = 15 and mode purity of 83.7% was obtained. Our work paves the way for future chip-scale OAM-based information multiplexing with more channels.

On-chip generation of single-photon circularly polarized single-mode vortex beams

Generation of single photons carrying spin and orbital angular momenta (SAM and OAM) opens enticing perspectives for exploiting multiple degrees of freedom for high-dimensional quantum systems. However, on-chip generation of single photons encoded with single-mode SAM-OAM states has been a major challenge. Here, by using carefully designed anisotropic nanodimers fabricated atop a substrate, supporting surface plasmon polariton (SPP) propagation, and accurately positioned around a quantum emitter (QE), we enable nonradiative QE-SPP coupling and the SPP outcoupling into free-space propagating radiation featuring the designed SAM and OAM. We demonstrate on-chip room-temperature generation of well-collimated (divergence < 7.5°) circularly polarized (chirality > 0.97) single-mode vortex beams with different topological charges (𝓁 = 0, 1, and 2) and high single-photon purity, g(2)(0) < 0.15. The developed approach can straightforwardly be extended to produce multiple, differently polarized, single-mode single-photon radiation channels and enable thereby realization of high-dimensional quantum sources for advanced quantum photonic technologies.

Nanophotonic Materials for Twisted-Light Manipulation

Twisted light, an unbounded set of helical spatial modes carrying orbital angular momentum (OAM), offers not only fundamental new insights into structured light-matter interactions, but also a new degree of freedom to boost optical and quantum information capacity. However, current OAM experiments still rely on bulky, expensive, and slow-response diffractive or refractive optical elements, hindering today's OAM systems to be largely deployed. In the last decade, nanophotonics has transformed the photonic design and unveiled a diverse range of compact and multifunctional nanophotonic devices harnessing the generation and detection of OAM modes. Recent metasurface devices developed for OAM generation in both real and momentum space, presenting design principle and exemplary devices, are summarized. Moreover, recent development of whispering-gallery-mode-based passive and tunable microcavities, capable of extracting degenerate OAM modes for on-chip vortex emission and lasing, is summarized. In addition, the design principle of different plasmonic devices and photodetectors recently developed for on-chip OAM detection is discussed. Current challenges faced by the nanophotonic field for twisted-light manipulation and future advances to meet these challenges are further discussed. It is believed that twisted-light manipulation in nanophotonics will continue to make significant impact on future development of ultracompact, ultrahigh-capacity, and ultrahigh-speed OAM systems-on-a-chip.

Selective high-order resonance in asymmetric plasmonic nanostructures stimulated by vortex beams

Orbital angular momentum (OAM) of light has the potential to induce high-order transitions of electrons in atoms by compensating for the OAM required. However, due to the dark spot situating at the focal center of the OAM beam, high-order transitions are typically weak. In this study, we demonstrate efficient and selective high-order resonances in symmetric and asymmetric plasmonic nanoparticles that are comparable in size to the waist radius of the OAM beam. In a symmetric nanoparticle configured with a complete nanoring lying on the focal center, there is a pure high-order resonance obeying the law of conservation of angular momentum during the interaction between OAM light and the nanosystem. In an asymmetric nanoparticle configured with an complete ring off the beam center or a splitting nanoring, there are multiple resonances whose resonance orders are influenced by the ring's geometry, position, orientation, and photon OAM. Thus, high-order resonances in the symmetric and asymmetric plasmonic nanostructures are selectively stimulated using vortex beams. Our results may help to understand and control OAM-involved light-material interactions of asymmetric nanosystems.

Single Photon Emitters with Polarization and Orbital Angular Momentum Locking in Monolayer Semiconductors

Excitons in monolayer transition metal dichalcogenide are endowed with intrinsic valley-orbit coupling between their center-of-mass motion and valley pseudospin. When trapped in a confinement potential, e.g., generated by strain field, we find that intralayer excitons are valley and orbital angular momentum (OAM) entangled. By tuning the trap profile and external magnetic field, one can engineer the exciton states at the ground state and realize a series of valley-OAM entangled states. We further show that the OAM of excitons can be transferred to emitted photons, and these novel exciton states can naturally serve as polarization-OAM locked single photon emitters, which under certain circumstance become polarization-OAM entangled, highly tunable by strain trap and magnetic field. Our proposal demonstrates a novel scheme to generate polarization-OAM locked/entangled photons at the nanoscale with a high degree of integrability and tunability, pointing to exciting opportunities for quantum information applications.

Highly-twisted states of light from a high quality factor photonic crystal ring

Twisted light with orbital angular momentum (OAM) has been extensively studied for applications in quantum and classical communications, microscopy, and optical micromanipulation. Ejecting high angular momentum states of a whispering gallery mode (WGM) microresonator through a grating-assisted mechanism provides a scalable, chip-integrated solution for OAM generation. However, demonstrated OAM microresonators have exhibited a much lower quality factor (Q) than conventional WGM resonators (by >100×), and an understanding of the limits on Q has been lacking. This is crucial given the importance of Q in enhancing light-matter interactions. Moreover, though high-OAM states are often desirable, the limits on what is achievable in a microresonator are not well understood. Here, we provide insight on these two questions, through understanding OAM from the perspective of mode coupling in a photonic crystal ring and linking it to coherent backscattering between counter-propagating WGMs. In addition to demonstrating high-Q (10⁵ to 10⁶), a high estimated upper bound on OAM ejection efficiency (up to 90%), and high-OAM number (up to l = 60), our empirical model is supported by experiments and provides a quantitative explanation for the behavior of Q and the upper bound of OAM ejection efficiency with l. The state-of-the-art performance and understanding of microresonator OAM generation opens opportunities for OAM applications using chip-integrated technologies.

Planar Optical Cavities Hybridized with Low-Dimensional Light-Emitting Materials

Low-dimensional light-emitting materials have been actively investigated due to their unprecedented optical and optoelectronic properties that are not observed in their bulk forms. However, the emission from low-dimensional light-emitting materials is generally weak and difficult to use in nanophotonic devices without being amplified and engineered by optical cavities. Along with studies on various planar optical cavities over the last decade, the physics of cavity-emitter interactions as well as various integration methods are investigated deeply. These integrations not only enhance the light-matter interaction of the emitters, but also provide opportunities for realizing nanophotonic devices based on the new physics allowed by low-dimensional emitters. In this review, the fundamentals, strengths and weaknesses of various planar optical resonators are first provided. Then, commonly used low-dimensional light-emitting materials such as 0D emitters (quantum dots and upconversion nanoparticles) and 2D emitters (transition-metal dichalcogenide and hexagonal boron nitride) are discussed. The integration of these emitters and cavities and the expect interplay between them are explained in the following chapters. Finally, a comprehensive discussion and outlook of nanoscale cavity-emitter integrated systems is provided.

Arbitrary-Order and Multichannel Optical Vortices with Simultaneous Amplitude and Phase Modulation on Plasmonic Metasurfaces

The highly localized and uneven spatial distribution of the subwavelength light field in metal metasurfaces provides a promising means for the generation of optical vortices (OVs) with arbitrary topological charges. In this paper, a simple and reliable way for generating multichannel OVs on gold nanoporous metasurfaces is reported. The instantaneous field of arbitrary-order OVs can be regulated and concentrated on the same focal surface by adapting photonic spin-orbit interaction (SOI) and geometric phase. The focal ring energy distribution of OVs along the conical propagation path is accurately calculated, and the double phase of units induced by spin rotation is confirmed. Based on the parameter optimization of the nanohole arrangement, the simultaneous amplitude and phase modulation of multichannel OVs has been realized. Furthermore, the average multichannel signal-to-noise ratio exceeds 15 dB, which meets the requirements of high resolution and low crosstalk. Our study obtains broadband and efficient OVs, which can contribute to improving the capacity storage and security of optical information and possess great application prospects in beam shaping, optical tweezers, and communication coding.

On-chip ultracompact multimode vortex beam emitter based on vertical modes

Free-space orbital angular momentum (OAM) communication is considered as one of the potential alternative on-chip optical interconnect solutions. The number of OAM modes determines the capacity of high-speed communication. However, existing integrated vortex beam emitters have a constraint relationship between the number of OAM modes and the emitter size, rendering it difficult to emit more OAM modes with a small-sized emitter. In view of the above, this study proposes an on-chip ultracompact multimode vortex beam emitter based on vertical modes, which permits more OAM modes without requiring an increase in the size of the emitter. Vertical modes in large-aspect-ratio waveguides are pointed out to enable multimode microrings with small radii because high-order vertical modes can maintain almost the same horizontal wave vector as that of the fundamental mode. Four-mode and five-mode vortex beam emitters with the same radius of 1.5 µm are designed and the effectiveness of these emitters is verified through simulation. Furthermore, a high-efficiency and low-crosstalk approach for high-order vertical mode coupling by varying the waveguide height is presented. This research not only promotes further integration of on-chip optical interconnection, but also provides a new strategy for optical waveguide mode selection in photonic integrated circuits design.

Broadband dispersion compensating ring-core fiber for orbital angular momentum modes

A well designed ring-core fiber can theoretically support numerous orbital angular momentum (OAM) modes with low crosstalk for space-division-multiplexing (SDM) data transmission, which is considered as a promising solution for overcoming the capacity crunch in optical communication network. However, the accumulated chromatic dispersion in OAM-fiber could limit the data speed and transmission distance of communication systems. A potential solution is to insert a dispersion compensation ring-core fiber with opposite-sign of dispersion in the transmission fiber along the fiber link. In this work, we propose a triple ring-core fiber with broadband negative dispersion. A highest negative dispersion of -24.47 ps/(nm·km) at 1550 nm and an average dispersion slope in the C band from -0.182 ps/(nm²·km) to 0.065 ps/(nm²·km) can be achieved to compensate multi-order dispersion. The effects of Ge-doping concentration fluctuation in the high-index ring core and fabrication errors on fiber geometric structures are also investigated. Furthermore, the effective mode area decreases as the widths of high-index rings increase due to the enhanced confinement ability. The designed triple ring-core fiber could offer potential for compensating OAM fiber links with positive dispersions.

On-chip scalable highly pure and indistinguishable single-photon sources in ordered arrays: Path to quantum optical circuits

Realization of quantum optical circuits is at the heart of quantum photonic information processing. A long-standing obstacle, however, has been the absence of a suitable platform of single photon sources (SPSs). Such SPSs need to be in spatially ordered arrays and produce, on-demand, highly pure, and indistinguishable single photons with sufficiently uniform emission characteristics to enable controlled interference between photons from distinct sources underpinning functional quantum optical networks. We report on such a platform of SPSs based on a unique class of epitaxial quantum dots dubbed mesa-top single quantum dot. Under resonant excitation, the spatially ordered SPSs (without Purcell enhancement) show single photon purity of >99% [g(2)(0) ~ 0.015], high two-photon Hong-Ou-Mandel interference visibilities of 0.82 ± 0.03 (at 11.5 kelvin, without cavity), and spectral nonuniformity of <3 nanometers, within established locally tunable technology. Our platform of SPSs paves the path to creating on-chip scalable quantum photonic networks for communication, computation, simulation, sensing and imaging.

Beam manipulation for quantum dot light-emitting diode with an Ag grating and a phase-gradient metasurface

The quantum dot (QD) light-emitting diode (LED) is a robust scheme for single photon source. However, the spontaneous emission of a QD LED has arbitrary directions and polarizations, which is disadvantage for photon collection and manipulation. We propose a QD LED integrated with an Ag grating and a phase-gradient metasurface. The circular patterned Ag grating is adopted to collimate the emission beam with right phase and improve its spatial coherence, therefore a phase-gradient metasurface can work for beam manipulation. The 10°, 20°, and 30° angle deflection as well as doughnut-pattern generation are demonstrated by numerical simulation. A small metasurface with the width of 6 µm can provide a collection efficiency of 25.9% at the deflection angle of 10°. Furthermore, only one single QD can be selected from a QD assembly with a low-density.

Direct Light Orbital Angular Momentum Detection in Mid-Infrared Based on the Type-II Weyl Semimetal TaIrTe4

The direct photocurrent detection capability of light orbital angular momentum (OAM) has recently been realized with topological Weyl semimetals, but it is limited to the near-infrared wavelength range. The extension of the direct OAM detection capability to the mid-infrared band, which is a wave band that plays an important role in a vast range of applications, has not yet been realized. This is because the photocurrent responses of most photodetectors are neither sensitive to the phase information nor efficient in the mid-infrared region. In this study, a photodetector based on the type-II Weyl semimetal tantalum iridium telluride (TaIrTe₄ ) is designed with peculiar electrode geometries to directly detect the topological charge of the OAM using the orbital photogalvanic effect (OPGE). The results indicate that the helical phase gradient of light can be distinguished by a current winding around the optical beam axis, with a magnitude proportional to its quantized OAM mode number. The topologically enhanced responses in the mid-infrared region of TaIrTe₄ further help overcome the low responsivity issues and finally render direct OAM detection capability. This study enables on-chip-integrated OAM detection, and thus OAM-sensitive focal plane arrays in the mid-infrared region.

Supercontinuum Induced by Filamentation of Bessel-Gaussian and Laguerre-Gaussian Beams in Water

In this paper, we study the characteristics of the supercontinuum (SC) induced by the filamentation of two typical vortex beams (i.e., Laguerre-Gaussian (LG) and Bessel-Gaussian (BG) beams) in water. By moving the cuvette filled with water along the laser propagation path, we measure the SC induced by the filamentation of the two vortex beams at different positions in water. The results show that the degree of spectral broadening induced by the filamentation of LG beams hardly changes with the change of position, while for BG beams, the spectral broadening induced by filamentation is weak on both sides and strong in the middle. The value of topological charge (TC) affects the length of the filament formed by BG beams; however, its effect on the spectral broadening induced by the filamentation of LG and BG beams is negligible.

Transverse Kerker Effect for Dipole Sources

Transverse Kerker effect is known by the directional scattering of an electromagnetic plane wave perpendicular to the propagation direction with nearly suppression of both forward and backward scattering. Compared with plane waves, localized electromagnetic emitters are more general sources in modern nanophotonics. As a typical example, manipulating the emission direction of a quantum dot is of vital importance for the investigation of on-chip quantum optics and quantum information processing. Herein, we introduce the concept of transverse Kerker effect for dipole sources utilizing a subwavelength dielectric antenna, where the radiative power of magnetic, electric, and more general chiral dipole emitters can be dominantly redirected along their dipole moments with nearly suppression of radiation perpendicular to the dipole moments. This type of transverse Kerker effect is also associated with Purcell enhancement mediated by electromagnetic multipolar resonances induced in the dielectric antenna. Analytical conditions of transverse Kerker effect are derived for the magnetic, electric, and chiral dipole emitters. We further provide microwave experiment validation for the magnetic dipole emitter. Our results provide new physical mechanisms to manipulate the emission properties of localized electromagnetic source which might facilitate the on-chip quantum optics and beyond.

Room-temperature on-chip orbital angular momentum single-photon sources

On-chip photon sources carrying orbital angular momentum (OAM) are in demand for high-capacity optical information processing in both classical and quantum regimes. However, currently exploited integrated OAM sources have been primarily limited to the classical regime. Here, we demonstrate a room-temperature on-chip integrated OAM source that emits well-collimated single photons, with a single-photon purity of g⁽²⁾(0) ≈ 0.22, carrying entangled spin and OAM states and forming two spatially separated entangled radiation channels with different polarization properties. The OAM-encoded single photons are generated by efficiently outcoupling diverging surface plasmon polaritons excited with a deterministically positioned quantum emitter via Archimedean spiral gratings. Our OAM single-photon source platform bridges the gap between conventional OAM manipulation and nonclassical light sources, enabling high-dimensional and large-scale photonic quantum systems for quantum information processing.

Guided mode meta-optics: metasurface-dressed waveguides for arbitrary mode couplers and on-chip OAM emitters with a configurable topological charge

Metasurface has achieved fruitful results in tailoring optical fields in free space. However, a systematic investigation on applying meta-optics to completely control waveguide modes is still elusive. Here we present a comprehensive catalog to selectively and exclusively couple free space light into arbitrary high-order waveguide modes of interest, leveraging silicon metasurface-patterned silicon nitride waveguides. By simultaneously engineering the matched phase gradient of the nanoantennas and the vectorial spatial modal overlap between the antenna near-field and target waveguide mode profile, either single or multiple high-order modes are successfully launched with high purity reaching 98%. Moreover, on-chip twisted light generators are theoretically proposed with configurable OAM topological charge ℓ from -3 to +2. This work may serve as a comprehensive framework for guided mode meta-optics and motivates further applications such as versatile integrated couplers, multiplexers, and mode-division multiplexing-based communication systems.

Steering Room-Temperature Plexcitonic Strong Coupling: A Diexcitonic Perspective

Plexcitonic strong coupling between a plasmon-polariton and a quantum emitter empowers ultrafast quantum manipulations in the nanoscale under ambient conditions. The main body of previous studies deals with homogeneous quantum emitters. To enable multiqubit states for future quantum computing and network, the strong coupling involving two excitons of the same material but different resonant energies has been investigated and observed primarily at very low temperature. Here, we report a room-temperature diexcitonic strong coupling (DiSC) nanosystem in which the excitons of a transition metal dichalcogenide monolayer and dye molecules are both strongly coupled to a single Au nanocube. Coherent information exchange in this DiSC nanosystem could be observed even when exciton energy detuning is about five times larger than the respective line widths. The strong coupling behaviors in such a DiSC nanosystem can be manipulated by tuning the plasmon resonant energies and the coupling strengths, opening up a paradigm of controlling plasmon-assisted coherent energy transfer.

Multidimensional phase singularities in nanophotonics

[Figure: see text].

Complex wavefront sensing based on coherent diffraction imaging using vortex modulation

Phase retrieval seeks to reconstruct the phase from the measured intensity, which is an ill-posed problem. A phase retrieval problem can be solved with physical constraints by modulating the investigated complex wavefront. Orbital angular momentum has been recently employed as a type of reliable modulation. The topological charge l is robust during propagation when there is atmospheric turbulence. In this work, topological modulation is used to solve the phase retrieval problem. Topological modulation offers an effective dynamic range of intensity constraints for reconstruction. The maximum intensity value of the spectrum is reduced by a factor of 173 under topological modulation when l is 50. The phase is iteratively reconstructed without a priori knowledge. The stagnation problem during the iteration can be avoided using multiple topological modulations.