Disordered optical metasurfaces: from light manipulation to energy harvesting

ABSTRACT Metasurfaces, the planar version of artificial structured media at sub-wavelength scale, provide the ability to manipulate light wave in a naturally unavailable way. They offer an unprecedented platform for a plethora of applications ranging from holography, imaging, optical communication to nonlinear light source and quantum computing. Conventionally and straightforwardly, metasurfaces are prepared in ordered configuration, aiming at reducing the geometric fluctuations to guarantee a good performance as designed. On the other hand, the inevitability of fabrication imperfection in nanophotonics and unique properties of disorder have been inspiring the exploration of the metasurfaces with novel design. To supplement the comprehensiveness in review for metasurfaces, here, we overview the mechanisms, characteristics and related applications of disordered metasurfaces, concentrating on recent progresses from light manipulation to energy harvesting and beyond. Besides reviewing the achievements in a wide range of applications with disordered metasurface, we provide an outlook on their future developments. With unique features, disordered metasurface may be a promising alternative for the ordered ones, especially for the practical requirement for large-scale production.


Introduction
Optical metasurfaces are artificial planar devices composed of subwavelength meta-atoms that can manipulate optical properties in a way unprecedented in nature.Metasurfaces are able to produce abrupt changes along the propagation direction, diminishing the dependence on the propagation effect in conventional bulky elements.The high flexibility in both local design and global arrangement of meta-atoms promises metasurfaces with unparalleled ability in tailoring the incident light.The planar configuration and ultrathin thickness make metasurfaces feasible for integration, satisfying the trend of miniaturization in optical systems.Consequently, metasurfaces have been applied in a wide range of fields, including optical imaging [1], optical communications [2], nonlinear optics [3], augmented reality (AR) display [4], quantum communication and computing [5], biomedical sensing [6] and energy conversion [7], to name a few.With judicious design, metasurfaces are used to manipulate the light by controlling its amplitude, phase and polarisation.Benefiting from the exploitation of propagation phase and geometric phase, light tailoring based on metasurfaces has become more precise.Thus, metasurfaces are further endowed with versatile optical functions.Many metasurface-based optical devices are demonstrated, such as metagratings [8], metalenses [9,10], metaholograms [11], q-plate for spin-orbit interaction [12,13], metasurface-based nonlinear sources [14].After rapid development during the past decades, the study of metasurfaces has been recognized as a pivotal topic in nano-photonics.Nowadays, increasing efforts are made to introduce novel concepts and phenomena into metasurfaces to explore their potential [15].
The optical properties of metasurfaces are directly decided by the shape and arrangement of meta-atoms, which subsequently determine the optical functions of the meta-devices.Therefore, to obtain the target optical function, minimizing the fluctuations in the shape and arrangement of meta-atoms is required.Metasurfaces in this configuration are ordered metasurfaces, which account for the majority at present.To prepare ordered metasurfaces, fabrication technologies with high precision are a prerequisite, such as electron beam lithography (EBL), ion beam lithography and direct laser writing.Recently, metasurfaces with disorder have aroused attention as an alternative.Without the demand for completely eliminating the fluctuations, disordered metasurfaces can be prepared by using easy and cost-effective fabrication technologies.The disorder itself also introduces new phenomena and mechanisms during the interaction between light and metasurfaces, stimulating research in a new path.Moreover, improved understanding of disorder may shed light on the optimisation of ordered metasurfaces if the influence of the fluctuations on their optical functions can be ascertained.
Disorder, regarded as another pole opposite to order, is a ubiquitous phenomenon in physical systems both artificially and naturally.While the disorder enhances the complexity of the system, it endows fascinating possibilities and properties unachievable in ordered ones.The explorations on disorder in photonics have a long history, dating back to Anderson's discovery of the suppression of diffusion in disordered crystals [16].The diffusion and unidirectional scattering of light by a random media are typical phenomena relevant to disorder [17].During the interaction with the disordered structures, the randomly distributed waves of light interfere with each other, resulting in complicated patterns of light field.It subsequently gives rise to a series of interesting optical properties.A series of theoretical frameworks have been proposed to describe the optical properties of disordered media, such as models describing the impact of the geometrical parameters of structures, and models for statistical description of the structural properties [18].A variety of applications based on disordered photonic structures have been proposed, including imaging [19], time-reversed optical focusing [20], light harvesting [21] and nano-lasing [22].
Ordered metasurface is the mainstream in meta-devices, whose mechanisms and applications have been concluded systematically by the previous reviews [1][2][3][4][5]7,23,24]. Disordered metasurface, as another indispensable branch of meta-devices, also exhibits unique optical properties and leads to appreciable applications.However, a comprehensive review for disordered metasurfaces is missing.In this article, we aim to provide an overview of disordered metasurfaces in the classical applications and beyond, concentrating on their configurations, functions and applications.We organize the main contents into six sub-sections, corresponding to various applications of disordered metasurfaces, including resonance manipulation, wavefront manipulation, structural colour generation, energy harvesting and manipulation, nonlinear optics, and disorder-induced transition.In addition, we prospect the future potential of disordered metasurfaces at the end of the article.Here, we deliberately focus on the disordered structures in metasurfaces, i.e. quasi-two-dimensional thin-flim-liked structures, while discussions of configurations beyond can be found in other reviews [17,[25][26][27][28].

Modelling the disorder in metasurface
In the following content, the degree of disorder is modelled using two different approaches.The first approach is based on the geometric parameters of meta-atoms, including their shape and x-y coordinates.Starting from a periodic lattice with identical meta-atoms, the disorder can be introduced in the size, the position and the both, as shown in Figure 1a.Random variables following Gaussian or uniform distributions are used as the perturbation to the geometric parameters.The standard deviation quantitatively represents the level of the disorder.
Alternatively, the disorder can alternatively be modelled statistically as the correlation function.In this case, different criteria of disorder can be differentiated, as depicted in Figure 1b.More details can be found in a recent review [18] and references therein.The choice of the type of the model depends on different morphologies of the metasurfaces and specific properties under investigation.

Resonance manipulation
Optical cavities supporting resonant modes are widely applied in lasers, optical parametric oscillators and interferometers.Apart from conventional structures, resonance has attracted great attention from nanophotonics, including the fields of optical metamaterials and metasurfaces.The optical properties of metasurfaces are strongly affected by resonances, such as surface plasmon resonance in metallic metasurfaces and Mie resonance in dielectric metasurfaces [29].The manipulation of resonances is of great importance for metasurfaces, affecting the subsequent light control and function realization.Compared with ordered counterparts, disordered metasurfaces with additional complexity provide new degrees of freedom for the resonance manipulation, which, on the other hand, is challenging and urgent to be explored.
The optical scattering from metasurfaces can be manipulated through engineering the arrangement and/or shape of meta-atoms.The resonance engineering becomes much more complicated for disordered metasurfaces.Correspondingly, many efforts have been devoted into developing models and methods for disordered metasurfaces to understand and predict the properties of resonances and relevant scattering behaviors.Albooyeh et al. proposed a model to predict scattering parameters applicable to any incident angle [30], as shown in Figure 2a.By considering the contributions of both electric and magnetic components of light, this model allowed to derive the effective surface polarizability tensors and subsequently obtained the scattering parameters of disordered metasurfaces.Moreover, an analytical model based on homogenization method was proposed to describe the electromagnetic properties of metasurfaces composed of nanoparticles with randomly fluctuating polarizabilities [34].The diffused scattering of the disordered metasurfaces could be increased when larger polarizability fluctuations of nano-particles were induced.
Quasicrystal metasurfaces (QCMs) have been reported as versatile platforms for resonance manipulation.As another type of configurations besides periodic and amorphous arrangements, structures with quasicrystal arrangement exhibit unique optical property.Kruk et al. introduced the concept of quasicrystal into optical metasurfaces and explored the impact of arrangement on their far-field property [35].It is revealed that the breaking of periodicity in quasicrystal metasurfaces leads to isotropic optical properties, while the existence of long-range order preserves the strength of optical properties.For QCMs with metal structures, surface plasmons (SPs) and their interaction plays vital roles in the scattering behaviors of light.To investigate the behaviors of SPs, the near-filed distributions of SPs on metal aperture arrays with periodic, quasicrystal and random arrangements were studied [31], as demonstrated in Figure 2b.Based on the characteristics of surfaces for the study of the interplay of order and disorder at oblique incidence.As shown in panel i, the photonic metasurface is composed of sandwich-type nanoparticles, with different arrangements, periodic, disordered and amorphous metasurfaces are prepared, which exhibit different optical response, for example, different transmission spectra in panel ii [30].(b) Metal split metasurfaces with quasicrystal, periodic and random arrangements for the study of near-field surface plasmons.The schematic diagrams of the three types of metasurfaces and corresponding simulated surface plasmon distributions at different frequency are presented in panel i, ii and iii, respectively [31], where the propagation modes of SPs for quasicrystal and periodic metasurfaces are denoted by the arrows and there is no propagation of regular SPs for random metasurface.directional propagation, metasurfaces arranged in three configurations were compared.The propagation mode of SPs in QCMs depended on the reciprocal vector, elucidated by the quasi-momentum conservation rule.Apart from the own influence of SPs, the interaction between SPs and transmission resonances also affects the scattering on QCMs.To investigate the related mechanisms, QCMs arranged with different rotational symmetries were prepared and characterized [36].Due to the coupling condition of resonances, SPs on QCMs were mainly determined by the reciprocal vectors, which subsequently changed the optical properties of QCMs, such as transmission minima.As a result, it paves the way to control the optical properties of QCMs with different arrangements.
For metasurfaces composed of high-refractive-index meta-atoms, Mie resonances play a vital role in the optical response through the induced electric and magnetic moments.With judicious design to control Mie resonances, disordered metasurfaces were endowed with different optical functions, including enhanced absorption and anti-reflection.Babicheva et al. proposed a semi-analytical model to analyze the reflective properties of Mie-resonancebased disordered metasurfaces, particularly addressing the influence of the substrate [37].The total reflection of the disordered metasurface was determined by the interference of the contributions of meta-atom array and substrate.To suppress the reflection from the substrate, it was found that a phase difference between the electric and magnetic dipole moments induced by the array must be satisfied.This model explained the characteristics of the reflection spectra related to Mie resonances.Mie-resonance-based disordered metasurfaces were also applied in anti-reflection coatings.In Figure 2c, a disordered metasurface that consists of SiGe islands on silicon was demonstrated [32].Mediated by Mie resonances in SiGe islands, the incident light was efficiently coupled into the substrate, inducing significantly reduced reflection.As a result, high-efficient light trapping with broad ranges in both spectrum and angle was achieved.
The influence of different kinds of disorders on resonances is also of substantial significance.Recently, Zakomirnyi studied collective lattice resonances in disordered metasurfaces where three types of disorder were considered, including positional disorder, size disorder and quasi-random disorder [33], as demonstrated in Figure 2d.By applying coupled dipole approximation, the influence of aforementioned disorders on the electric dipole and magnetic dipole coupling in collective lattice resonances was analyzed systematically.And the relationship between lattice configuration and resonances was clarified.

Wavefront manipulation
Wavefront control of light is a topic that has attracted great attention of researchers in different disciplines, with relevant optical elements invented and utilised in a wide range of applications.With the rapid development of optical systems and devices, the requirements in miniaturization and integration become indispensable.On this issue, conventional optical elements are restricted because they are usually bulky and heavy, and metasurfaces have been proved as a promising alternative.Hence, metasurfaces used in wavefront manipulation have been developed rapidly, stimulating the emergence of versatile meta-devices.For the sake of obtaining specific optical functions in wavefront manipulation, the arrangement of metasurfaces is usually ordered, which demands precise and high-quality fabrication such as EBL, ion beam lithography or direct laser writing.On the contrary, the fabrication methods of some disordered metasurfaces are relatively easy and cost-effective.Moreover, disorder can be a new tool for the design of metasurfaces, providing a new degree of freedom in wavefront manipulation.Consequently, disordered metasurfaces have been proposed and realised for wavefront manipulation, demonstrating alternative controllability of angular momentum, chirality, convergence or split of light, and so on.
Spin is one of the intrinsic properties of light, which is related to the polarisation and angular momentum of photons.Spin-based metasurfaces have already been widely studied in both the linear and nonlinear regime, leading to applications such as the manipulation of spin-orbit interaction [12], generation of optical vortex beam [38], spin-based multi-channel holography [39], and so on.Recently, disordered metasurfaces have been attractive alternatives in the spin-related manipulations due to the unique properties in optical response.Veksler et al. proposed to use random gradient metasurfaces with a custom-tailored geometric phase for multiple wavefront shaping [40].With judicious design of the orientation angle distribution of meta-atoms, phase-matching condition was fulfilled, enabling the control of light in near field and far field.The anisotropic meta-atoms provided spin-dependent channels for light manipulation, subsequently realizing spin-controlled wavefront-shaping and the generation of structured light.Moreover, disordered metasurface based on geometric phase is an ideal platform to investigate light-matter interaction in nano-scale media.Wang et al. investigated stochastic photonic spin Hall effect from disordered metasurfaces composed of ferromagnetic meta-atoms [41].As demonstrated in Figure 3a, instead of anisotropic dimensions, the Berry-Zak phases were induced by disordered magneto-optical effects under an external magnetic field.A spin-dependent light shift between two opposite spin components was obtained, for precise detection of a five-nanometre fluctuations in the meta-atoms.While spin-dependency is a feature that can be utilised in the design of disordered metasurfaces, however, in particular cases, it is expected to be eliminated.For disordered QCMs that consist of anisotropic meta-atoms, it was discovered that the spin-dependency affect the light-matter interaction, leading to different scattering behaviors [48].By inserting defects in the geometric phase, the spin-dependency of QCMs can be removed.The chirality in optical response of metasurfaces has been studied in both linear and nonlinear regions, stimulating the emergence of related applications, such as ultrathin circular polarizers [49], multi-functional nonlinear sources [50] and multi-channel holography [51].The complexity in disordered metasurfaces can induce unique chiral response in sharp contrast with their ordered counterpart.Based on a single-layer metallic QCMs with coupled response from periodic perturbations, Amin et al. realised chirality response in the reflected fields, exhibiting broadband response and high conversion efficiency [52].The QCMs operated as HWP and QWP, realizing the conversion between the opposite circular polarisations and the conversion between linear and circular polarisations, respectively.Apart from the single-layer metasurfaces, Fasold et al. achieved chiral optical response with bilayer-disordered metasurfaces [42], as presented in Figure 3b.The bilayer metasurfaces were composed of two layers of anisotropic meta-atom array with linear birefringence, resulting in strong circular dichroism.By introducing rotational disorder in the arrangement of meta-atoms, the linear birefringence was eliminated while the circular dichroism was preserved.The eigenvectors of the Jones matrices of the bilayer-disordered metasurfaces were analyzed, which are proved as left-and right-circularly polarized states.
Versatile functional elements based on metasurfaces have been demonstrated, including metalenses, metagratings, meta-holograms and so on.Disordered metasurfaces exhibit distinctive optical properties, which, on the other hand, affect their performance as functional elements.Hence, strategies to optimise the design of disordered metasurfaces have been proposed to endow them with better performance and functionalities in with permission from REF [43].Copyright 2018, American Chemical Society.(d) Metasurfaces with engineered noise for high-capacity polarisation multiplexing.Panel i illustrates the design of the polarisation-multiplexed metasurfaces.The SEM images of the fabricated metasurface are shown in panel ii (scale bars from left to right: 500 nm, 250 nm).The measured holographic images in 11 independent channels of linear polarisation states are shown in panel iii, where the polarisation orientations are denoted by the arrows.Panel (d) is from REF [44], reprinted with permission from AAAS.(e) Metasurface in randomly flipped configuration for antiglaring application.Panel i demonstrates the concept of antiglaring.The schematic diagrams of the metasurface is shown in panel ii, whose imaging results in transmission and reflection directions are presented in panel iii, verifying the antiglaring function [45].(f)Disordered metasurface with long-range-order via a topology optimisation approach.The disordered supercell metasurface and topology-optimised freeform metasurface as well as their diffraction intensity distributions are shown in panel i and ii, respectively, confirming that the phase fluctuation is suppressed by applying the topology optimisation.Panel (f) is adapted with permission from REF [46].Copyright 2022, Wiley-VCH GmbH.(g) Perfect optical diffusers via dielectric Huygens' metasurfaces with positional disorder.The pair correlation functions and SEM images of the three types of disordered metasurfaces are presented, indicating that the long-range order is maintained by the perturbed array while the uniformtypes metasurfaces exhibit ideal-gas-like long-range order and limited short-range order.Panel (g) is adapted with permission from REF [47].Copyright 2021, Advanced Materials published by Wiley-VCH GmbH.
wavefront manipulation.In an example of disordered metasurfaces composed of randomly placed anisotropic meta-atoms, the factors that influence their performance were investigated [53].For 1D disordered metalenses, their performance was mainly affected by the density of meta-atoms, while the performance of 2D metalenses was mostly governed by both the density and the near-field coupling.
Yannai et al. realised a multifunctional element for spectrum-dependent disguise, holographic tagging and imaging of a target object by interleaving the ordered and disordered systems in an all-dielectric metasurface [43], as demonstrated in Figure 3c.A group of meta-atoms were arranged according to analytic-phase profiles (e.g.lens) and were spatially interleaved with another group of randomly arranged meta-atoms.Based on this interleaving strategy, the metasurfaces were able to camouflage the object at a specific wavelength, while image it at another wavelength.Multiplexing can also be achieved by introducing disorder into metasurfaces.Xiong et al. realised high-capacity polarisation multiplexing by introducing engineered noise into optical metasurfaces, achieving 11-channel independent holographic imaging [44], as shown in Figure 3d.Traditional polarisation manipulation via single-layer metasurfaces is limited by the precise solution of Jones matrix elements.This work brake the bottleneck by introducing engineered noise, including correlated noise and noncorrelated noise with random distribution.Incident light with different polarisation states generated different holographic images in 11 independent channels, where the inter-channel coupling was greatly reduced by the designed disorder (noise).
By controlling the coupling and local light-matter interaction in disordered metasurfaces, the properties of light in far-filed can be manipulated.For example, the far-field wavefront of light generated from nonlinear luminescence process was controlled by using disordered metallic metasurfaces [54].By shaping the phase of femtosecond excitation, the local nonlinear luminescence was optimised, and the magnitude of the nonlinear luminescence signal was enhanced by two orders.An inverse design algorithm was proposed to obtain size-free disordered metasurfaces with desired angular multi-functionality [55].The meta-atoms in the disordered metasurfaces were analyzed as scatters forming a disordered network, where the coupling between each element was considered to predict the radiation profile in near and far fields.
Apart from the aforementioned applications, disordered metasurfaces have been utilised to manipulate other processes of the light-matter interaction.Various functions have been realized by using disordered metasurface, ranging from non-reciprocal transmission, unidirectional transmission to scattering reduction.Chen et al. realized quasiperiodic dendritic cluster set metasurfaces through electrochemical deposition method, with ease of fabrication compared to ordered metasurfaces [56].To predict the optical response of the quasiperiodic metasurfaces with random-phase profiles, a model based on Gerchberg-Saxton algorithm was established.Moreover, as predicted in the model, when the resonance condition was satisfied, the quasiperiodic metasurfaces were able to produce abnormal-phase shift within the range of visible light.
Metasurfaces with disordered arrangement were also applied in the field of display such as antiglare coatings, which requires the reflection on the surfaces is reduced when the quality of transmission is not affected.Chu et al. proposed an antiglare coating utilising a bilayer metasurface with randomly flipped components [45], as shown in Figure 3e.Globally random-phase profile was introduced for the reflection, diffusing the light reflected from the metasurface.However, the non-reciprocity of the bilayer configuration preserves the transmitted light to be distortion-free.By using the bilayer metasurface, the reflected light was effectively blurred, while the transmitted images remained clear, meeting the requirements of antiglare coatings and was able to be applied in screens for display.
Taking consideration of long-range order in disordered system is also important for the design of metasurfaces.Disorder engineering and topology optimisation were combined in the design of all-dielectric metasurfaces with anisotropic meta-atoms [46].By engineering the disorder parameters in the supercell of metasurfaces, homogeneous-phase fluctuations were introduced, resulting in a unidirectional transmission.The topology optimisation significantly improved the conversion efficiency of the disorder metasurfaces for circularly polarized light.The comparison between disorder metasurfaces with and without topology-optimisation is demonstrated in Figure 3f.
Owing to the characteristics of randomness, disordered metasurfaces provide an ideal platform for the diffusion of light, with easier integration compared with conventional optical diffusers.Disordered hyperuniform metasurfaces with V-shaped meta-atoms were demonstrated as promising optical diffusers, where the disordered distribution was induced as an additional degree of freedom to enhance the diffusion, leading to effective reduction of the radar cross section [57].It was shown that hyperuniform disorder made the metasurface perform better in the operation bandwidth compared with periodic one.Dielectric Huygens' metasurfaces with disorder were also proved as alternatives for optical diffusers [47], which are attributed to the low absorption losses and nearly Lambertian scattering profiles.As shown in Figure 3g, the positional disorder was carefully tailored to balance the scattering strength of electric and magnetic moments, thus scattering performance is improved.

Structural colour generation
Structural colours are resulted from the light-matter interactions in micro/nanostructures [58][59][60], where the colour is determined by not only the material but also by geometric features comparable with the optical wavelengths.Compared to conventional colouration based on dyes or pigments, colours produced from micro/nano structures have demonstrated promising potential due to their unprecedented stability and spatial resolution of a single pixel.Period configurations including photonic crystals, plasmonic nanostructures and Mie resonators have been widely utilised to produce colours, continuously achieving higher colour purity and better spatial resolution.However, the sophisticated fabrication of periodic structures severely becomes the Achilles' heel for usage in practical applications where cost is a major concern.
The disordered metasurfaces free from the lithography process may provide an alternative solution.Intuitively, the disorder, or the geometric fluctuation, can broaden the optical resonance of the structure, equivalently reducing the quality of the produced colour.Recently, Mao et al. proposed a mechanism to utilise the broadband absorption of disordered plasmonic metasurface to produce colours with purity comparable to periodic counterparts [61].A transparent Fabry -Perot (FP) cavity was introduced beneath the metasurface, which can protect the photons around resonance from absorption.A back reflector was introduced to improve the light confinement of the cavity.Colours with high purity were achieved by varying the thickness (resonance) of the Fabry-Perot cavity, as shown in Figure 4a.Compared to conventional methods using the optical resonances for absorption, the structure reflected (generated) the colour at the resonance.Meanwhile, increasing broadband absorption of the disordered metasurface at critical coupling leads to better colour purity.Or equivalently, fabrication imperfection (disorder) counter-intuitively improved the performance of the systems.
Similar configurations (disordered metasurface + Fabry -Perot cavity) were achieved by disordered metasurfaces fabricated with different techniques [65][66][67] that significantly reduced the complexity of the nanofabrication.Taking full advantage of the disorder, Franklin et al. achieved angle-independent structural colours [65], a task that is extremely difficult for ordered structures.Wu et al. embedded disordered metasurfaces into perovskite composite films for backlit displays [67].
Tunability is one of the most desired features for structural colour generation.The resonance-determined colour from the disordered metasurface provides a feasible way to control the colour by changing the thickness or refractive indices [61].Jung et al. replaced the dielectric FP-cavity with a chitosan hydrogel, which changed the thickness according to external humidity conditions [62], as demonstrated in Figure 4b.Vivid colours were realised with a single structure under different humidities, which can be used as a humidity sensor.Later, the same group used another polymer, polyvinyl alcohol, to improve the time response to less than 1 s [68].Recently, a similar configuration produced at large scale was reported, where disordered structures offered opportunity to obtain structural colour with angle-and polarisation-independencies [69].
The impact of an order-to-disorder transition of the metasurfaces was investigated comprehensively [70], even with the aid of a deep neural network for the prediction [71].Vynck et al. further demonstrated that tailoring the disorder in the metasurface can control quasi-independently the colours of the specular and diffuse components [63].The simulated visual appearance of macroscopic objects covered by different manipulated disordered metasurfaces is demonstrated in the top panel of Figure 4c, combined with corresponding experimental realisations.To reach the designed degree of disorder in the metasurfaces, top-down lithography techniques were used, compromising the controllability in practical applications.
In nanophotonics, the optical response is determined by both the geometry and the material.Conventionally, the colours from structures with a fixed geometry vary with the type of materials.For example, nanostructures (such as nanospheres, nanocubes or nanorods) demonstrate different colours for Au or Ag-based ones.By virtue of the randomised coupling and scattering in the metasurface, Mao et al. demonstrated the possibility to produce the same colour with different materials [64].Utilising the configuration by combining a disordered metasurface and an FP-cavity, vivid colours were achieved based on a library of materials including silver, gold, platinum, palladium (experimentally) and aluminium, chromium, copper, nickel, ruthenium, and titanium nitride (numerically).Figure 4d illustrates the indiscernible colours produced from four different metals.Taking the unparalleled advantages of the disorder, metasurfaces with abundant materials such as Cu or Al may further reduce the cost for nanostructure-based colouration.
Disordered metasurfaces themselves can generate structural colours by tuning geometric properties of meta-atom (nanoparticles), with a limited colour coverage in the CIE map [72,73].However, the back-reflector-free structures offered the possibility for image multiplexing, as shown in ref [73].Meanwhile, some 3D disordered nanostructures have been discovered or proposed for the generation of vivid colours [74][75][76].

Energy harvesting and manipulation
Efficient capture and utilisation of energy in an environmentally friendly way are of paramount significance.Among the different types of energy available, solar energy originating from the blackbody radiation of sun probably is the most promising one.Apart from the harvesting, the manipulation of energy also plays a vital role in energy-related devices.In optical region, it refers to the conversion of photon energy to other forms, such as photon-photon, photon-phonon and photon-electron, as well as empowering the energyrelated devices with specific functions.Naturally, nanophotonic structures with enhanced light-matter interaction and designed optical functions have been ones of the most pivotal components in energy-related applications.While impressive progress has been made in introducing nanostructures to energy harvesting systems ranging from photovoltaics, photo-thermal conversion to photocatalysis [7,77], disordered structures have been less investigated compared to their ordered counterparts [78].
Besides the ease of fabrication, disordered structures bear another intrinsic feature that significantly benefits solar energy harvesting -the broadband response insensitive to wavelength and polarisation of incident light.The undesired fabrication imperfection in conventional devices including shape deformation, size fluctuation and misalignment of the element endows disordered structures with a broadband absorption covering the spectrum from ultraviolet to infrared and insensitivity to the initial conditions including polarisation and incident angle [79][80][81].Meanwhile, the disorder-enhanced absorption is a geometric effect, which can be used for both dielectric and metallic platforms.
Besides EBL [84,87], disordered structures can be realised through simple bottom-up methods, including attaching pre-synthesised nanoparticles on the substrate [86], a one-step anodisation process on Al thin film [84].With judicious optimisation, broadband absorption covering a large portion of the solar spectrum was reported, with averaged absorption of 66.5% for silicon from 400 to 1500 nm [87] and >90% absorption for a hybrid structure utilising TiN as absorbing material from 300 to 2500 nm [84].Figure 5a demonstrates a lithography-free metasurface composed of TiN covering anodic aluminium oxide nanotemplates, with the ability to capture a large portion of solar radiation.Yildirim et al. utilised metasurfaces composed of tin oxide (ITO) nanorods with emissivity (absorption) of 0.968 from 2.5 to 25 μm, working as an optical solar reflector requiring high reflection in the solar spectrum and perfect absorption in the thermal infrared [94].Alternatively, near-perfect absorption was reported using the ordered structure embedded in a disordered medium [95].A chaotic cavity [96] worked as disordered medium, facilitating the absorption of metasurface at microwave frequency.Recently, the photo-thermal conversion inside a disordered metasurface was wisely used as anti-fogging coatings [97].The structure was optimised for good transparency in the visible but absorptive in the infrared, generating heat locally to protect flexible and foldable surfaces such as windows, shields or glasses away from fogging.
Disordered structures have also been designed to trap light and create optical modes for the next stage usage.Vynck et al. proposed and realised a disordered photonic slab, dielectric membrane perforated with cylindrical air holes acting as scattering centres for efficient light coupling from light source with broadband spectrum and wide-angle incidence [89,98] (Figure 5b).Strong light localisation can also be realised through engineering the confinement and the mutual interaction of disordered modes [99].Petoukhoff et al. investigated the interaction between disordered metasurface and absorber, where organic semiconductor was introduced on the top of the silver plasmonic metasurface [90], as shown in Figure 5c.Three different modes including localised and propagating surface plasmons and an absorptioninduced scattering mode were realised, demonstrating the ability of light manipulation for optoelectronic applications.
Light capture is the first step for solar energy harvesting.For most practical applications, the energy stored in absorbed photons needs to be transferred to other forms of energy such as electric or chemical ones.Disordered metasurfaces have been proposed to enhance the absorption in photovoltaic platforms and consequently improve the external quantum efficiency (EQE) [100][101][102][103][104]. While theoretical/numerical analysis in optics demonstrated prominent improvement of the performance [102,104], the enhancement from the experimental results was comparatively limited [100,101,103], probably due to the negative impact on the charge carrier transport as the introduction of disordered structures.
Photocatalysis, including water splitting and carbon dioxide reduction, provides an alternative way of energy harvesting other than photovoltaics, directly transforming solar energy to chemical fuels.For this purpose, the disordered metasurfaces play a dual role -an absorber to harvest photons and a catalyst to facilitate desired chemical reactions.Shi et al. utilised a hybrid structure (metasurface/spacer/mirror) for efficient water splitting [91].Tailoring the disordered metasurface composed of Au nanoparticles and the thickness of the space can lead to a strong coupling where photons were efficiently captured (see the black colour of the sample in Figure 5d).Titanium dioxide was chosen as the material for the spacer, together with gold as classical catalyst for water splitting.Combined with a two-electrode system with a potentiostat, hydrogen and oxygen were produced through the water.Recently, a disordered metasurface composed of p-type chromium oxide (CrO x ) for photoelectrochemical hydrogen generation [105].The semiconductor metasurfaces worked as both absorber optically and photocathode chemically.
Time reversal in classical electromagnetism implies a good solar energy harvester may have the ability to manage light emission.Recently, disordered metasurfaces were proposed and realised as a low-cost and efficient light extraction layer for light emitting diodes [92,93].Fusella et al. introduced a disordered metasurface composed of pre-synthesised silver nanocubes on top of OLEDs [92], as shown in Figure 5e.Besides improving the light extraction efficiency, the plasmonic structures were able to improve the stability of the device through enhancing the decay rate.Mao et al. fabricated Ag-based disordered metasurfaces through a single-step process on top of a commercialised GaN LED, improving the EQE from 30% to 50% [93].In addition, a comprehensive numerical investigation was implemented to demonstrate the effect of disorder.The randomness of the metasurface endows the system with the ability to operate for different colours (or broad spectrum).Meanwhile, the top shape of the meta-atom played a significant role in light extraction, as demonstrated in Figure 5f.

Nonlinear optics and disorder induced transition
Nonlinearity is ubiquitously observed in different systems including photonics, and the nonlinear optical response of materials has been stimulating a plenty of promising applications.Besides the ability to control the light in linear regime, metasurfaces have been a unique platform for enhancement of nonlinearities [3], owing to strong light-matter interaction at sub-wavelength scale.Nonlinear metasurfaces have been proposed and realised for a repertoire of applications ranging from nonlinear imaging [106], manipulation of nonlinear spin-orbit interaction [12] to nonlinear holography and image encoding [107].Compared with conventional metasurfaces with well-defined periodicity, disordered metasurfaces provide an alternative platform with unique properties for nonlinear optics.
Plasmonic metasurface is a feasible platform to demonstrate the nonlinear photoluminescence (NPL), whose properties are determined by the localized and delocalized plasmonic modes.Such plasmonic modes can be tuned by the structural disorder in metasurfaces.Roubaud et al. demonstrated the statistical linkage between NPL signal and the morphology of plasmonic metasurfaces [108], as demonstrated in Figure 6a.
The concept of quasicrystals without perfect periodicity inspired new possibilities in nonlinear optics.In such systems, the properties of the generated nonlinear waves are influenced by both the local and global symmetries.Recently, Tang et al. investigated the effect of second harmonic generation by using nonlinear quasicrystal metasurfaces [109], which consist of geometric-phase-controlled plasmonic meta-atoms.The meta-atoms were arranged according to the Penrose tiling and the hexagonal quasicrystalline tiling, while the local rotational symmetry is manipulated by endowing the meta-atoms in the unit-cell with different orientation angles.Consequently, the radiation properties of the second-harmonic waves were found to be determined by global tilting schemes of metasurfaces as well as the local symmetry of the meta-atoms.The arrangement methods and corresponding experimental results are shown in Figure 6b.
Disorder-induced state/phase transition is a ubiquitous phenomenon in many different systems including photonics [25,111,112], with pivotal importance in fundamental physics.In photonics, disorder-induced transition has been reported both numerically and experimentally in different platforms including topological photonics and optical skyrmions [113][114][115][116].The unique features of metasurfaces offer a novel platform to investigate the disorder-induced transition in photonics [110,117,118].
Maguid et al. demonstrated a transition from spin Hall to random Rashba effect [110].The disorder was introduced into the Pancharatnam -Berry phase (geometric phase) of the metasurface, through the orientation angle of each nanoantenna.As the disorder increased from weak to strong one, the systems experienced a transformation from a photonic spin Hall effect to spin-split modes in momentum space, a random optical Rashba effect, as shown in Figure 6c.The critical point was confirmed where structure's anisotropy axis vanishes through the study of the momentum space entropy.Rahimzadegan et al. reported a phase transition in the transmission of dielectric Huygens' metasurfaces made from silicon nanocylinders [117].The increment of the disorder in the particle positions induced the phase angle of the transmitted light to switch from normal dispersion to anomalous dispersion.Zhang et al. demonstrated order-to-disorder transitions on optical responses of aluminium plasmonic metasurfaces [118].The disorder in the location or size of each Al nanodisk induced the broadening and reduction of their plasmonic resonances, while random rotation led to the decreased polarisation dependence.

Outlook and perspective
Although the interaction between metasurfaces and disorder is still in its infancy, it offers a novel platform for both fundamental science and practical applications.Recent works in this area have highlighted the opportunities for working as an alternative or even an improvement of ordered metasurfaces in some applications.
The disordered metasurfaces can be classified into two groups: the metasurface with deliberately introduced (engineered) disorder by top-down patterning techniques and the metasurfaces with disorder intrinsically induced from bottom-up technologies with inevitable larger fabrication error.The metasurfaces with engineered disorder can be treated as a generalised one, releasing all constraints from ordered metasurfaces (such as perfect periodicity or fixed shape of meta-atoms).Consequently, the parameter space is gigantically enlarged for the metasurface design, providing an unlimited possibility and additional difficulties simultaneously.Instead of a physical intuitive-based design, the numerical optimisation can be used for exploring the global minimum in the enlarged space.Optimisation algorithms such as the adjoint method [119] that do not require additional computational resources as the increment of the number of parameters will be ideal for the design of engineered disordered metasurfaces.More importantly, utilising the advancement in deep learning of the photonic design may further improve the optimisation efficiency and the performance of the metasurfaces [120,121].From the view of fabrication, the disordered metasurfaces with complex patterns may not largely increase the cost of lithographybased technology.A metasurface with engineered (optimised) disorder may offer better figure of merit, while shares similar cost as the ordered one (after expensive training, the computational cost is limited).We may foresee that more disordered structures will be utilised in metalens and flat optics, taking full advantage of the dramatically enlarged degrees of freedom.On the other hand, the metasurface with intrinsic disorder may become a competitive candidate for applications requiring large-scale production.Sacrificing the controllability of single meta-atoms, the lithography-free techniques endow intrinsic-disordered metasurfaces with unparalleled ease of fabrication.Device size beyond centimetre can be feasibly achieved, meeting some industrial-level applications.For example, a light extraction metasurface has already been applied to commercialised LEDs for better energy efficiency [93].
For metasurfaces with either engineered or intrinsic disorder, the simulations become much more sophisticated and time-consuming compared to the ordered counterpart (only modelling a single-unit cell).In the future, the conventional Maxwell's equations solvers based on FDTD or FEM method may be replaced by more efficient solvers [122] or even deep learning aided predictors [123,124].The progress in computational electromagnetics may refresh the design methodology in disordered metasurfaces.
In the past decade, we have been witnessing rapid and continuous advances in the field of metasurfaces, introducing new concepts and configurations, X, into the platform, where X ranges from reconfigurable/active metasurfaces [125,126], multi-layer/twisted metasurfaces [127,128] to quantum metasurfaces [5,129,130] and nonlocal metasurfaces [131,132].We may soon see the research on disordered X metasurfaces for the synergic effect between the unique physics in disorder and different metasurfaces with various configurations, features and functions.
In a nutshell, we hope this review may arouse further advancement of the interweaving between the concept of disorder and metasurface-based optical devices, expanding the metasurface design toolbox with new approaches and possibilities.Specifically, the investigation of harnessing the intrinsic disorder may transform the fabrication imperfect into a positive factor and facilitate the large-scale production of metasurfaces towards commercial success in some applications such as colouration and broadband energy harvesting.Meanwhile, the evolution from ordered to engineered disordered metasurface with exponentially enlarged parameter space may lead to devices with unprecedented performance and/or functionalities.
In this review, our focus is on harnessing the disorder in metasurfaces as a beneficial factor to enhance device performance.However, fabrication imperfections are typically detrimental to most optical devices.In addition to improve the fabrication resolution to eliminate (or at least minimise) the disorder, considerable efforts have been dedicated to discovering or designing robust systems immune to moderate geometric fluctuations, with topologyinduced protection being the key to success.The initial and subsequent realizations of topological insulators have demonstrated back-scattering-free edge states robustness to local disorder [133][134][135].Furthermore, investigations into global disorder in topological metamaterials have demonstrated strong robustness against refractive index fluctuations numerically [113].Till now, a repertoire of configurations with non-trivial topology in bandstructures have been investigated and verified, exhibiting strong robustness against disorder [115,[136][137][138][139][140][141], thereby offering a promising future for practical devices.Meanwhile, the concept of bound states in the continuum (BIC) [142,143] has provided an alternative possibility for improving disorder immunity, by emerging topological charge in momentum space into high-order [144] or Brillouin zone folding [145].Apart from the topology in momentum space, non-trivial topology in optical skyrmions can protect their feature against strong fluctuations in both amplitude and phase [116].Recently, in a metasurface-lens system, a topologically protected polarisation singularity is demonstrated by forming a four-dimensional synthetic space [146].In addition, an increase in the refractive index of periodic dielectric nanostructures can enhance the robustness, due to the transition from photonic crystal to Mie resonance [147,148].
Last but not least, reviewing with a broader vision, the progresses of research on disordered structures and systems have great impact on not only optical metasurfaces but also other dimensions of photonics.The investigations on light scattering properties of disordered structures have grown rapidly over the past few decades, which have been developing new applications in numerous areas, such as biological imaging, lasing and solar cell [17].The transverse localization of light in photonic systems incorporating disorder is also an attractive topic, explorations on which have enhanced our understanding of fundamental optical phenomena, such as hyper-transport and localization with quantum-correlated photons [149].The more comprehensive investigation on disorder may also be beneficial to the design of disorder-immune systems, which are important in emerging regions of photonics, such as photonic quantum computing, topological/non-Hermitian photonics and photonic machine-learning [150].

Figure 1 .
Figure 1.Modelling of disorder in metasurfaces based on (a) geometric parameters and (b) correlation.(a) Illustration of ordered metasurface and metasurfaces with disorder in the size, position and both, respectively.(b) Illustration of statistically modelled disorder with different correlation [18].

Figure 2 .
Figure 2. Disordered metasurfaces for resonance manipulation applications.(a) Resonant meta-surfaces for the study of the interplay of order and disorder at oblique incidence.As shown in panel i, the photonic metasurface is composed of sandwich-type nanoparticles, with different arrangements, periodic, disordered and amorphous metasurfaces are prepared, which exhibit different optical response, for example, different transmission spectra in panel ii[30].(b) Metal split metasurfaces with quasicrystal, periodic and random arrangements for the study of near-field surface plasmons.The schematic diagrams of the three types of metasurfaces and corresponding simulated surface plasmon distributions at different frequency are presented in panel i, ii and iii, respectively[31], where the propagation modes of SPs for quasicrystal and periodic metasurfaces are denoted by the arrows and there is no propagation of regular SPs for random metasurface.(c) SiGe random metasurfaces based on Mie resonances for antireflection.By modifying the fabrication process, three samples with different statistical distribution of particle diameter are prepared (panel i), resulting in different reflectance and transmission spectra, as shown in panel ii and iii.Panel (c) is adapted with permission from REF [32].Copyright 2018, American Physical Society.(d) Collective lattice resonances in metasurfaces with different types of disorder.Four types of disorder are studied (panel i).As the extinction spectra shown in panel ii, different types of disorder lead to diverse resonances.Panel (d) is adapted with permission from REF [33].Copyright 2019, the Optical Society.
Figure 2. Disordered metasurfaces for resonance manipulation applications.(a) Resonant meta-surfaces for the study of the interplay of order and disorder at oblique incidence.As shown in panel i, the photonic metasurface is composed of sandwich-type nanoparticles, with different arrangements, periodic, disordered and amorphous metasurfaces are prepared, which exhibit different optical response, for example, different transmission spectra in panel ii[30].(b) Metal split metasurfaces with quasicrystal, periodic and random arrangements for the study of near-field surface plasmons.The schematic diagrams of the three types of metasurfaces and corresponding simulated surface plasmon distributions at different frequency are presented in panel i, ii and iii, respectively[31], where the propagation modes of SPs for quasicrystal and periodic metasurfaces are denoted by the arrows and there is no propagation of regular SPs for random metasurface.(c) SiGe random metasurfaces based on Mie resonances for antireflection.By modifying the fabrication process, three samples with different statistical distribution of particle diameter are prepared (panel i), resulting in different reflectance and transmission spectra, as shown in panel ii and iii.Panel (c) is adapted with permission from REF [32].Copyright 2018, American Physical Society.(d) Collective lattice resonances in metasurfaces with different types of disorder.Four types of disorder are studied (panel i).As the extinction spectra shown in panel ii, different types of disorder lead to diverse resonances.Panel (d) is adapted with permission from REF [33].Copyright 2019, the Optical Society.

Figure 3 .
Figure 3. Disordered metasurfaces for wavefront manipulation applications.(a) Ferromagneticdisordered metasurfaces for investigating the stochastic photonic spin Hall effect (PSHE).Panel i illustrates the PSHE from a spatially fluctuated metasurface under an external magnetic field, which has a typical spin shift.The calculated probability distributions of PSHEs along with spin shift and the measured standard deviation of PSHEs as a function of the spatial fluctuation are shown in panel ii and iii, respectively.Panel (a) isadapted with permission from REF [41].Copyright 2020, Wang B. et al, under exclusive licence to Springer Nature Limited.(b) Bilayer plasmonic metasurfaces with optical chirality induced by rotational disorder.The schematic diagrams and scanning electron microscopy (SEM) images of the bilayer metasurfaces are shown in panel i, whose chirality is proved by the measured transmittance and ellipticity spectra.Panel (b) is adapted with permission from REF [42].Copyright 2018, American Chemical Society.(c) Multifunctional silicon metasurface based on order -disorder interleaving approach.In panel i, the schematic diagrams and SEM images of the interleaving metasurface are presented.The results of wavelength-dependent imaging at 600 nm and 820 nm are shown in panel ii.Panel (c) is adapted

Figure 4 .
Figure 4. Structural colour realized by using disordered metasurfaces.(a) High-purity colour display achieved by combining a disordered nanostructure system and an external Fabry-Perot cavity.The schematic of the clusters-on-spacer sample is shown in panel i, which is endowed with a rainbow-like photonic image (panel ii) by varying the thickness of the spacer.The generated colour in CIE 1931 ×y chromaticity diagram is shown in panel iii [61].(b) Disordered metasurfaces with FP-cavity configuration for generating humidity-responsive structural colour.Panel i demonstrates the schematics of the chitosan hydrogel FP-cavity in humid and dry states, whose reflectance spectra vary with the humidity in the environment (panel ii) [62].(c) the quasiindependent control of the colours of the specular and diffuse components by using disordered metasurfaces.The schematics of the disordered metasurface for visual appearance is shown in panel i, whose SEM images and photographs under different incident and detection angles are presented in panel ii.Panel (c) is adapted with permission from REF [63].Copyright 2022, Vynck K. et al, under exclusive licence to Springer Nature Limited.(d) Material-insensitive structural colours achieved by using plasmonic disordered metasurfaces.The SEM image of the cross-section of the disordered metasurface is shown in panel i, and panel ii demonstrates the colour-map of the corresponding metasurfaces with different materials and thicknesses of the spacer.Panel (d) is adapted with permission from REF [64].Copyright 2021, Wiley-VCH GmbH.

Figure 5 .
Figure 5. Energy related applications of disordered metasurfaces.(a) Broadband perfect absorbers based on disordered titanium nitride metasurface.Panel i demonstrates the schematic and the top-view SEM image of the perfect meta-absorber.The absorption spectra of the disordered metasurfaces with and without TiN coating are shown by cyan and black curves, respectively, in panel ii [84].(b) Enhanced broadband photon confinement in two-dimensional disordered media.The schematic of the random film is shown in panel i.The absorption spectra of the bare and random films are shown by black and blue curves, respectively, in panel ii.Panel iii presents the distributions of the electromagnetic energy density in the sample at different frequencies.Panel (b) is adapted with permission from REF [89].Copyright 2012, Springer Nature Limited.(c) Absorber-coated plasmonic metasurfaces with abundant optical modes.The planar Ag with rough organic coating in panel i supports Mie-absorption-induced scattering.The scattering modes supported by AgNPA/Ag metasurface with semi-crystalline and amorphous organic coatings are shown in panel ii and iii, respectively [90].(d) a hybrid disordered metasurface for tailoring optical modes to facilitate water splitting reactions.The schematic and photographs of hybrid disordered metasurfaces are shown in panel i.Panel ii illustrates the photoelectrochemical measurement system for water splitting assisted by the disordered metasurface.The corresponding evolution of H 2 and O 2 as well as the H 2 -evolution action spectrum are presented in Panel iii.Panel (d) is adapted with permission from REF [91].Copyright 2018, Shi X. et al.(e) an OLED with improved device stability obtained by enhancing the decay rate using a disordered metasurface.The schematic of the disordered metasurface and the atomic-force micrograph of the Ag nanocubes on the top are shown in panel i and ii, respectively.The measured results of the lifetime and efficiency of the three samples are presented in panel iii.Panel (e) is adapted with permission from REF [92].Copyright 2020, Fusella M. A. et al, under exclusive licence to Springer Nature Limited.(f) a EQE-improved GaN LED assisted by the Ag disordered metasurface.The field distributions of the bare substrate and the three different metasurfaces are shown in panel i.The far-field intensity and the transition spectra at different incident angles for the three different metasurfaces are shown in panel ii and iii, respectively [93].

Figure 6 .
Figure 6.Applications of disordered metasurfaces in nonlinear optics and state/phase transition.(a) Tunable nonlinear photoluminescence generation via disordered metasurfaces.The distributions of normalized standard deviations for the three samples are shown in panel i.The average NPL intensity distributions for metasurfaces with gold filling fractions of 0.29 and 0.58 associated with local field enhancements (panel ii), interfering delocalized plasmonic modes (panel iii) and combined results (panel iv) are also presented .Panel (a) is adapted with permission from REF [108].Copyright 2021, American Chemical Society.(b) Manipulating the radiation of secondharmonic generation via nonlinear quasicrystal metasurfaces.Panel i shows the SEM images of the three quasicrystal metasurfaces, whose measured linear and nonlinear diffractions are also presented in panel ii and iii, respectively.Panel (b) is adapted with permission from REF [109].Copyright 2019, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.(c) the transition from spin Hall to random Rashba effect realized by increasing the disorder in geometric phase metasurfaces.In panel i, the photonic transition is illustrated by using ordered, weakly disordered and strongly disordered metasurfaces.Panel ii illustrates the obtaining of the twisted metasurface by combining the helical phase and disordered phase.The metasurfaces with distorted helical phase distributions and corresponding measured intensities in momentum space are shown in panel iii.Panel (c) is from REF [110], reprinted with permission from AAAS.