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The ability to solve differential equations represents a key step in the modeling and understanding of complex systems. There exist several analytical and numerical methods for solving differential equations, each with their own advantages and limitations. Physics-informed neural networks (PINNs) offer an alternative perspective. Although PINNs deliver promising results, many stones remain unturned about this method. In this paper, we introduce a method that improves the efficiency of PINNs in solving differential equations. Our method is related to the formulation of the problem: Instead of training a network to solve an nth order differential equation, we propose transforming the problem into the equivalent system of n first-order equations in phase space. The target of the network is to solve all equations of the system simultaneously, effectively introducing a multitask optimization problem. We compare both approaches empirically on various problems, ranging from second-order differential equations with constant coefficients to higher-order and nonlinear problems. We also show that our approach is suited for solving partial differential equations. Our results show that the system approach performs equal or better in most experiments performed. We analyze the learning process for the few runs that did not perform well and show that the problem stems from conflicting gradients during training, effectively obstructing multitask learning. The result of this paper is a straightforward heuristic that can be incorporated into any subsequent research that builds on PINNs solving differential equations. Moreover, it also shows how to make PINNs even more efficient by implementing techniques from multitask learning literature.

Recent research has indicated considerable potential for millimeter-waves in copper access, with data-rate estimations up to 1 Terabit per second for a reach of 100 m. This line of research exploits millimeter-waves and their corresponding higher-order propagation modes inside the twisted pair cable binder. Unlike the conventionally used transmission-line mode (currents through copper wires), the approach relies on the copper and plastics present in these cables to form a low-loss waveguide. Here, we take a closer look at the potential of millimeter-wave propagation in twisted pair cables by refining the idealized assumptions, used by Cioffi et al., and identifying the limiting factors. To this end, we introduce the concept of transformation optics as an efficient method of calculating the propagating modes on a twisted pair. Leveraging this technique allows us to calculate modal propagation using realistic material parameters, exposing an important trade-off between loss and confinement. Our modeling results yield achievable data rates that are orders of magnitude lower than those achieved under idealized assumptions. According to our results, 1 Terabit per second can be achieved up to a distance of about 10 m over a twisted-pair with a plastic sheath.

Machine learning offers the potential to revolutionize the inverse design of complex nanophotonic components. Here, we propose a novel variant of this formalism specifically suited for the design of resonant nanophotonic components. Typically, the first step of an inverse design process based on machine learning is training a neural network to approximate the non-linear mapping from a set of input parameters to a given optical system’s features. The second step starts from the desired features, e.g. a transmission spectrum, and propagates back through the trained network to find the optimal input parameters. For resonant systems, this second step corresponds to a gradient descent in a highly oscillatory loss landscape. As a result, the algorithm often converges into a local minimum. We significantly improve this method’s efficiency by adding the Fourier transform of the desired spectrum to the optimization procedure. We demonstrate our method by retrieving the optimal design parameters for desired transmission and reflection spectra of Fabry–Pérot resonators and Bragg reflectors, two canonical optical components whose functionality is based on wave interference. Our results can be extended to the optimization of more complex nanophotonic components interacting with structured incident fields.

Orbital angular momentum (OAM) from lasers holds promise for compact, at-source solutions for applications ranging from imaging to communications. However, conjugate symmetry between circular spin and opposite helicity OAM states (±l) from conventional spin–orbit approaches has meant that complete control of light’s angular momentum from lasers has remained elusive. Here, we report a metasurface-enhanced laser that overcomes this limitation. We demonstrate new high-purity OAM states with quantum numbers reaching l = 100 and non-symmetric vector vortex beams that lase simultaneously on independent OAM states as much as Δl = 90 apart, an extreme violation of previous symmetric spin–orbit lasing devices. Our laser conveniently outputs in the visible, producing new OAM states of light as well as all previously reported OAM modes from lasers, offering a compact and power-scalable source that harnesses intracavity structured matter for the creation of arbitrary chiral states of structured light. A metasurface laser generates orbital angular momentum states with quantum numbers reaching l = 100. Simultaneous output vortex beams, with Δl as great as 90, are demonstrated in the visible regime.

Optical resonators enable the generation, manipulation, and storage of electromagnetic waves. The physics underlying their operation is determined by the interference of electromagnetic waves, giving rise to the resonance spectrum. This mechanism causes the limitations and trade-offs of resonator design, such as the fixed relationship between free spectral range, modal linewidth, and the resonator's refractive index and size. Here, we introduce a new class of optical resonators, generating resonances by designing the optical path through transverse mode coupling in a cascaded process created by mode-converting mirrors. The generalized round-trip phase condition leads to resonator characteristics that are markedly different from Fabry-Perot resonators and can be tailored over a wide range. We confirm the existence of these modes experimentally in an integrated waveguide cavity with mode converters coupling transverse modes into one supermode. We also demonstrate a transverse mode-independent transmission and show that its engineered spectral properties agree with theoretical predictions., Resonators are key components in optics. In this work, the authors introduce a class of optical resonators with distinctly different properties from conventional resonators, allowing fundamental design trade-offs to be circumvented.

Our ability to generate new distributions of light has been remarkably enhanced in recent years. At the most fundamental level, these light patterns are obtained by ingeniously combining different electromagnetic modes. Interestingly, the modal superposition occurs in the spatial, temporal as well as spatio-temporal domain. This generalized concept of structured light is being applied across the entire spectrum of optics: generating classical and quantum states of light, harnessing linear and nonlinear light-matter interactions, and advancing applications in microscopy, spectroscopy, holography, communication, and synchronization. This Roadmap highlights the common roots of these different techniques and thus establishes links between research areas that complement each other seamlessly. We provide an overview of all these areas, their backgrounds, current research, and future developments. We highlight the power of multimodal light manipulation and want to inspire new eclectic approaches in this vibrant research community., Under review in J. Opt

We introduce a new class of on-chip optical tweezers with high trapping efficiency, compact footprint, and broadband operation by integrating free-form micro-reflectors and micro-lenses to the facets of waveguides to generate the strong three-dimensional optical field gradient for optical trapping. We demonstrate the design, fabrication, and measurement of both reflective and refractive micro-optical tweezers. The reflective tweezers feature a remarkably small trapping threshold power, and the refractive tweezers are handy for multi-particle trapping and inter-particle interaction analysis. This new class of tweezers is promising for on-chip sensing, cell assembly, particle dynamics analysis, and ion trapping.

We introduce a new class of on-chip optical tweezers with high trapping efficiency, compact footprint, and broadband operation by integrating free-form micro-reflectors to the facets of waveguides.

In addition to the celebrated numerical techniques, such as Finite-Element and Finite-Difference methods, it is also possible to predict the scattering properties of optical components using artificial neural networks. However, these machine-learning models typically suffer from a simplicity versus accuracy trade-off. In our work, we overcome this trade-off. We train several neural networks with an indirect goal. Instead of training the net to predict scattering, we try to train it the laws of Optics on a more fundamental level. In this way, we can increase the predictive power and robustness while maintaining a high degree of transparency in the system.

The electromagnetic near field enables subwavelength applications such as near-field microscopy and nanoparticle manipulation. Present methods to structure the near field rely on optical antenna theory, involving nanostructures that locally convert propagating waves into confined near-field patterns. We developed a theory of remote rather than local near-field shaping, based on cascaded mode conversion and interference of counterpropagating guided waves with different propagation constants. We demonstrate how to structure at will the longitudinal and transverse variation of the near field, allowing for distributions beyond the conventional monotonic decay of the evanescent field. We provide an experimental realization that confirms our theory. Our method applies to fields with arbitrary polarization states and mode profiles, providing a path toward three-dimensional control of the near field.

A particular class of focused, pulsed light beams can propagate self-similarly in free space at a fixed group velocity. Now, scientists present a law of refraction that determines how the group velocity of these beams changes as they refract at an interface between two materials.

Twisted light carrying orbital angular momentum (OAM) has given rise to many developments ranging from optical manipulation to optical communications. Generating twisted light from solid-state lasers was initially achieved by amplitude and dynamic phase control, and more recently by manipulating the geometric phase of light. These lasers have been limited to generate superposition of OAM modes as well as scalar modes with OAM e = 10. Here we incorporate a metasurface device into a visible solid-state laser to control the angular momentum of light by arbitrary spin-to-orbit coupling. We demonstrate the generation of pure Laguerre Gaussian modes with OAM up to e = 100. Modal decomposition measurements of the output beams reveal the higher purity of the generated modes can reach up to 96% for e = 1 and 88% for e = 100. our approach offers a new route for high brightness OAM states at the source.

The use of beams carrying orbital angular momentum (OAM) has become ubiquitous and topical in a variety of research fields. More recently, there has been a growing interest in exotic OAM carrying beams with spatially variant polarisation, so called Poincare sphere beams. Structuring these beams at the source gives rise to compact solutions for a myriad of applications, from laser materials processing to microscopy. Here we present a visible laser that control's the angular momentum of light by arbitrary spin-orbit (SO) momentum conversion using novel metasurface devices. Further, we outline how to generate high purity OAM states in a deterministic manner with charge up to 100. Finally, we demonstrate the generation of symmetric and non-symmetric vector vortex beams from the same source with a large OAM differential between modes of up to 90. The performance and versatility in design of our approach offers a route to control light's angular momentum at the source.

Current day high speed optical communication systems employ photonic circuits using platforms such as silicon photonics. In these systems, the polarization state of light drifts due to effects such as polarization mode dispersion and nonlinear phenomena generated by photonic circuit building blocks. As the complexity, the number, and the variety of these building blocks grows, the demand increases for an in-situ polarization determination strategy. Here, we show that the transfer of the Belinfante momentum to particles in the evanescent field of waveguides depends in a non-trivial way on the polarization state of light within that waveguide. Surprisingly, we find that the maxima and minima of the lateral force are not produced with circularly polarized light, corresponding to the north and south poles of the Poincaré sphere. Instead, the maxima are shifted along the great circle of the sphere due to the phase differences between the scattered TE and TM components of light. This effect allows for an unambiguous reconstruction of the local polarization state of light inside a waveguide. Importantly, this technique depends on interaction with only the evanescent tails of the fields, allowing for a minimally invasive method to probe the polarization within a photonic chip.

Metasurfaces allow for agile manipulation of incoming light using a single layer of resonators. Despite recent progress, it remains difficult to generate new spectral components using nonlinear surfaces, because of the limited interaction length of a pulse traveling through a single surface. Time-dependent surfaces offer an exciting alternative to overcome this limitation, but a self-consistent framework to describe this mechanism is lacking. Based on an analytical model and finite-difference time-domain numerical simulations, we obtain physical insight into the frequency-shifting process that occurs when broadband electromagnetic pulses interact with time-varying surfaces. In particular, we find that there is an intriguing relationship between the bandwidth of the incident pulse, the targeted frequency shift, and the number of Lorentzian resonators that need to be implemented. We also demonstrate that in certain parameter regimes, pulse distortion and a deviation of pulse amplitude cannot be avoided. These results are independent of the mechanism that generates the time dependence. They are also independent of the frequency, the geometry, size, and material of the unit cell, but they are rather a direct result of the subtle interplay between time-dependent Lorentzian resonances.

Since its advent in the 1970s, optical tweezers have been widely deployed as a preferred non-contact technique for manipulating microscale objects. On-chip integrated optical tweezers, which afford significant size, weight, and cost benefits, have been implemented, relying upon near-field evanescent waves. As a result, these tweezers are only capable of manipulation in near-surface regions and often demand high power since the evanescent interactions are relatively weak. We introduce on-chip optical tweezers based on freeform micro-optics, which comprise optical reflectors or refractive lenses integrated on waveguide end facets via two-photon polymerization. The freeform optical design offers unprecedented degrees of freedom to design optical fields with strong three-dimensional intensity gradients, useful for trapping and manipulating suspended particles in an integrated chip-scale platform. We demonstrate the design, fabrication, and measurement of both reflective and refractive micro-optical tweezers. The reflective tweezers feature a remarkably low trapping threshold power, and the refractive tweezers are particularly useful for multiparticle trapping and interparticle interaction analysis. Our integrated micro-optical tweezers uniquely combine a compact footprint, broadband operation, high trapping efficiency, and scalable integration with planar photonic circuits. This class of tweezers is promising for on-chip sensing, cell assembly, particle dynamics analysis, and ion trapping.

Recently, it was shown that a Mie particle in an evanescent field ought to experience optical forces that depend on the helicity of the totally internally reflected beam. As yet, a direct measurement of such helicity-dependent forces has been elusive, as the widely differing force magnitudes in the three spatial dimensions place stringent demands on a measurement's sensitivity and range. In this study, we report the simultaneous measurement of all components of this polarization-dependent optical force by using a 3D force spectroscopy technique with femtonewton sensitivity. The vector force fields are compared quantitatively with our theoretical calculations as the polarization state of the incident light is varied and show excellent agreement. By plotting the 3D motion of the Mie particle in response to the switched force field, we offer visual evidence of the effect of spin momentum on the Poynting vector of an evanescent optical field.

Transformation optics asks, using Maxwell's equations, what kind of electromagnetic medium recreates some smooth deformation of space? The guiding principle is Einstein's principle of covariance: that any physical theory must take the same form in any coordinate system. This requirement fixes very precisely the required electromagnetic medium. The impact of this insight cannot be overestimated. Many practitioners were used to thinking that only a few analytic solutions to Maxwell's equations existed, such as the monochromatic plane wave in a homogeneous, isotropic medium. At a stroke, transformation optics increases that landscape from 'few' to 'infinity', and to each of the infinitude of analytic solutions dreamt up by the researcher, there corresponds an electromagnetic medium capable of reproducing that solution precisely. The most striking example is the electromagnetic cloak, thought to be an unreachable dream of science fiction writers, but realised in the laboratory a few months after the papers proposing the possibility were published. But the practical challenges are considerable, requiring meta-media that are at once electrically and magnetically inhomogeneous and anisotropic. How far have we come since the first demonstrations over a decade ago? And what does the future hold? If the wizardry of perfect macroscopic optical invisibility still eludes us in practice, then what compromises still enable us to create interesting, useful, devices? While three-dimensional (3D) cloaking remains a significant technical challenge, much progress has been made in two dimensions. Carpet cloaking, wherein an object is hidden under a surface that appears optically flat, relaxes the constraints of extreme electromagnetic parameters. Surface wave cloaking guides sub-wavelength surface waves, making uneven surfaces appear flat. Two dimensions is also the setting in which conformal and complex coordinate transformations are realisable, and the possibilities in this restricted domain do not appear to have been exhausted yet. Beyond cloaking, the enhanced electromagnetic landscape provided by transformation optics has shown how fully analytic solutions can be found to a number of physical scenarios such as plasmonic systems used in electron energy loss spectroscopy and cathodoluminescence. Are there further fields to be enriched? A new twist to transformation optics was the extension to the spacetime domain. By applying transformations to spacetime, rather than just space, it was shown that events rather than objects could be hidden from view; transformation optics had provided a means of effectively redacting events from history. The hype quickly settled into serious nonlinear optical experiments that demonstrated the soundness of the idea, and it is now possible to consider the practical implications, particularly in optical signal processing, of having an 'interrupt-without-interrupt' facility that the so-called temporal cloak provides. Inevitable issues of dispersion in actual systems have only begun to be addressed. Now that time is included in the programme of transformation optics, it is natural to ask what role ideas from general relativity can play in shaping the future of transformation optics. Indeed, one of the earliest papers on transformation optics was provocatively titled 'General Relativity in Electrical Engineering'. The answer that curvature does not enter directly into transformation optics merely encourages us to speculate on the role of transformation optics in defining laboratory analogues. Quite why Maxwell's theory defines a 'perfect' transformation theory, while other areas of physics such as acoustics are not apparently quite so amenable, is a deep question whose precise, mathematical answer will help inform us of the extent to which similar ideas can be extended to other fields. The contributors to this Roadmap, who are all renowned practitioners or inventors of transformation optics, will give their perspectives into the field's status and future development. QC 20180614

We reveal the surprising polarization dependence of lateral forces acting on reso-nant Mie particles in evanescent fields. This effect can be utilized to retrieve the polarization state of light inside birefringent waveguides.

Artificially structured metasurfaces make use of specific configurations of subwavelength resonators to efficiently manipulate electromagnetic waves. Additionally, optomechanical metasurfaces have the desired property that their actual configuration may be tuned by adjusting the power of a pump beam, as resonators move to balance pump-induced electromagnetic forces with forces due to elastic filaments or substrates. Although the reconfiguration time of optomechanical metasurfaces crucially determines their performance, the transient dynamics of unit cells from one equilibrium state to another is not understood. Here, we make use of tools from nonlinear dynamics to analyze the transient dynamics of generic optomechanical metasurfaces based on a damped-resonator model with one configuration parameter. We show that the reconfiguration time of optomechanical metasurfaces is not only limited by the elastic properties of the unit cell but also by the nonlinear dependence of equilibrium states on the pump power. For example, when switching is enabled by hysteresis phenomena, the reconfiguration time is seen to increase by over an order of magnitude. To illustrate these results, we analyze the nonlinear dynamics of a bilayer cross-wire metasurface whose optical activity is tuned by an electromagnetic torque. Moreover, we provide a lower bound for the configuration time of generic optomechanical metasurfaces. This lower bound shows that optomechanical metasurfaces cannot be faster than state-of-the-art switches at reasonable powers, even at optical frequencies.

Significance Using a highly sensitive particle-tracking scheme, we report an observation of elliptical motion of microparticles driven by a single evanescent field. We show that this behavior is highly tunable and predictable using a theoretical model that accounts for Mie scattering and hydrodynamic drag. The significance is twofold. First, our work represents an important step in understanding the detailed dynamics of microparticles in a fluidic environment—especially near a surface—a prerequisite for effective application of optically driven microparticles as functional elements in optofluidic devices. Second, our method could complement current structured-light approaches for inducing orbital motion in microparticles, with readily tunable parameters of motion including orbital frequency, radius, and ellipticity.

The enhancement of optical forces has enabled a variety of technological applications that rely on the optical control of small objects and devices. Unfortunately, optical forces are still too small for the convenient actuation of integrated switches and waveguide couplers. Here we show how the optical gradient force can be enhanced by an order of magnitude by making use of gauge materials inside two evanescently coupled waveguides. To this end, the gauge materials inside the cores should emulate imaginary vector potentials for photons pointing perpendicularly to the waveguide plane. Depending on the relative orientation of the vector potentials in neighboring waveguides, i.e., pointing away from or towards each other, the conventional attractive force due to an even mode profile may be enhanced, suppressed, or may even become repulsive. This and other new features indicate that the implementation of complex-valued vector potentials with non-Hermitian waveguide cores may further enhance our control over mode profiles and the associated optical forces.

Recently, it was demonstrated that metamaterials can be used to harvest ambient electromagnetic energy [1]. However, the choice of the rectifying circuit for highly efficient energy conversion of RF electromagnetic power to DC power remains an important challenge [2,3]. Therefore, it is important to design a metasurface integrating a rectifier circuit in combination with a proper matching impedance to optimize the power transfer.

Transformation optics makes use of coordinate transformations to explore the possibilities offered by artificially structured metamaterials for the manipulation of a wide variety of electromagnetic phenomena. Since a decade, transformation optics has consistently extended its scope. Initially, coordinate transformations were only applied to the transformation of light in the simplest of optical setups, i.e., empty space [1,2]. In particular, straight light trajectories were being transformed into curved ones, e.g., avoiding a specific region in space (invisibility devices), bending over a specific angle (benders and splitters), and focussing to a particular point (lenses). Today, research efforts go beyond the transformation of empty space, allowing metamaterials to further enhance our control on electromagnetic phenomena other than light propagation. In this contribution, I will review our work on transformation optics to manipulate guided modes along and optical forces between metamaterial waveguides [3,4], the emission of Cerenkov radiation due to fast charged particles [5,6], and the Goos-Hanchen shift at metamaterial surfaces [7].

Transformation optics has revolutionized our approach to material design in several scientific disciplines by determining the material properties that implement the desired effects of a coordinate transformation. Unfortunately, the performance of several coordinate-based devices, such as beam splitters and invisibility cloaks, suffers from the necessary implementation of singularities with extreme material parameters. Here, we make use of transformation optics to eliminate these singularities in an isotropic way for the improvement of coordinate-based metamaterial waveguides. In particular, singularities that lead to vanishing material properties are softened with a global rescaling of the coordinates, while singular terms that lead to infinite material properties are strategically replaced by well-behaved curve factors. Detailed full-wave simulations confirm that the resulting waveguide devices are as efficient as their singular counterparts despite the fact that they consist of materials with much more moderate optical properties.

Electromagnetic energy harvesting, i.e., capturing energy from ambient microwave signals, may become an essential part in extending the battery lifetime of wearable devices. Here, we present a design of a microwave energy harvester based on a cut-wire metasurface with an integrated PN junction diode. The cut wire with a quasistatic electric-dipole moment is designed to have a resonance at 6.75 GHz, leading to a substantial cross-section for absorption. The external microwaves create a unidirectional current through the rectifying action of the integrated diode. Using an electrical-circuit model, we design the operating frequency and the resistive load of the cut wire. Subsequently, by optimizing our design using full-wave numerical simulations, we obtain an energy harvesting efficiency of 50% for incident power densities in agreement with the typical power density of WiFi signals. Finally, we study the effect of connecting adjacent unit cells of the metasurface in parallel by a thin highly inductive wire and we demonstrate that this allows for the collection of current from all individual cells, while the microwave resonance of the unit cell is not significantly altered, thus solving the wiring problem that arises in many nonlinear metamaterials.

Transformation optics, a geometrical recipe for metamaterial design, was originally conceived as a tool to smoothly modify the trajectory of light using continuous coordinate transformations. Here, we show how discontinuous transformations can be used as a geometric framework to understand and manipulate phenomena at the surface of nanophotonic structures. In particular, we show how the Goos-Hanchen shift - a lateral shift exhibited by totally reflected beams - can be tailored and we provide a classification and complete analytical description of this effect in existing complex media.

Surrounded by electromagnetic radiation coming from wireless power transfer to consumer devices such as mobile phones, computers and television, our society is facing the scientific and technological challenge to recover energy that is otherwise lost to the environment. Energy harvesting is an emerging field of research focused on this largely unsolved problem, especially in the microwave regime. Metamaterials provide a very promising platform to meet this purpose. These artificial materials are made from subwavelength building blocks, and can be designed by resonate at particular frequencies, depending on their shape, geometry, size, and orientation. In this work, we show that an efficient electromagnetic energy harvester can be design by inserting a nonlinear element directly within the metamaterial unit cell, leading to the conversion of RF input power to DC charge accumulation. The electromagnetic energy harvester operating at microwave frequencies is built from a cut-wire metasurface, which operates as a quasistatic electric dipole resonator. Using the equivalent electrical circuit, we design the parameters to tune the resonance frequency of the harvester at the desired frequency, and we compare these results with numerical simulations. Finally, we discuss the efficiency of our metamaterial energy harvesters. This work potentially offers a variety of applications, for example in the telecommunications industry to charge phones, in robotics to power microrobots, and also in medicine to advance pacemakers or health monitoring sensors.