WO2020212925A1

WO2020212925A1 - A method for the design and manufacture of an optical device including an aperiodic matrix of nanostructures for near-field optical modulation and optical devices based on an aperiodic matrix of nanostructures obtainable by means of said method

Info

Publication number: WO2020212925A1
Application number: PCT/IB2020/053653
Authority: WO
Inventors: Roman Krahne; Mario Miscuglio; Davide SPIRITO; Beatriz MARTIN-GARCIA; Remo PROIETTI ZACCARIA; Nicholas Jon BORYS; Alexander Weber-Bargioni; Peter James Schuck
Original assignee: Fondazione Istituto Italiano Di Tecnologia; The Regents Of The University Of California
Priority date: 2019-04-19
Filing date: 2020-04-17
Publication date: 2020-10-22
Also published as: WO2020212925A8

Abstract

A computer-implemented method of designing an optical device that comprises an aperiodic matrix of nanostructures (NR) made of a predetermined material, adapted to modulate a near field incident optical radiation and examples of optical devices obtainable by means of the method are described. A precursor array of nanostructures (NR) that comprises a regular periodic arrangement of replicas of a unit cell is defined (100) in a predetermined two-dimensional reference system associated with a substrate. Starting from the precursor array, a configuration pattern of the aperiodic matrix of nanostructures (NR) is calculated by modifying (200, 300) at least one spatial parameter of the nanostructure unit cell in the regular periodic arrangement of replicas of the unit cell according to a predetermined mathematical model. A method for manufacturing an optical device comprises controlling a predetermined technique of structuring the substrate in accordance with the calculated configuration model.

Description

A method for the design and manufacture of an optical device including an aperiodic matrix of nanostructures for near-field optical modulation and optical devices based on an aperiodic matrix of nanostructures ohtainable by means of said method The present invention generally relates to optical technologies, in particular manufacturing systems for modulating optical radiation, more particularly modulation systems based on matrices of nanostructures, preferably forming plasmonic metasurfaces.

More specifically, the invention relates to a method for designing an optical device that comprises an aperiodic matrix of nanostructures for optical modulation in the near field as per the preamble of claim 1, a method for producing an optical device and optical devices obtainable by means of said method.

It is known that the phase, intensity, directionality, dispersion and polarization of optical radiation can be modulated by diffusive metal nanostructures (antennae) having dimensions smaller than the wavelength of the radiation, and orderly groups (arrays) of these nanostructures form what are known as plasmonic metasurfaces.

The overall effects of a group of nanostructures are that of controlling the wavefront of incident optical radiation and adapting the properties of reflective or refractive optical beams. For example,“V” nanoantennae described in the article by Yu N. et al., entitled "Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction" which appeared in Science 334, 333 (2011) have been designed to focus the light in the far field.

The manufacture of optical systems for modulating incident optical radiation based on plasmonic metasurfaces is an advantageous solution for replacing conventional optical bulk elements such as lenses and polarizers with planar elements in which ultrathin surfaces are covered with periodic or aperiodic arrays of metal or dielectric nanostructures.

The incident optical radiation is modulated in the near field at the interface of the nanostructure surface. Though the manufacture of optical elements adapted to shape an incident optical radiation in the far field is a technique that has been explored, for example for achromatic focusing, the change in polarization, the manufacture of perfect absorbers or reflectors, the optical characterization in the near field of optical elements that are based on the essential properties of the metasurfaces and the ability to control incident optical radiation in the near field by means of these elements is a very promising direction of study.

International patent application WO 2011/050272 A2 entitled“Nanoantenna arrays for nanospectroscopy, methods of use and methods of high-throughput nanofabrication” describes arrays of nanoantennae and methods for the production thereof. These arrays, which can be periodic or aperiodic, comprise cruciform plasmonic nanostructures that are arranged in relief on the surface of a substrate and are adapted to cause the collective excitation of surface plasmons. These arrays of plasmonic nanostructures can be designed in a uniform manner with respect to the surface of the substrate such that the collective resonances modify the quality factor or the growth properties in the near field of the resonance.

Patent application US 2018/0059440 A1 entitled“Systems and methods for active photonic devices using correlated perovskites” describes coverings having a variable emissivity that use a layer of plasmonic metasurfaces comprising arrays of antennae having a cruciform aperture or created binary metal structures using inverse design techniques, such as binary research algorithms or genetic algorithms. Cruciform apertures in particular have been selected by virtue of their suitability for use with light having arbitrary polarization states. Antennae having a cruciform aperture of different dimensions resonate at different wavelengths, in particular larger apertures interact in a resonant manner with light having a longer wavelength, and the wavelength of the transmitted or reflected optical radiation can be controlled by means of the dimensions of the antennae. These aperture antennae can be formed on a silicon wafer by means of electron-beam lithography. The covering having variable emissivity that uses binary metasurfaces shows the ability of the metasurfaces to tune the distributions of the near-field optical radiation to different wavelengths. Nanostructured planar optics therefore make it possible to form completely new optical elements that act in the plane of the metasurface comprising the nanostructures, or near thereto. For example, optical elements of this type can be used as couplers for effectively interfacing optoelectronic elements on a nanometric scale. Size and phase modulation by means of metasurfaces having plasmonic nanoresonators could also be directly used to shape the near field and to therefore design concentrator devices, switch devices or optical trap devices that act in the vicinity of said metasurface.

US patent US 9 373 740 entitled "Wavelength converting structure for near-infrared rays and solar cell comprising the same" describes a structure for converting radiation having near-infrared wavelengths into electrical energy by means of plasmonic phenomena.

US patent application US 2015/288318 entitled "Refractory plasmonic metamaterial absorber and emitter for energy harvesting" describes absorbers and emitters based on metamaterials having structures deriving from the arrangement of plasmonic and dielectric structures.

International patent application WO 2012/103289 A1 entitled "Optical devices with spiral aperiodic structures for circularly symmetric light scattering" describes optical devices based on aperiodic arrays of plasmonic nanoparticles having a spiral geometry.

Disadvantageously, none of the cited documents comprises an efficient method for designing and manufacturing optical elements that modulate incident radiation in the near field and are based on plasmonic metasurfaces having specific nanoresonator spatial distributions.

The present invention has the object of providing optical devices based on nanostructures, which are adapted to modulate the near field of an incident optical radiation.

According to the present invention, this object is achieved by a method for designing an optical device, which comprises an aperiodic matrix of nanostructures having the features specified in claim 1. The invention further relates to a method for manufacturing an optical device having the features specified in claim 13 and an optical device based on an aperiodic matrix of nanostructures having the features specified in claim 14. Particular embodiments form the subject matter of the respective dependent claims, the content of which is intended to be an integral part of the present description.

In summary, the subject of the present invention is a method for designing metasurfaces, preferably and in the following, plasmonic metasurfaces, which are adapted to modulate the near field of an incident optical radiation, in which a specific arrangement of the nanoresonator elements constituting the metasurface is calculated - as a function of the desired modulation of the incident optical radiation - by acting on spatial parameters of said nanoresonator elements, and especially on their position, planar orientation and dimension. The formation of three different aperiodic arrays of nanoresonators is described, which form respective configurations of plasmonic metasurfaces adapted to provide respective modulation functions of a beam of an incident optical radiation, including the concentration of the radiation at the center of the device, the generation of optical vortices and the switching of the intensity distribution of the radiation, which are controllable as a function of the wavelength of said incident optical radiation or the linear polarization angle thereof.

The manufacturing of the different plasmonic metasurface configurations, that is the design of the particular optical modulation function, is obtained by:

defining a precursor array of nanoresonators in a two-dimensional reference system associated with a substrate adapted to receive said nanoresonators, which can be a reference system expressed in polar or Cartesian coordinates, in which the precursor array comprises a regular periodic arrangement of preferably identical nanoresonators, that is a regular periodic arrangement of replicas of a common structure (unit cell) of a nanoresonator; and applying a modulation of the position and planar orientation of the nanoresonators and/or a modulation of the dimensions of the nanoresonators, that is the modification of the position (period of the array) and planar orientation of the nanoresonators and/or of the dimensions of the nanoresonators in the above-mentioned reference system in accordance with a predetermined mathematical model.

The predetermined mathematical model for modulating the precursor array is a function of the desired modulation of the incident optical radiation.

The optical near field resulting from tire modulation of the incident optical radiation by means of a designed plasmonic metasurface configuration is suitably evaluated by means of modeling, for example by means of numerical simulation techniques, such as the finite element method, and the result is compared with a reference optical near field, such as an expected or ideal optical field, with any difference being used as feedback information to optimize the design of the plasmonic metasurface by means of the parameters characterizing the relative function of modulating the position and/or orientation of the nanoresonators and/or the relative function of modulating the dimensions of the nanoresonators, in order to obtain a consequent optical near field corresponding to the expected optical near field within a predefined threshold tolerance.

Preferably, within the application of the present invention, aperiodic arrays of rhomboid nanoresonator elements (unit cells) have been designed from a periodic arrangement of these elements on a substrate, in which the arrangement of the elements is represented by the values that a physical parameter that is representative of the overall metasurface assumes in the region of the device, for example the electric permittivity of the material of the metasurface comprising the substrate and the nanoresonators arranged thereon, in the reference system.

The values that the physical parameter that is representative of the overall metasurface assumes in the region of the device define a configuration function of the metasurface that is indicative of the spatial localization of the nanoresonators in a predetermined reference coordinate system , which is applicable to a process of manufacturing an optical device based on aperiodic matrices of plasmonic nanostructures according to the invention, for example for controlling process parameters. Advantageously, the physical parameter that is representative of the overall metasurface, that is the configuration function of the metasurface, can be expressed in Fourier series, that is as a linear combination of sinusoidal functions forming an orthogonal base, and the relative function of modulating the position and orientation of the nanoresonators is a function that is representative of the trend of the spatial parameters of the repetition period and orientation of the common structure (unit cell) of the nanoresonator in the two- dimensional space of extension of the metasurface.

Advantageously, the above -mentioned modulation function can be represented by modulating the grating vector of the Fourier series.

Advantageously, the configuration function representative of the plasmonic metasurface is preferably discretized by associating one of a pair of predefined values for the physical parameter representative of the metasurface with each coordinate of the configuration of the metasurface according to the resolution of the production process. Each value for the physical parameter of the configuration function is used by a computer-aided design system for controlling a process for structuring a planar substrate, for example for defining a lithographic mask to be used in the production of the metasurface by means of electron-beam lithography, or for controlling the deposition or removal of a metal or dielectric by means of an electron beam directed towards the substrate of the metasurface. The association of one of a pair of predefined values for the physical parameter representative (of the material) of the metasurface with each coordinate of the metasurface configuration is formed by comparing it with a threshold value, and advantageously by comparing it with an adaptive threshold value, in which the threshold is a function of the spatial coordinate, thereby forming a function of modulating the dimensions of the nanoresonators.

In the case in which the physical parameter representative of the metasurface is the electric permittivity of the material of the metasurface, the pair of predefined values comprises the electric permittivity value of the material from which the nanoresonators are made, and the electric permittivity value of the air. In summary, the approach to designing plasmonic metasurfaces based on aperiodic arrays of nanoresonators has made it possible to build, for example, three specific types of optical devices based on corresponding configurations of aperiodic arrays of nanoresonators, whose functions of modulating an incident optical radiation have been demonstrated:

the“twisted bulls-eye’' configuration adapted to concentrate the power of a beam of an incident optical radiation in the center of the device and rotate the linear polarization direction of the incident optical radiation;

the“twisted gaussian” configuration adapted to shape a beam of incident optical radiation according to a distribution of power that is focused at the center or extended in the shape of a circular crown as a function of the linear polarization angle of the incident radiation, also creating optical vorticity in the vicinity of the surface of tire optical device; and

a“tangential” configuration adapted to rotate the distribution of the power of an incident optical radiation in the optical near field between a dipolar distribution and a quadrupolar distribution as a function of the wavelength of the incident optical radiation.

Advantageously, a polygonal, preferably rhomboid, plasmonic nanostructure in a predetermined region, whether it be metal or dielectric, is preferable as a common structure (unit cell) of a nanoresonator and it exhibits separate dipolar resonance in at least two different directions, preferably in orthogonal directions, at different wavelengths. Plasmonic metasurfaces formed by arrays of nanoresonators having a polygonal shape in a predetermined region can be designed using a deterministic mathematical algorithm in order to obtain a desired function of the metasurface. The nanoresonators are made of a material that can diffuse the light, preferably a plasmonic material, such as a noble metal, copper, aluminum, graphene or two-dimensional semiconductors, more preferably a noble metal, and even more preferably in the embodiment described here, gold.

In the currently preferred embodiment, gold rhomboid nanoresonators are used since, in accordance with their dimensions, they can support dipolar plasmonic resonance of predetermined wavelengths that affect a large portion of the near-infrared spectrum, exhibiting fundamental modes at different frequencies in orthogonal directions. Furthermore, the pointed shape at the vertices of the rhomboid structure promote the plasmonic coupling between nearby elements. The essential modes are dipolar; the subsequent harmonics are quadrupoles, and each of them exhibits intense fields at the vertices in the polarization direction of the incident optical radiation.

Additional features and advantages of tire invention will be explained in more detail in the following detailed description of one embodiment thereof, given by way of non-limiting example and with reference to the attached drawings, in which:

Fig. 1 is a schematic view of a system for manufacturing optical devices based on aperiodic matrices of plasmonic nanostructures for near field optical modulation according to the invention;

Fig. 2 is a flow diagram of a method for designing plasmonic metasurfaces including an aperiodic matrix of plasmonic nanostructures (nanoresonators) according to the invention;

Fig. 3 is a schematic view of a preferred plasmonic nanostructure and the arrangement thereof in an array;

Fig.4 shows the wavelength of the dipolar and quadrupolar plasmonic resonance that set up in the nanostructure of Fig. 3 as a function of the geometric dimensions of said nanostructure;

Fig. 5a, 5b and 5c are exemplary views of three plasmonic metasurfaces that are obtainable by means of the method according to the invention;

Fig. 6a and 6b show the functions of modulating the orientation and dimensions of the nanoresonators of tire plasmonic metasurface in Fig. 5a;

Fig. 7a and 7b show the functions of modulating the orientation and dimensions of the nanoresonators of the plasmonic metasurface in Fig. 5b;

Fig. 8a and 8b show' the functions of modulating the orientation and dimensions of the nanoresonators of the plasmonic metasurface in Fig. 5c;

Fig. 9 shows images obtained using dark-field optical microscopy of 2 x 2 arrays of the metasurfaces in Fig. 5a, 5b and 5c; Fig. 10a-lOd and Fig. 11a-11c show experimental measurements and simulations of the modulation of a near field optical radiation, obtained by means of the plasmonic metasurface in Fig. 5a;

Fig. 12a-12d show experimental measurements of the modulation of a near field optical radiation obtained by means of the plasmonic metasurface in Fig. 5b;

Fig. 13a and 13b show experimental measurements of the modulation of a near field optical radiation obtained by means of the plasmonic metasurface in Fig. 5c; and

Fig. 14a, 14b and 14c respectively show a plasmonic metasurface covered with a nanometric layer of photoluminescent nanoparticles, such as fluorescent nanoparticles, the spectral characteristics of a layer of PbS nanocrystals and experimental measurements of the modulation of a near field optical radiation obtained by means of the plasmonic metasurface in Fig. 5a that is covered with a layer of PbS nanocrystals.

Fig. 1 schematically shows a system for manufacturing optical devices based on plasmonic metasurfaces having an aperiodic matrix of nanostructures for use in the modulation of an incident optical radiation. The system comprises a processing station 10 or similar processing means, including distributed means, programmed to carry out a method of manufacturing aperiodic matrices of plasmonic nanostructures according to the invention, which will be described in more detail in the remainder of the present discussion. The processing station 10 is provided for executing a computer program adapted to implement a predefined algorithm for designing a plasmonic metasurface characterized by predetermined design parameters, the result of which is a two-dimensional or three-dimensional configuration model of a matrix of plasmonic nanostructure. The design parameters include the shape and dimensions of a predetermined unitary plasmonic nanostructure and the spatial configuration of a precursor array or matrix of plasmonic nanostructures, which array or matrix is made up of an orderly plurality of replicas, preferably identical replicas, of said predetermined nanostructure (unit cell) arranged in a predetermined region corresponding to the surface of a substrate on which the optical device is to be formed. The plasmonic nanostructures of the precursor array are arranged according to a regular two-dimensional period, which is repeated a finite number of times, according to a Cartesian (x, y) or polar (r,q) coordinate system. The design parameters also include coefficients of modulating the position, planar orientation and dimensions of the nanostructures on the extension of the precursor array.

Reference numeral 12 indicates processing means provided for executing a program for simulating the optical near field emerging from an optical device based on a designed plasmonic metasurface configuration. Although indicated separately, the processing means 12 can suitably be integrated in the processing station 10. The processing means 12 receive the two-dimensional or three-dimensional configuration model of a designed matrix of plasmonic nanostructures from the processing station 10 and carry out the simulation of the optical near field resulting from the modulation of an incident optical radiation by means of said matrix of plasmonic nanostructures, for example by means of numerical techniques, such as the finite element method.

The result is compared with the characteristics of the expected or ideal optical field at the processing means 12 or at the processing station 10, and differences that are too high in comparison with a predetermined increase threshold value for the induced electrical field are reported to the processing station 10 in order to redesign the configuration of the matrix of plasmonic nanostructures, in particular for adjusting one or more design parameters. The processing station 10 is also connected to a plant 14 for manufacturing nanostructures, for example for the selective deposition or removal of material from a substrate, and more specifically said plant is provided for controlling the emission of an electron beam B when scanning the surface of a substrate SUB covered with a resist film R according to electron- beam lithography, in orderto write a two-dimensional patter on the resist film in accordance with a configuration mathematical model of a previously designed matrix of plasmonic nanostructures. Alteratively, the plant 14 for producing nanostructures 12 can be a plant for emitting a focused ion beam, which plant is provided for controlling the emission of an ion beam when scanning the surface of the substrate SUB, in order to write a three-dimensional patter on the substrate in accordance with a configuration mathematic model of a previously designed matrix of plasmonic nanostructures. According to further alteratives, the plant 14 for manufacturing nanostructures can be designed to apply a nanoimprint lithography or ion etching technique without thereby departing from the principle of the invention. Fig. 2 is a flow diagram of the steps carried out by the system in Fig. 1 for manufacturing a planar optical device including a plasmonic metasurface. In order to create a desired preset modulation function for an incident optical radiation, a corresponding configuration of a plasmonic metasurface is mathematically designed from a precursor array of nanoresonators defined at the step 100, tuning the position, orientation and dimensions of the nanoresonators of the precursor array at the steps 200 and 300, respectively. The modulation of the phase and dimensions of the nanostructures can be derived from a desired modulation function of the incident optical radiation by means of complex qualitative factors or analytical mathematical relations derivable from an inverse scattering problem having specific boundary conditions. The modulation of the incident optical radiation by the mathematically designed plasmonic metasurface is simulated at the step 400 and the result is compared with the expected modulation at the step 500. If the difference is greater than a predefined threshold tolerance, the tuning of the position, orientation and dimensions of the nanoresonators of the precursor array is repeated at the steps 200 and 300. If the difference fells within said threshold tolerance, the plasmonic metasurfece is manufactured by controlling the selective structuring of the nanoresonators on a substrate at step 600 by means of the mathematical model representative of the tuned array of nanoresonators .

In a preferred embodiment, each nanoresonator comprises a rhomboid metal structure having an aspect ratio of between 1:1.5 and 1:4, preferably 1:2, a preferably constant thickness of between 20 nm and 200 nm, preferably 30 nm. Fig. 3 shows - from left to right - a nanoresonator NR (characteristic of a unit cell of the array) having characteristic dimensions indicated by D (major axis) and d (minor axis), a periodic array of nanoresonators NR arranged to form a rectangular matrix, the period of which is a function of the spatial

coordinates (similarly, an arrangement forming a radial matrix having the period

would be possible), and shows modulation of the orientation of the nanoresonators NR as a function of the spatial coordinates (similarly, a modulation as a function of the polar

coordinates is possible), and shows modulation of the dimensions of the

nanoresonators (that influences the load factor of the metasurface) as a function of the spatial coordinates FF(x,y) (similarly, a modulation as a function of the polar coordinates FF(p,q) would be possible). The lateral dimensions D, d of each nanoresonator are selected as a function of the wavelength at which the plasmonic metasurface is to operate. Fig. 4 shows the dependency between a dimension characteristic of the rhomboid nanoresonators (major axis D) on the X-axis and the optical wavelength in which plasmonic resonance are established on the Y-axis, for an operation in the visible and near-infrared spectral range. To be specific, A indicates the evolution curve of the first resonance (dipolar) that is parallel to the major axis of the nanoresonator, B indicates the evolution curve of the second resonance (quadrupolar) that is parallel to the major axis of the nanoresonator, C indicates the evolution curve of the first resonance (dipolar) that is perpendicular to the major axis of the nanoresonator, and D indicates the evolution curve of the second resonance (quadrupolar) that is perpendicular to the major axis of the nanoresonator. More specifically, the mathematical approach of the invention for designing the plasmonic metasurface configurations is as follows.

At the step 100, a periodic grid of nanoresonators, that is a periodic array of identical replicas of a specific unit cell, is defined in radial coordinates (r,q) or Cartesian coordinates (x,y). Advantageously, the unit cell is defined mathematically in the three-dimensional space by defining the vertices of the polygonal structure of the nanoresonator in a predefined portion of space— four in the case of the currentiy preferred rhomboid structure.

The periodic array of nanoresonators can be defined mathematically by means of an expansion in Fourier series according to an expression that is a weighted sum of the base function eⁱ _kn ^r where k_n is the grating vector, in which

, in which L is the period of the array, and r is the position vector, which can be expressed as a combination of the Cartesian coordinates x, y or as a combination of the polar coordinates p, q.

By associating the individual terms of the Fourier series with values for the electric permeability' e of the plasmonic metasurface to be built, a function έ(r) is obtained in real space and is representable as:

where the weight of the coefficients a„ can be numerically calculated by means of a Former transform (FFT)

Phase modulation (that is the position, or period, of the array, and the orientation) of the array is obtained at the step 200 by introducing the function of the gradient of the phase F(r), that is

Therefore, the expression of the electric permeability e of the aperiodic array of nanoresonators can be expressed in Fourier series as:

In order to determine the configuration of the metasurface, only the real part of e(r) is considered

where is a continuous function.

The function is preferably discretized in view of the physical structuring of the

substrate at the step 300, for example by introducing at least one threshold G

(r) that, in the case in which the physical parameter is the electric permeability, can be expressed as:

whereby the function associates with each coordinate either the value of the electric

permeability of the material used, which is gold in the example,

at the wavelength of the incident radiation or the value of the electric permeability of air, corresponding to the nanostructures not being deposited.

The significance of the threshold function F(r) that is dependent on the spatial coordinate of the metasurface is that of modulating the dimension of the unit cell, that is of the base element of the metasurface .

The modulation functions and G(r) are design parameters of the metasurface.

The efficacy of the modulation of the incident radiation is evaluated by means of a simulation of the optical near field emerging from the metasurface, for example by means of a finite element method, and the design parameters are tuned on the basis of the results of the simulation.

The values assumed by the function are used in a computer-aided design process,

for example for defining lithographic masks to be used in the manufacturing of the array of nanoresonators by means of techniques for structuring a planar surface, for example by means of a selective electron beam deposition or removal technique on a substrate, or the like. The resolution of the configuration is defined by the number of terms of the Fourier series and a sufficiently high number of these are selected (preferably only the first 10 terms for a balance between estimating the value for the physical parameter and computing resources) so that is as close to as possible.

In accordance with the present invention, three different plasmonic metasurface configurations have been designed, that is configurations of aperiodic arrays of nanoresonators, respectively shown in Fig. 5a, 5b and 5c, which show the versatility of the method of the invention for manufacturing optical devices.

Fig. 5a shows a“Twisted Bulls-Eye” configuration based on a radial coordinate reference system, which is designed to rotate the polarization of a linearly polarized incident optical radiation and to centrally concentrate the light intensity of a collimated light beam in a spot having a smaller diameter than the diffraction limit.

Fig. 5b shows a“Twisted Gaussian” configuration based on an aperiodic array in Cartesian coordinates adapted to generate optical vorticity, whereby an incident optical radiation is focused for a preset direction of the linear polarization (linear polarization parallel to the major axis of the rhomboids), while, for a perpendicular polarization direction (linear polarization perpendicular to the major axis of the rhomboids), the near field is concentrated at the edges of the metasurface, forming a donut distribution. Switching between a focused distribution and a donut distribution is obtained by modifying the polarization angle of the incident radiation in the plane of the metasurface.

Fig. 5c shows a‘tangential” configuration based on a tangential mathematical function that implements a rotation of the intensity distribution of an incident optical radiation in the near field depending on the wavelength; in particular, with reference to a resonance wavelength of the quadrupolar distribution, for smaller wavelengths the intensity distribution of the transmitted radiation is oriented parallel to the polarization of the incident radiation, whereas, for longer wavelengths the intensity distribution of the transmitted radiation is oriented perpendicularly to the polarization of the incident radiation. In particular, for short wavelengths, for example red wavelengths of approximately 700 nm, the optical power distribution in the near field is dipolar, oriented parallel to the linear polarization of the incident optical radiation, while, as the wavelength increases, the optical power distribution in the near field becomes quadrupolar, with intensity confined to the comers of the metasurface configuration, until a newly dipolar optical power distribution is obtained, that is however oriented perpendicularly to the linear polarization of the incident optical radiation for large wavelengths, for example approximately 1000 nm. With reference to Fig. 5a, the precursor array of the“Twisted Bulls-Eye” configuration is a regular radial arrangement comprising concentric rings of replicas of rhomboid nanoresonators, in which the rings have a regular circumferential arrangement of an equal number of nanoresonators, and the“Twisted Bulls-Eye” configuration can be described as a succession of concentric rings of said rhomboid nanoresonators (only some of which are indicated by reference NR so as not to compromise the comprehensibility of the drawing) having characteristic dimensions that increase towards the outside, the dimension of the nanoresonators being modulated according to a Gaussian function of the plane, the position of the nanoresonators being modulated so that each nanoresonator ring is rotated anti- clockwise with respect to the adjacent inner ring by 30° with respect to the radial arrangement of the precursor array (in which the nanoresonators are aligned in respective radial directions).

To be specific, the modulation equation for the orientation of the nanoresonators - expressed in polar coordinates - is:

where x, y are the Cartesian coordinates in the plane of the two-dimensional reference system and z is the Cartesian coordinate that is perpendicular to the plane of the reference system, and

and is shown graphically in Fig. 6a, while the modulation equation for the dimensions or load factor - expressed in Cartesian coordinates for simplicity reasons - is

where

25

is the position of the expected value or peak of the Gaussian function in x,y

and

is a diagonal matrix representative of the standard deviation of the Gaussian

function that defines how sharply the load factor is modulated,

and is shown graphically in Fig. 6b. With reference to Fig. 5b, the precursor array of the“Twisted Gaussian” configuration is a regular Cartesian grid arrangement of replicas of rhomboid nanoresonators, and the“Twisted Gaussian” configuration can be described as a succession of wavy rows of said rhomboid nanoresonators (only some of which are indicated by the reference NR so as not to compromise the comprehensibility of the drawing), wherein the nanoresonators comprise characteristic dimensions that increase towards the radial outside of the arrangement. The orientation and the dimensions of the nanoresistors are both modulated according to a Gaussian function of the plane.

To be specific, the modulation equation for the orientation of the nanoresonators - expressed in Cartesian coordinates - is

where

is the position of the maximum modulation in x,y and

is a diagonal matrix representative of the standard deviation of the Gaussian

function that defines the modulation steepness of the orientation,

and is shown graphically in Fig. 7a.

The modulation equation for the dimensions or load factor - expressed in Cartesian coordinates - is

25

where

is the position of the expected value or peak of the Gaussian function in x,y

and is a diagonal matrix representative of the standard deviation of the Gaussian

function that defines how sharply the load factor is modulated,

and is shown graphically in Fig. 7b. With reference to Fig. 5c, the precursor array of the‘Tangential” configuration is a regular

Cartesian grid arrangement of replicas of rhomboid nanoresonators, and the“Tangential” configuration can be described as a succession of nested rows of said rhomboid nanoresonators (only a few of which are indicated by the reference NR so as not to compromise the comprehensibility of the drawing) oriented along hyperbolic directrix curves, having double axial symmetry or being centrosymmetrical, in which the nanoresonators have characteristic dimensions that increase towards the outside of the arrangement. The orientation of the nanoresonators is modulated according to a tangential three-dimensional function of the plane, while the dimensions of the nanoresonators are modulated according to a Gaussian function of the plane.

To be specific, the modulation equation for the orientation of the nanoresonators - expressed in Cartesian coordinates - is:

and is shown graphically in Fig. 8a.

The modulation equation for tire dimensions or load factor - expressed in Cartesian coordinates - is:

where

25

is the position of the expected value or peak of the Gaussian function in x,y, and

is a diagonal matrix representative of the standard deviation of the Gaussian

function that defines how sharply the load factor is modulated,

and is shown graphically in Fig. 8b. The above-mentioned nanoresonator arrays are made by means of techniques for structuring a planar surface, and, in a preferred embodiment, the substrates used are made of CaF2 (operating in transmission) or of Si/SiOz (operating in reflection). In general, any substrate can be used that can support the formation of nanostructures on its surface for deposition or erosion, including glass substrates, substrates made of plastic material and flexible substrates. If used for building an optical device operating by transmission of the incident optical radiation, the substrate of course has to be transparent in the spectral bandwith of the processed optical radiation, and in this case its thickness can be between several hundred microns to a few millimeters, for example. If used for building an optical device operating by reflection of the incident optical radiation, there are no limits to the thickness of the substrate.

Different aperiodic arrays of nanoresonators have been constructed to obtain different preset functions for modulating an incident optical radiation, which include the concentration of the radiation at the center of the distribution, the generation of optical vortices and the switching of the intensity distribution of the optical radiation in the near field by varying the wavelength or linear polarization angle of the incident radiation.

Purely by way of example, configurations of nanoresonators defined according to the method of the invention have been made by evaporating a thin layer that is 5 nm thick of titanium and subsequently a layer that is 25 nm thick of gold on silicon substrates having an SiOz covering that is 285 nm thick, for example thermally grown by electron beam evaporation at the rate of 0.2 A/s and at an operating pressure of approximately 2x10^-6 mbar, which substrates have previously been washed in acetone and rinsed in isopropanol for 2 minutes and plasma cleaned for 5 minutes at 100 W in a 100 % O2 atmosphere in order to remove any organic residue, therefore baked at 120 °C for 2 minutes to remove residual solvents. The configurations of nanoresonators calculated as described above have been built by means of electron-beam lithography techniques, used for a selective exposure of a 495-K poly(methyl methacrylate) resist that is 160 nm thick and is deposited on the substrate by means of spin-coating and baked for 7 minutes on a hot plate at 180 °C, with a beam accelerating voltage of 20 kV and with an exposure current of 35 pA; the resist is therefore developed using a suitable solution for times and temperatures that are dependent on the solution itself, for example for a time of 30 s in a cold mixture (8 °C) of MIBK:IPA 1:3.

The optical properties of the produced nanoresonator arrays, that is of the plasmonic metasurfaces of the present invention, that is to say the modulation effects of the incident optical radiation in the near field, were characterized by the following laboratory tests.

The modulation of the incident optical radiation in the vicinity of the surface of the optical device has been investigated using dark-field microscopy.

Fig. 9 shows focused and defocused images obtained by means of dark-field optical microscopy of 2 x 2 arrays of the designed and produced devices. In particular, the upper row shows the focused images, the lower row the defocused images, the left-hand column the“Twisted Bulls-Eye” (TBE) configuration, tire central column the“Twisted Gaussian” (TG) configuration and the right-hand column the“Tangential” (T) configuration.

In the focusing condition, it is possible to discern that the configuration of the metasurface and the intensity of the light reflects the degree of diffusion from the individual nanoresonators. The out-of-focus images provide information about the spatial distribution of the intensity of the emergent optical radiation near to the metasurfaces and exhibit completely different behavior. For the TBE configuration (left-hand image), the maximum intensity is distributed in the shape of a circular crown and a vortex distribution that spreads out towards the outside can be identified. The vorticity is also clearly present in the out-of- focus image of the TG configuration (central image) and the intensity distribution has a pinwheel shape having four petals. Lastly, the out-of-focus image of the T configuration (right-hand image) shows that the incident optical radiation diverges towards the four comers of the configuration, making the hyperbolic intensity lines clearly identifiable.

Two-photon photoluminescence microscopy (TPPL) experiments were carried out to take images and to directly measure the spatial distribution of the optical modulation in the near field generated by the designed metasurfaces (samples). Advantageously, the signal of the TPPL microscopic analysis is proportional to the square of the intensity of the incident optical radiation and therefore this is extremely sensitive to local increases in the near field.

The TPPL microscopy experiments, the illumination of the samples and the collection of the TPPL signals were carried out using a 100x objective lens having a numerical aperture of 0.90. A pulsed laser device based on titanium-doped sapphire (140 fs, and pulse emission frequency of 80 MHz) emitting at wavelengths of between 680 and 1080 nm has been used as the excitation source. In order to obtain an almost homogenous illumination of the metasurfaces, the laser beam was expanded in a spot dimension having a diameter of 10 microns by focusing the laser on the rear focal plane of the objective. The light collected by each sample was transmitted via two low-pass filters having cut-off wavelengths of 633 and at 650 nm to remove the excitation light from the TPPL signal, thereby ensuring that only the fight emission produced by the plasmonic metasurfaces in the range of between 450 and 600 nm is picked up.

Fig. 10a shows light intensity maps recorded when taking images using a widc-ficld TPPL technique for a TBE configuration having a diameter of 1.25 micron for different excitation wavelengths (700 nm, 750 nm and 800 nm, respectively) of the incident optical radiation that is linearly polarized in a horizontal direction.

For an excitation having a wavelength of 700 nm, two lobes that are clearly defined on the horizontal axis (parallel to the polarization of the light) can be observed. At an excitation wavelength of 750 nm, the intensity of these peaks decreases and a third peak emerges at the center of tire configuration. At the excitation wavelength of 800 nm, the central intensity peak dominates and the lateral lobes have substantially disappeared. While the configurations caused by excitations at wavelengths of 700 nm and 750 nm substantially have specular symmetry with respect to the horizontal and vertical axes, the configuration caused by the excitation at the wavelength of 800 nm shows rotational symmetry. Fig. 10b shows a simulation of the intensity distribution of the electrical near field calculated by means of the finite element method, and this confirms that, for incident radiation of 700 nm, the distribution in the near field assumes a dipolar nature, for incident radiation of 750 nm, the distribution in the near field gathers intensity at the center of the configuration, and for incident radiation of 800 nm, the dipolar nature is significantly reduced and the concentration of the intensity is dominant at the center. Fig. 10c and lOd show the trend of the extinction cross section calculated by the finite elements simulation model and of the TPPL signal obtained by simulation and experimentally as a function of the wavelength of the incident excitation radiation. The similar qualitative aspect thereof confirms that the TPPL intensity can be used as a measure of the optical near field.

At the excitation wavelength of 700 nm, the dipolar resonances along the major axis of the rhomboid nanoresonators resonate with the incident optical radiation and therefore the regions in which the nanoresonators are located, and in which the major axes of the nanoresonators are co-aligned with the polarization of the incident optical radiation manifest an increase in the near field, which brings about the formation of the two lobes shown in Fig.

10a. The simulation in Fig. 10b, in which the polarization of the incident radiation is directed along the horizontal axis, shows that the electric field is largely confined to the empty regions of the metasurface between the vertices of the major axes of the rhomboid nanoresonators, the major axes of which are aligned in parallel with the horizontal direction. This confinement of the field between the vertices resembles the behavior of the plasmonic bow- tie antennae. The rhomboid nanoresonators that have the major axis in the vertical direction (i.e. perpendicular to the polarization of the incident optical radiation) show maximum degrees of intensity of the electric field in all four vertices, which can be considered representative of a quadrupolar resonance. As the wavelength of the incident excitation radiation increases, the coupling to the nanoresonators having smaller dimensions previously not in resonance increases, which corresponds to focusing the electric field in self-similar antennae towards the smaller elements. This coupling caused by the self-similarity characteristics in the TBE configurations focuses the energy inwards towards a common point at the center, which, for excitation having a wavelength of 800 run, causes a largely localized peak in the near field with a full width at half maximum (FWHM) of 300 nm. The simulation in Fig. 10b confirms the channeling of the intensity towards the center by means of the coupling of the self-similar elements. Fig. 1 la-11c show - by means ofTPPL images recorded at different excitation wavelengths and for devices having different nanoresonator dimensions - that the effect of modulating the incident optical radiation by means of the TBE configuration is dependent on the dimensions of the nanoresonators and on the wavelength of the incident optical radiation.

Specifically, Fig. 1 la shows a succession ofTPPL images for different wavelengths of the incident optical radiation (x-axis) and different diameters of the TBE configuration (y-axis); Fig. lib shows a succession of TPPL images for different linear polarization directions of the incident optical radiation (expressed in angles of between 0° and 180° with respect to a horizontal polarization direction), and Fig. 11c is a diagram showing the integrated light intensity recorded in the TPPL images for the above-mentioned different linear polarization directions.

In general, the overall lateral dimensions of the nanoresonator configurations under experimentation, to which reference is made above, could vary by from 1 micron to 7 microns. More extensive configurations have a similar distribution of the electrical field for longer wavelengths. Devices having larger dimensions therefore resonate at longer wavelengths and therefore the concentration of the radiation emitted in a circular spot at the center of the configuration is achieved at longer wavelengths compared with what happens for smaller devices. From an engineering perspective, the minimum dimensions of a configuration are limited by the resolution of the technique used to produce it, for example by the resolution of the lithographic method when producing the nanoresonators.

Devices having different dimensions were obtained by linearly scaling a reference overall configuration predetermined with the method of the invention.

Fig. 12a-12d show TPPL intensity maps for the“Twisted Gaussian” (TG) configuration.

Fig. 12a shows TPPL intensity maps for the TG configuration at different incident optical radiation wavelengths, which are recorded using incident optical radiation that is linearly polarized according to a horizontal direction in the drawing. For shorter wavelengths (700 nm), the intensity profile shows a highly focused circular crown shape, which extends in a substantially diagonal direction that corresponds to the main direction of orientation of the major axis of the rhomboid nanoresonators for longer wavelengths. The variation in the distribution in the near field as a function of the wavelength is illustrated schematically in Fig. 12b.

The TPPL distribution of the intensity of the optical radiation emerging from the device having a TG configuration of the metasurface is strongly dependent on the angle of the linear polarization of the incident optical radiation, as shown in Fig. 12c. For linear polarization directed vertically in the figure, a circular spot that is highly focused is obtained, while, for linear polarization directed horizontally in the figure, the intensity distribution has a wide circular crown shape with greater intensity at the left-hand and right-hand side. Fig. 12d is an enlarged view of the radiation resulting for these two polarization ends. This optical device configuration can therefore be used to change the intensity distribution of the near field by means of controlling the polarization angle of the incident optical radiation.

Fig. 13a and 13b show TPPL intensity maps for the“Tangential” (T) configuration.

At shorter wavelengths (700 nm) of the incident optical radiation, the TPPL intensity maps for the tangential configuration display a distribution whereby the intensity is focused in triangular zones that are aligned in the vertical direction in the drawing. The increase in the wavelength (750 nm) of the incident optical radiation leads to the intensity being confined at the four comers of the structure, and for even longer wavelengths (800 nm) the intensity is focused in aligned triangular zones in the horizontal direction in the drawing. The variation in the distribution in the near field as a function of the wavelength is shown schematically in Fig. 13b.

Advantageously, the effect of modulating the incident optical radiation can be transferred to the emission of photoluminescent nanoparticles (for example fluorescent nanoparticles) deposited on a plasmonic metasurface.

On an experimental basis, the TBE configuration was covered with a layer of lead sulfide (PbS) nanocrystals, as shown in Fig. 14a. SUB indicates the substrate, C indicates the plasmonic configuration of nanoresonators, A indicates a layer of an aluminum oxide (AI2O3) covering and D indicates the layer based on PbS nanocrystals. R1 represents incident optical radiation having a wavelength of 680 nm and R2 represents reflected optical radiation having a wavelength of 930 nm.

The synthesis of colloidal sulfur nanocrystals was carried out according to a known protocol by mixing lead oxide, oleic acid and octadecene, therefore injecting a mixture of hexamethyldisiloxane and ODE, thus obtaining a growth of the crystals that is controlled and stopped after a predetermined amount of time to obtain monodispersed PbS nanocrystals having small dimensions covered with oleic acid. In order to prepare the nanometric layer, a suspension of PbS nanocrystals obtained by the synthesis process was purified and dispersed in toluene.

The aluminum oxide covering layer (AI2O3), for a preferred thickness of 10 nm, was deposited on the surface of the sample in order to avoid suppression of the emission of the nanocrystals caused by the metal of the nanoresonators. The solution of PbS nanocrystals was deposited on the aluminum oxide layer by spin-coating, thus making it possible to obtain a covering having a thickness of 20 nm. Such layers of PbS nanocrystals have an absorption peak of approximately 800 nm and an emission peak (caused by exciton recombination) centered at 930 nm, as shown in the graph in Fig. 14b, in which the curve indicated by A represents the absorption spectrum (in arbitrary units) as a function of the wavelength, and the curve indicated by B represents the photoluminescence intensity spectrum as a function of the wavelength.

Fig. 14c compares the TPPL intensity maps of TBE configurations having different dimensions (overall diameters of between 1.5 microns and 2.5 microns in the x-axis, the scale bar shown in the drawing corresponding to the dimension of 1 micron), which were covered with the layer of PbS nanocrystals and excited by incident optical radiation having a wavelength of 680 nm (upper row), with the TPPL intensity maps of the TBE configurations that were not covered and excited by incident optical radiation having a wavelength of 1000 nm (lower row). If the plasmonic metasurface responded to the incident optical radiation in both cases, the modulation of this radiation would have to be different due to the different excitation wavelength. However, if the metasurface instead interacts with the emission radiation of the PbS nanocrystals at 930 nm, distributions of the near field are expected in both cases. Fig. 14c shows that the emission configurations of the device provided with a layer of PbS nanocrystals that covers the plasmonic metasurface when the dimensions increase substantially follow the TPPL intensity distributions of the device devoid of a layer of nanocrystals. This observation confirms that the modulation effect of the optical near field can be transferred to photoluminescent nanoparticles (for example fluorescent nanoparticles) deposited on metasurfaces formed according to the invention.

Since the coupling mechanism of the optical radiation between photoluminescent nanoparticles (for example fluorescent nanoparticles) and the metasurface on the basis of the effect described is independent of the configuration of the metasurface, it is possible to state that the effect would also happen with the other configurations“Twisted Gaussian” and “Tangential” described above, and although the resonance would change, and consequently the distribution of the electric field, the exciton-plasmon coupling effect would be maintained. A metasurface to which the invention relates, which can be obtained according to the method of the invention, would enter into resonance with the incident optical radiation or with the optical radiation emitted by the emitter layer arranged above the metasurface, but in both cases the modulation effect of the optical radiation is present.

Advantageously, the metasurfaces described that are obtainable by means of the method according to the invention can be used as substrates for tip-enhanced Raman spectroscopy (TERS). In combination with microfluidic devices, the switching of the incident optical radiation that is obtained with metasurfaces having a Twisted Gaussian and Tangential configuration can be used to trap and release (free) particles or to switch the direction of flow thereof. Furthermore, the projection of three-dimensional periodic functions on two- dimensional aperiodic arrays of nanoresonators can be used in the field of anti-counterfeiting for products and processing of information. Lastly, it is advantageously possible to generate light radiation having chirality that can be used in quantum photonics. The focusing of optical power in a largely confined central spot, which is obtainable by the Twisted Bulls-Eye and Twisted Gaussian configurations, can suitably largely improve the coupling of an optical signal to the tips of a spherical point in TERS spectroscopy and NSOM microscopy applications. Furthermore, the control of the distribution of the optical near field by means of the polarization or wavelength of the incident optical radiation is an interesting possibility for creating optical tweezers applications. For example, by means of the Twisted Gaussian configuration, particles can be confined to the donut-shape by means of horizontally polarized light and can be released by means of vertically polarized light. The tangential configuration can be used to control the direction (horizontal, vertical) of optical channels by controlling the wavelength of the incident optical radiation possibly by passing through an intermediate configuration in which both the horizontal and vertical channels are open.

Furthermore, the plasmonic metasurfaces for the modulation of the optical near field can be used in integrated device structures and coupled to other functional elements formed on a nanometric scale, for example nanostructures having light-emitting quantum dots. This concept makes use of the optical near field for the processing of the information, the protection against counterfeiting and the storing of data. Lastly, the generation of optical vortices and the chirality of the light obtainable by means of the Twisted Bulls-Eye and Twisted Gaussian plasmonic metasurface configurations can be used to form spin-orbit couplings in quantum photonics.

Of course, without prejudice to the principle of the invention, the embodiments and the implementation details can be largely varied with respect to that described and illustrated purely by way of non-limiting example, without thereby departing from the scope of protection of the invention, as defined by the attached claims.

Claims

1. A computer-implemented method of designing an optical device including an aperiodic matrix of nanostructures (NR) of a predetermined material, adapted to modulate in the near field an incident optical radiation, comprising:

defining (100) a nanostructure (NR) precursor array of in a predetermined two- dimensional reference system associated with a substrate adapted to receive said nanostructures (NR), wherein the nanostructure (NR) precursor array includes a regular periodic arrangement of replicas of a nanostructure (NR) unit cell;

starting from said nanostructure (NR) precursor array , calculating a configuration patter of said aperiodic matrix of nanostructures to produce a target modulation of the incident optical radiation by modifying at least one spatial parameter of the nanostructure unit cell in said regular periodic arrangement of replicas of the nanostructure unit cell, wherein, modifying at least one spatial parameter of the nanostructure unit cell in said regular periodic arrangement of replicas of the nanostructure unit cell includes modulating (200, 300) said at least one spatial parameter of the nanostructure (NR) unit cell in the aforementioned two-dimensional reference system according to a predetermined mathematical model that is a function of the planar coordinates of said two-dimensional reference system.

2. The method according to claim 1, wherein defining (100) a nanostructure precursor array in a predetermined two-dimensional reference system and calculating a configuration patter of said aperiodic matrix of nanostructures from said precursor nanostructure (NR) array comprise defining a mathematical function for configuring said nanostructures in said two-dimensional reference system that is representative of a physical parameter of the nanostructures.

3. The method according to claim 2, wherein said physical parameter is the electric permittivity.

4. The method according to any one of the preceding claims, wherein said at least one spatial parameter of the nanostructure unit cell of said regular periodic arrangement of replicas of the nanostructure unit cell includes at least one of the position, dimensions or planar orientation of the nanostructure unit cell in the aforementioned two-dimensional reference system.

5. The method according to any one of the preceding claims, further comprising simulating (400) an optical field emerging from said configuration pattern of said aperiodic matrix of nanostructures, which is calculated as a result of a predetermined incident optical radiation, comparing (500) said simulated emerging optical field with an expected reference optical field, and modifying at least one characteristic of said predetermined mathematical model if a difference between the simulated emerging optical field and the expected reference optical field is greater than a predefined threshold tolerance.

6. The method according to claim 2, wherein said mathematical function for configuring said nanostructures in said two-dimensional reference system is expressed in a Fourier series and the predetermined mathematical model for modulating said at least one spatial parameter of the nanostructures includes a modulation of the grating vector of the Fourier series expression of the mathematical function for configuring the nanostructures.

7. The method according to claim 6, wherein said modulation of the grating vector of the Fourier series expression of the mathematical function for configuring the nanostructures is defined, in polar coordinates r, Q, as

8. The method according to claim 6, wherein said modulation of the grating vector of the Fourier series expression of the mathematical function for configuring the nanostructures is defined, in Cartesian coordinates x, y, as where

is the position of the maximum modulation in x,y and

is a diagonal matrix representative of the standard deviation of the function

9. The method according to claim 6, wherein said modulation of the grating vector of the Fourier series expression of the mathematical function for configuring the nanostructures is defined, in Cartesian coordinates x, y, as

10. The method according to claim 2, wherein said mathematical function for configuring said nanostructures is discretized by associating one of a pair of predefined values for the physical parameter of the nanostructures with each coordinate of said two- dimensional reference system depending on the comparison with a threshold value that is a function of said coordinate.

11. The method according to claim 10, wherein said threshold value is defined, in Cartesian coordinates x, y, as

where

is the position of the expected value or peak of the Gaussian function in x,y

25 and

is a diagonal matrix representative of the standard deviation of the FF(x) function.

12. The method according to claim 10, wherein, when said physical parameter is the electric permittivity, said pair of predefined values comprises the electric permittivity value for the material of said nanostructures and the electric permittivity value for air.

13. A method for manufacturing an optical device including an aperiodic matrix of nanostructures (NR) of a predetermined material, adapted to modulate in the near field an incident optical radiation, comprising:

performing a method (100-500) for designing said optical device according to any one of the preceding claims for calculating a configuration pattern of said aperiodic matrix of nanostructures: and

controlling (600) a manufacturing process of the optical device according to a predetermined technique of structuring said substrate in accordance with said calculated configuration model.

14. An optical device obtainable by means of a method according to claim 13, comprising:

a substrate (SUB); and

an aperiodic matrix (C) of nanostructures (NR) made of a predetermined material, having a configuration that corresponds to a configuration pattern of said aperiodic matrix of nanostructures (NR) calculated from a precursor array of nanostructures (NR) comprising a regular periodic arrangement of replicas of a nanostructure (NR) unit cell in a predetermined two-dimensional reference system that is associated with the substrate (SUB), by means of modifying at least one spatial parameter of the nanostructures (NR) in said regular periodic arrangement of replicas of the nanostructure (NR) unit cell, said modification including modulating at least one spatial parameter of the nanostructures (NR) according to a predetermined mathematical model that is a function of the coordinates of said two-dimensional reference system.

15. The optical device according to claim 14, wherein said precursor array of nanostructures (NR) is a regular radial arrangement comprising concentric rings of replicas of a nanostructure (NR) unit cell, and said aperiodic matrix configuration pattern comprises a succession of concentric rings of said nanostructure (NR) unit cells having modulated characteristic dimensions that increase towards the outside, wherein the position of said nanostructure (NR) unit cells is modulated so that each ring of said nanostructure (NR) unit cells is rotated in an anticlockwise direction with respect to the adjacent inner ring, preferably by 30° with respect to said radial arrangement of the precursor array.

16. The optical device according to claim 14, wherein said precursor array of nanostructures (NR) is a regular Cartesian grid arrangement of replicas of a nanostructure (NR) unit cell, and said configuration pattern of the aperiodic matrix comprises a succession of wavy rows of said nanostructure (NR) unit cells that is obtained by modulating the position of said nanostructure (NR) unit cells, wherein said nanostructure (NR) unit cells have modulated characteristic dimensions that increase radially outwards of the arrangement.

17. The optical device according to claim 14, wherein said precursor array of nanostructures (NR) is a regular Cartesian grid arrangement of replicas of a nanostructure (NR) unit cell, and said configuration patter of the aperiodic matrix comprises a centrosymmetric series of nested rows of said nanostructure (NR) unit cells oriented along hyperbolic directrix curves, which series is obtained by modulating the position of said nanostructure (NR) unit cells, wherein said nanostructure (NR) unit cells have modulated characteristic dimensions that increase towards the outside of the arrangement.

18. The optical device according to any one of claims 14 to 17, wherein said nanostructures (NR) are polygonal nanostructures.

19. The optical device according to claim 18, wherein said polygonal nanostructures are rhomboid nanostructures (NR) each having an orthogonal major axis (D) and minor axis (d), which exhibit separate dipolar resonance in the two axial directions at different wavelengths.

20. The optical device according to claim 19, wherein said rhomboid nanostructures (NR) are nanostructures of a noble metal - preferably of gold, copper nanostructures, aluminum nanostructures, graphene nanostructures or nanostructures of a two-dimensional semiconductor.

21. The optical device according to any one of claims 14 to 20, wherein the aperiodic matrix (C) of nanostructures (NR) comprises a covering layer (D) of photoluminescent material.