Method for generating a lattice of bessel beams with arbitrary geometries and optical properties with one single optical element

Abstract

The present invention relates to the field for generating structured light and in particular it relates to a computer-implemented method for generating a phase profile of a lattice of Bessel beams with properties independent from each other. The invention also relates to a method for preparing an optical element configured for generating a lattice of Bessel beams, an optical element obtainable by such method and a method which provides the use of such optical element for generating a lattice of Bessel beams.

Description

[0001] "Method for generating a lattice of Bessel beams with arbitrary geometries and optical properties with one single optical element"

[0002] FIELD OF THE INVENTION

[0003] The present invention relates to the field for generating structured light and in particular it relates to a computer-implemented method for generating a phase profile of a lattice of Bessel beams with properties independent from each other. The invention also relates to a method for preparing an optical element configured for generating a lattice of Bessel beams, an optical element obtainable by such method ed a method which provides the use of such optical element for generating a lattice of Bessel beams.

[0004] STATE OF ART

[0005] The development of optics and photonics and of their capability in monitoring with increasingly high precision the properties of light has a fundamental role in many modern systems, from technologies of mobile phones to manufacturing by laser, to the optical communication systems up to the missile systems.

[0006] The monitoring of the properties of a light beam is currently based upon three main tools: i) a theoretical and computational basis which describes and predicts which light properties have to be modified and how they have to be modified in order to obtain the wished light beam; ii) a theoretical and computational basis which describes and predicts the light interactions with the material and defines which geometries and materials have to be selected to modify the light properties in order to obtain the wished beam; iii) a technology which allows to develop and manufacture the optical element or the optical system capable of integrating the capabilities of modulating the light properties in order to obtain the wished light beam.

[0007] The development of increasingly compact and precise systems pushed the optical technology to produce increasingly thin optical elements: the so-called “flat optics”, that is optical elements characterized by thicknesses comparable with the light wavelength. This technological breakthrough allowed not only to overcome the limitations linked to the intrinsic thicknesses of the classical optics, but even to overcome the limits of diffraction by allowing sub-diffractive focalizations and the generation of non-diffractive beams. Over the last decades, the idea of generating Bessel beams as alternative to the Gaussian beams has widely widespread in order to be able to obtain lighting profiles not subject to the typical diffraction degradation associated to the beam propagation. The transverse profile of the Bessel beams consists in a narrow and brighter central peak, and side lobes with decreasing intensity and concentric to the central peak. With respect to a Gaussian beam having similar numerical aperture (NA), these Bessel beams have a narrower central peak which remains unchanged in intensity and size for many Rayleigh lengths. This property makes the intensity profile along the propagation axis of a beam Bessel more elongated and narrower than that of Gaussian beam.

[0008] This type of light beams, obtained by manipulating the phase or width of the electromagnetic field, found a wide range of possible applications such as photolithography, the laser processing of materials, accelerators of electrons, optical trapping and bioimaging. Generally, the key advantages provided by this type of beam are: i) decoupling of focus depth from the transversal extension of the beam; ii) their non-diffractive nature thanks to which their intensity profile remains unchanged along the propagation axis.

[0009] The generation of Bessel beams, with one single optical element with thicknesses comparable with the radiation wavelength, can be implemented (i) with Fresnel technology, such as an axicon or a lattice, o (ii) with the technology of the metasurfaces such as super-oscillatory lens (SOL), Dammann gratings, prism-lens or axicons, or at last (iii) with flat masks such as super- oscillatory lens (SOL) or grating.

[0010] In response to an increasing need for parallelization of the processes, the technology didn’t just find strategies for generating single Bessel beams, but made an additional effort to find solutions which allowed to generate parallelly several Bessel beams in a lattice, by implementing a series of identical Bessel beams from the point of view of the optical properties, distributed according to the scheme of a regular lattice on the plane transversal to propagation and all with the same focusing distance. In order to obtain these regular lattices of structured light with multiplexing of the original beam, typically several optical elements and methods are used for monitoring the phase and intensity of light to manipulate the electromagnetic field of incoming light and then in order to be able to obtain as output the wished structured light. The optical paths for the generation of lattices of Bessel beams typically consist of three fundamental components:

[0011] 1. An element for the beam multiplexing;

[0012] 2. An element for the Bessel beam formation;

[0013] 3. A target for forming the structured light in the space of the sample to be illuminated.

[0014] However, this type of approach based on traditional optical elements requires a quite complex and not very flexible optical path, not suitable for small-scale integration.

[0015] The technology of metasurfaces has recently demonstrated to be able to integrate in one single element all manipulations required for the generation of lattices of Bessel beams: to this purpose, metasurfaces based upon lattices of Dammann (method-i), and metasurfaces based upon prism-lenses (method-ii), in particular, are known.

[0016] The method-i consists in a system based upon the phase modulation through a mask given by the overlapping of a mask for generating the Bessel beam through an axicon or a cylindrical lens and a lattice (grating) for multiplexing such beam.

[0017] The method-ii, instead, is based upon a metasurface for generating the phase profile of a lens-prism combined with a filter or a lighting system dedicated to structure the lighting on the metasurface and so as to obtain the beam multiplexing.

[0018] Both these approaches produce a regular lattice of Bessel beams, all with the same identical properties: all beams are distributed at a predefined precise distance from the lens exit surface along the direction of propagation of the radiation, that is they have the same focal length and have the same focus depth, that is longitudinal extension. Moreover, with the current methods a compromise has to be reached between the number of beams which can be generated and the area in which they can be generated, and ultimately both these figures are limited by the diameter of the optical window of the mask which encodes the phase of the Bessel beam. Among these approaches and technologies, only the metasurfaces based upon the lattices of Dammann (method-i) demonstrated to be able to generate a lattice of Bessel beams without requiring to structure the incoming light with additional elements.

[0019] Although the flat optics, and in particular the metasurfaces, represent a promising tool to integrate optical elements capable of generating small-scale lattices of Bessel beams compatible with the CMOS (complementary metal-oxide semiconductor manufacturing processes, there are still some fundamental limits in the state of art relating the generation of lattices of Bessel beams: 1. The level of monitoring the properties of each single beam composing the lattice (beam position, beam and focus depth); 2. Optimization of effectiveness in the distribution of the light intensity among the several beams composing the lattice; 3. Generation of a wide number of sub-diffractive beams with micrometric distances.

[0020] Therefore, the need for methods and optical elements, being capable of allowing the generation of lattices of Bessel beams according to the geometries overcoming the regular organization of the lattice, remains unrealized in the field.

[0021] SUMMARY OF THE INVENTION

[0022] The task underlying the present invention is to provide a new method for generating a lattice of Bessel beams allowing to obviate the problems found in the field, with particular reference to the possibility of generating lattices of Bessel beams, wherein each single beam of the lattice could have different optical features (focus position, focus depth, relative intensity and transversal placement), while maintaining the fundamental property of being a Bessel beam.

[0023] As it will be illustrated in detail in the experimental section of the description (Example 1), the present invention is based upon the consideration that each Bessel beam in a lattice could be generated independently by creating a ring with a phase mask for a converging lens, with a ringmask combination for each single beam of the lattice (Figurel a and b). The authors of the present invention found that it is possible to extend the above-mentioned approach and to implement on the same plane a total phase mask capable of generating several Bessel beams simultaneously, each one with optical properties independent from the other ones and, as mentioned, a lattice of Bessel beams according an arbitrary non-regular geometry.

[0024] Therefore, the authors of the invention have devised a new method for processing a phase profile of a lattice of Bessel beams, in the present case designated as , which can be implemented in a same plane for generating a lattice of Bessel beams with optical properties independent from each other.

[0025] According to a preferred embodiment, the phase profile <pT0Tgenerated by the method devised by the authors of the present invention is the result of the argument of the sum of fields each one corresponding to a Bessel beam and each one as defined independently with the wished parameters of the ring and wherein are spatial parameters which define respectively the transverse position focus depth and relative intensity (P;- and A;) and the focus longitudinal position of j beam in the lattice.

[0026] This approach allows to produce a structured light of arbitrary Bessel beams in 3D, not only with different focus positions in the transverse plane, but even along the longitudinal direction. In particular, the various Bessel beams of the lattice can be generated with different focal depths and with relative arbitrary intensities. For example, the transverse positions can be selected so as to optimize the destructive interference between the secondary lobes of the Bessel beams and to optimize the constructive interference of the central peaks, with a not previously accessible flexibility level.

[0027] Alternatively, as shown by way of example in Figures 1 (e-h), the total phase profile can be processed so as to allow the generation of one-dimensional lattices of Bessel beams.

[0028] As mentioned, the phase profile processed by the method of the present invention can be implemented in one single optical plane, in reality by using one single flat optical element, by allowing the generation of a lattice of Bessel beams with properties independent from each other and in arbitrary positions.

[0029] In an aspect, the present invention then provides even a method for generating a lattice of Bessel beams, which provides the use of one single flat (preferably having a sub-millimetric thickness) and light (i.e. grams) optical element and it allows to monitor the properties of each beam of the lattice independently. Such approach, for example, allows to monitor the position, the focal length and the focus depth of each beam of the lattice by allowing the generation of arbitrary lattices 1 D, 2D, or 3D.

[0030] In a preferred aspect, the present invention in particular provides a method for generating a lattice of Bessel beams which provides the use of un optical system comprising the following elements: a beam of incoming plane waves and a surface of a flat optical element configured so as to allow to monitor the phase of the incoming light according to a predetermined spatial distribution of the phase. This allows to be able to transform an incoming Gaussian beam with the purpose of implementing a lattice of Bessel beams at the exit of the del optical system.

[0031] One of the main advantages provided by the present invention then is represented by the possibility of integrating in one single optical element all optical manipulations required to generate one lattice of Bessel beams. The invention in this way allows to generate a lattice of Bessel beams in which each beam can be defined with independent focal length, focus depth and position with respect to the other beams in the lattice.

[0032] A second important advantage of the present invention consists in the possibility of producing a lattice of Bessel beams without having to structure the incoming light nor masking it. Moreover, the present invention allows to produce a number of beams which can be extended independently from the numerical aperture of the single beam.

[0033] Thanks to the above-illustrated advantages, the embodiments, the present invention relates to, surely offer new perspectives for the development of applications which require the use of structured light with multiple beams, with greater focus depth and reduced width of the beam with respect to solutions based upon conventional optics, for example within the processing of materials, telecommunications and microscopy.

[0034] Other advantages and features of the present invention will result evident from the following detailed description.

[0035] BRIEF DESCRIPTION OF FIGURES

[0036] Figure 1. Operating principle of the invention: generation of a lattice of Bessel beams with one single optical element - a, The modulation annular map of the amplitude of the electric field, annular mask (a-above), is multiplied by the map of the phase modulation corresponding to a converging lens (a-below) to produce a phase profile with a ring shape (b); the ring-like phase profile produces the intensity profile corresponding to a Bessel type beam, c, Phase profile of the Bessel beam in the plane XY transversal to the beam propagation direction, d, Intensity profile of the Bessel beam in the plane XZ of the beam propagation, e, The lattice of Bessel beams is generated by the translation and overwriting of each phase ringt0obtain the total phase profile (e- above). The regions highlighted in e-below show the phase profile wherein the rings overlap in the periphery area of the mask (e centre) and in the central area (e down), f e g, Intensity profile of the lattice of beams generated with the mask described in e. f, Intensity profile of the lattice of Bessel beams in the plane XY. g, Intensity profile of the lattice of Bessel beams in the plane XZ. h, Total phase profile of the lattice of Bessel beams obtained by the argument of the sum of the fields of the single Bessel beams each one generated starting from a phase profile The regions highlighted in h-below show the phase profile in the peripheral area (h centre) and in the central area of the total phase profile (h down), i and I, Phase profile of the lattice of Bessel beams generated by the sum of the fields Intensity profile of the lattice of Bessel beams in the plane XY. I, Intensity profile of the lattice of Bessel beams in the plane XZ.

[0037] Figure 2. Generation of a lattice of Bessel beams with different focal length - a, Phase mask obtained by the sum of the fields of four Bessel beams with different focal lengths. Each ring in the mask was designed for sake of simplicity with the same internal and external radius ; b, the beam propagation (profile XZ) shows the profiles of the four beams obtained with different focal lengths:

[0038] Figure 3. Bessel beam based upon metasurfaces: principle and characterization, a, Scheme of the meta-atom of the metasurfaces (MS), b, Photograph of a typical optical element manufactured with the technology of the metasurfaces (MS). The structures MS appear in reflectance as light grey stripes on the device surface, c, A wide-field optical image of one of the metasurfaces for the excitation with lattice of Bessel beams extended transversally (1 D). MS shows the characteristic rings of the principle> each ring behaves simultaneously both as converging lens and as annular mask, thus transforming each annular mask into a Bessel beam with the optical features of the converging lens, d, Enlarged view of the central region marked in (c): map of MS (on the left) and corresponding image at the scanning electron microscope (SEM) (on the right), e, Enlarged view of the peripheral region marked in (c): map MS (on the left) and corresponding image SEM (on the right). The images SEM show the nanometric scaling of the square transverse section of the meta-atoms. f, Simulation of the intensity profiles for a lattice 1 D generated with our MS at different positions along the propagation direction (on the left); measured profiles of lattice 1 D experimentally generated by the manufactured MS (on the right) and the corresponding intensity profiles along the orthogonal direction. The width at half height (FWHM) of the profiles 1D measured along the orthogonal axis results to be <7 pm. Scale bars: 50 pm (c and f), 2 pm (d and e). Experimental parameters: power laser P < 0.5 mW; wavelength of source A = 488 nm; camera exposition time Exp = 50 ms.

[0039] Figure 4. Bessel lattice light sheet microscope based on MS. a, The proposed microscope consists of two independent arms: the horizontal lighting module, where the light sheet is generated by the system of metasurface (MS) with a Bessel lattice 1 D and the vertical imaging column, wherein the fluorescence emission signal is collected by a system including lens, emission filter and sensor CMOS. We designed this column with two independent and parallel arms: an axis (light grey) supports the heaviest components which have not to be moved, the second axis (dark grey) supports the lens for imaging and allows to manoeuvre the micrometric positioning thereof along the three axes, b, The biological sample (zebra fish in this example) is incorporated in the agarose inside a millimetric well on a rotating support fixed on a three-axis manual micro-positioner. The optical device with MS is mounted parallelly to the sagittal plane of the zebra fish (grey insert) with a custom-made precision support on a three-axis manual stage (grey insert).

[0040] Figure 5. Numeric simulation of the response of the metaatoms to the silicon nitride, a, Lateral view of the meta-atom of silicon nitride (on the right) of the transverse section of the metasurface (on the left). The port 1 is the input port to the interface between the substrate of silicon dioxide and air. The port 2 is the output port on the upper portion of the simulated air volume which surrounds the structure of meta-atom of silicon nitride (SiNx). The transmittance and the phase of the outgoing electric field are extracted through the scattering parameter S21. b, Top view of the meta-atom SiNx (dark blue), c, 3D-view of the mesh net generated by the simulation program COMSOL to simulate the modulation of the electromagnetic field of each meta-atom of SiNx. d, Phase (grey points) and Transmittance (black points) of each meta-atom simulated in COMSOL. The increase in the transverse section of the meta-atom (diameter D) induces an increase in the relative phase delay accumulated by the incoming light.

[0041] Figure 6. Layout of the metasurface for generating a Bessel lattice 1D. a, layout of the mask designed for generating a Bessel lattice 1 D. Bar of scale 20 pm. b, two details of the layout of the corresponding mask which show the scaling of the transverse section of the meta-atoms required to monitor the relative phase delay of the incoming light.

[0042] Figure 7. Process for manufacturing the metasurface. A silicon nitride film having thickness of 520 nm is deposited with a chemical vapour deposition (CVD) process on a substrate of silicon dioxide having thickness of 525 pm. A film of resist AZ ECI 3007 having thickness of 1.5 pm is applied by rotation on the surface of SiNx as protective layer for the wafer cutting phase. After a cleaning phase to remove the resist, a layer of Chrome (Cr) is created by cathode bombarding (sputter coater). Such Crfilm acts both as reference surface for the EBAM machine and as stiff mask for the phase for etching the SiNx. 200 nm of hydrogen silsesquioxane (HSQ) is applied as positive resist for the electronic beam lithography. The structured surface of the metasurfaces appear on the resist after the resist development. The structure of the metasurface is then transferred into the layer of Cr and SiNx by a first ion beam bombardment phase (I BE) followed by a reactive plasma bombardment phase. At last, the residues of Cr and resist are removed by attack with acid solution.

[0043] Figure 8. Inspections of metasurfaces after electronic beam lithography and the development of HSQ. Three metasurfaces (MS) for the generation of Bessel LLS. Focal length 2000 pm. Scale bar: 100 pm (on the left); 10 pm (on the right).

[0044] Figure 9. Inspections at scanning electron microscope (SEM) of the meta-atoms of the metasurface, a, and b, scale bar 200 nm. c, scale bar 300 nm.

[0045] Figure 10. Scheme of the metasurface characterization setup. The radius laser is aligned perpendicularly to the plane of the metasurface (MS). MS is fixed in the vertical slot of a custom- made support mounted on a micropositioner with three axes XYZ. The detection portion consists of: Olympus 20x (NA = 0.4) objective, a tublens (focal distance f = 200 mm) and camera (XIMEA, MQ013RG-ON) connected to the acquisition workstation. The objective is mounted on a motorized micropositioner to acquire automatically the beam profile which propagates along the axial direction (Z). Experimental parameters: wavelength of the source A = 488 nm; laser power P = 0.2 mW; camera exposition time Exp = 50 ms; rear-projected pixel size = 216 nm.

[0046] GLOSSARY

[0047] The terms used in the present description are as generally understood by the person skilled in the art, unless otherwise indicated.

[0048] In any point of the present description and claims, under the term Bessel beam an electromagnetic radiation field is meant, whose width follows a zero order Bessel function. Experimentally, a Bessel beam is approximated by the convolution of an ideal Bessel beam and of a Gaussian beam, whose physical result is a radiation field with extension limited in space. Said extension is defined by the Gaussian component of the experimental beam, and it derives physically from the finished sizes of the optical elements composing the system which generates the beam. For the zero order Bessel radius, the width is maximum in the origin, whereas a radius of higher order has an axial phase singularity in the origin where the width disappears so that the function demonstrates it. A real Bessel beam is not subject to diffraction, unlike normal behaviour of the light waves which disperse when focalized.

[0049] In the context of the present invention, under lattice of independent Bessel beams, a lattice 1 D, 2D or 3D is meant, comprising one or more Bessel beams, each one having independent properties with respect to the other different beams in the lattice, that is properties which can be defined and controlled independently with respect to those of the different beams in the lattice, in particular independent focal length, focus depth, relative intensity and focus position with respect to the other beams in the lattice. Under lattice 1 D in particular a lattice of beams distributed transversely along a privileged direction is meant; under lattice 2D in particular a lattice of beams distributed transversely along the inside of a plane is meant, parameterized by a system of cartesian coordinates; under lattice 3D, in particular a lattice 2D is meant with the characteristic that the different beams of the lattice have a focus distance different from one another.

[0050] In any point of the present description and claims, under the expression “suitable for configuring an optical element” with reference to a phase profile a phase profile is meant suitable for configuring an optical element according to any one of the variants described in the present one for generating a lattice of independent Bessel beams, that is suitable to be implemented on a same flat surface of said optical element for generating a lattice of independent Bessel beams.

[0051] In the context of the present invention, under the term metasurface a two-dimensional metamaterial is meant, that is with thickness comparable to the working wavelength and consisting of a spatial modulation of the refractive index, so as to modify locally width and phase of an incident wave front with a lower spatial resolution than the working wavelength (i.e. sub-wavelength). The simplest implementation of the metasurfaces, from a point of view of the design and manufacturing, consists in an inhomogeneous array of nano-structures. Each nano-structure depending upon one of its geometrical parameters has a different response (in width and / or phase) to incident light, so that the whole metasurface impresses a determined profile for correcting the phase and / or width to the transmitted field. The principle underlying the optical metasurfaces is very similar to those of arrays of radio frequency transmitting / receiving antennas, consisting of an array of 2D resonant elements capable of modifying locally both the phase and the width of an incident electromagnetic wave.

[0052] In the context of the present invention, under the expression “implementing the phase profile in a surface of an optical element”, the transfer on a same plane is meant, that is on one single optical surface all functions of an excitation target, of the apodization mask for the generation of a profile of Bessel beams and of the beam multiplexing, so as to impress a different phase spatial distribution to a beam of plane electromagnetic waves which strikes the optical element, and then to return, after the beam propagation of incident waves through the optical element, a lattice of Bessel beams with independent properties.

[0053] As used in the present description, the term “focus position” or “focal length” relates to the distance along the axis of the beam propagation between the exit surface of the radiation from the metasurface and the position wherein the beam has the minimum transverse section. The terms “focus position” and “focal length” can be used interchangeably in the present description and claims.

[0054] As used in the present description, the term “focus depth” relates to the spatial range, in the propagation direction, wherein the intensity of the central lobe of the beam is kept higher than 50% with respect to the maximum and the transverse section is kept lower than twice the minimum section of the beam. In any point of the present description and claims, the term “comprising” can be replaced by “consisting of”.

[0055] DETAILED DESCRIPTION OF THE INVENTION

[0056] Different embodiments and variants of the invention and portions thereof will be described hereinafter, based upon different aspects thereof usable separately or in combination.

[0057] In the following description, embodiments and additional variants with respect to embodiments and variants already treated in the same description will be illustrated limitedly to the differences with what already explained.

[0058] Moreover, as said, the different embodiments and variants described hereinafter are likely to be used in combination.

[0059] As previously mentioned, the present invention provides a computer-implemented method allowing to process a specific mask or total phase profile of a lattice of Bessel beams. As it will be better illustrated in details hereinafter, the present invention also provides a method (also herein defined as method 1) which allows to implement on the same plane, that is on the same plane surface of an optical element, said specific mask or total phase profile for generating a lattice of Bessel beams according to an arbitrary geometry 3D.

[0060] In a particular embodiment, the present invention also provides a method (also herein defined as method 2) for encoding said mask or total phase profile with the purpose of generating linear lattices of Bessel beams (lattice 1D), or a lattice of Bessel beams contained in one single plane as defined by a transverse axis and a longitudinal one with respect to the direction of propagation of the radiation, suitable in practice to form an ideally uniform light plane, in which case allowing an easier implementation of the (more general) method 1 .

[0061] Advantageously, by creating a ring with a phase mask for a converging lens as it will be illustrated hereinafter, the methods, the present invention relates to, allow to generate each Bessel beam of the lattice independently, that is with independent focal length, focus depth and position with respect to the other beams in the lattice.

[0062] Therefore, a first aspect of the present invention relates to a computer-implemented method for generating a phase profile of a lattice of Bessel beams, suitable for configuring an optical element, which method comprises the following steps: a) generating or providing as input a phase profile of a converging lens with radius and focal length represented by the following formula: wherein: j is a natural number ranging from 1, 2, up to N, corresponding to the index of a single Bessel beam in the lattice, wherein N corresponds to the total number of Bessel beams in the above- mentioned lattice; Pj and Qj are spatial parameters which define the position of a single Bessel beam of index j in the lattice; and the values of Rj and determine, respectively, the focus position and the effective numerical aperture (NAj) of a single Bessel beam of index j in the lattice and, then, even the maximum thickness of the beam and the minimum focus depth of the beam; b) generating or providing as input a ring phase mask represented by the following formula: wherein: represents the external radius of the ring; represents the thickness of the ring; and wherein the radius of the ring has the same value of as defined in in step a); and c) multiplying the phase profile generated or provided in step a) for the phase mask generated or provided in step b) so as to obtain a phase profile with a ring shape, represented by the following formula: wherein are as defined as in a) and b); d) generating or processing said total phase profile starting from said phase profiles

[0063] In particular, the phase profile <pT0Trepresents the phase correction required for generating a lattice of Bessel beams.

[0064] In a more general embodiment (method 1), in step d) of a method according to any one of the herein described embodiments, said phase profile is obtained by calculating the electromagnetic fields generated by said phase profiles determined in step c), by summing up such fields and by calculating the argument of such sum.

[0065] The binary mask defined in step b) of the method according to the invention allows the incoming light to pass through a ring as defined by the external radius and thickness . The narrower the ring, that is the smaller A7, the narrower the central peak of the generated Bessel beam is and longer its focus depth is.

[0066] Each phase profile as determined in step c) of a method according to the present invention produces the intensity profile of a wished beam Bessel of the lattice.

[0067] As represented by pure way of example in Figure 1 (i-n), according to an aspect of the invention, the phase profile determined in step d) of a method of the invention, is the result of the argument of the sum of fields each one corresponding to a Bessel beam and each one as defined independently with the wished parameters of the ring according to the following expressions: wherein is a real number greater than zero, and wherein are the spatial parameters which define the position of the beam j in the lattice.

[0068] Therefore, according to an aspect of the invention, step d) of a method according to any one of the herein described embodiments comprises the following steps: e) calculating the field distribution of an electromagnetic wave characterized by the above-mentioned phase profile equal to the phase profile obtained in step c) and with width given by , represented by the following formula: f) repeating each one of the steps a)-c) and e) for a number N of times corresponding to the number of wished beams in the lattice, wherein, at each iteration j, said length and said radius have different values, and wherein, at each iteration of the index j, said spatial parameters and Qj have different values such that the position of each beam of index j in the lattice results to be translated with respect to that of a beam of different index j. In other terms, in step f), at each iteration of the index j the phase profile is defined with a different length thickness of the ring and radius , in particular wherein, at each iteration j, the beam position is translated by the wished parameters ; g) calculating said total phase profile fiT0Tas the argument of the sum of the fields each one corresponding to a Bessel beam translated by the values in the lattice, that is translated by in Y of the lattice, according to the following formula: , up to N and N is the number of total beams.

[0069] As shown by pure way of example in Figure 1 (e-h), the total phase profile fiT0Tdetermined in step d) of a method according to the present invention can be generated even overwriting consecutively each phase ring so as to generate linear lattices of Bessel beams.

[0070] In the specific case (limited to a) of a linear lattice of Bessel beams, said phase profile fiT0Tis obtained by considering the phase of each single ring j, and by constructing the total phase profile so that, for any point of the domain corresponds to the phase of one and only one of the rings (method 2).

[0071] Therefore, according to an aspect of the invention, step d) of a method according to any one of the herein described embodiments comprises the following steps: e’) repeating steps a)-c) of a method according to any one of the herein described embodiments; and f’) processing or determining said total phase profile so that, for any point of the domain corresponds to the phase of one and only one of the rings Said correspondence is solved in the overlapping points of the rings that is in all points (x,y) of the codomain belonging to more than one ring through a sorting function, hereinafter explained analytically, which always selects the ring of lower index, thus implementing a complete sorting of the rings.

[0072] In other terms, the total phase profileisprocessed so as to correspond to the phase values of all single rings and, in all points (x,y) of the codomain belonging to more than one ring assumes always the phase values of the ring of lower index.

[0073] The phase profile in this case is defined as: wherein j is a natural number less than or equal to N, as defined by the relation wherein are as defined as in a) e b), and k is a natural number support index less than or equal to N.

[0074] In step e’) of the method of the invention, at each iteration of the index j the phase profile of each beam is defined with the wished focal length thickness of the ring and radius in particular, at each iteration j, the position of each beam is translated by the wished parameter P (by assuming Qj = 0) or the wished parameter (by assuming Pj = 0).

[0075] Therefore, according to an additional aspect of the invention, step d) of a method according to any one of the herein described embodiments comprises the following steps: e’) repeating steps a)-c) of a method according to any one of the herein described embodiments N times to calculate a number of phase profiles wherein N corresponds to the number of wished beams in the lattice; wherein, at each iteration each phase profilecanbe as defined with wished focal length a thickness and radius of the ring (that is equal or different) and wherein, at each iteration of the index j, its position can be translated by the wished parameters in particular wherein said spatial parameters Pj e Qj have different values such that the position of each beam of index j in the lattice results to be translated with respect to that of a beam of different index j; and f’) calculating said total phase profile as: wherein j is a natural number less than or equal to N, as defined by the relation: wherein are as defined as in a) and b) according to any one of the herein described embodiments, and k is a support index, natural number and as .

[0076] According to an aspect of the invention, the computer-implemented method according to any one of the herein described embodiments allows to generate the phase profile of a lattice 1 D, 2D or 3D of Bessel beams with properties independent from each other, in particular with different position, different focal length and different focus depth from one another.

[0077] According to an aspect of the present invention, the computer-implemented method according to any one of the herein described embodiments allows to generate the phase profile of a lattice comprising an arbitrary number N of Bessel beams limited only by the transverse extension of the input beam, for example a number of beams comprised between 1 and 10.

[0078] According to the present invention, the various Bessel beams of the lattice can be generated with different focal lengths and arbitrary transverse positions.

[0079] According to an aspect of the invention, the transverse positions of the beams in the lattice, detected by the spatial parameters P}and Q ., are selected so as to optimize the destructive interference between the secondary lobes of the Bessel beams and to optimize the constructive interference of the central peaks.

[0080] An additional aspect of the present invention is represented by a computer program, comprising a list of instructions which, when executed on an electronic computer, implement the steps of a method for generating a total phase profile of a lattice of Bessel beams according to any one of the herein described embodiments.

[0081] The present invention also relates to a method for preparing an optical element configured for generating a lattice of independent Bessel beams, when struck by a beam of plane electromagnetic waves emitted by a light source, comprising the following steps: a) generating a total phase profile of a lattice of Bessel beams according to any one of the herein described embodiments; and b) encoding or making the total phase profile generated in a) in a surface of a flat optical element.

[0082] A flat optical element suitable to be used in a method according to the present invention is any optical element known in the field to be used in the generation of structured light. Preferably, the optical element has a sub-millimetric thickness, for example comprised between 1 and 600 pm. The weight of the optical element depends upon the material and upon the surface which one wished to structure and is generally in the order of few grams. By pure way of example, a chip of SiNx (density 3.17 g / cm3), on SiO2 (density 2.65 g / cm3) which, according to the present invention, has ten metasurfaces distributed on a surface of 1 cm x 1 cm, it weighs about 0.1 grams.

[0083] According to an aspect of the invention, said optical element comprises or consists of a substrate made of an optically transparent material, preferably made of a material selected among silicon dioxide or sapphire (i.e. sapphire).

[0084] In a preferred embodiment according to the present invention, said optical element comprises or consists of silicon dioxide, in particular has a thickness comprised between 1 pm and 600 pm, preferably equal to 525 pm. As a person skilled in the art knows, the thickness of a substrate or optical element suitable for the generation of structured light usually is simply as defined / constrained by the thickness of the wafers available on the market but which can be made even thinner (up to 100 nm), for example by digging it before starting the process for manufacturing the metasurface.

[0085] According to an aspect of the invention step b) of a method for preparing an optical element according to any one of the herein described embodiments comprises converting said total phase profile <pT0Tin a discrete phase map of a metasurface, in particular characterized by a suitable distribution of meta-atoms on the surface of the flat optical element.

[0086] Alternatively, step b) of a method for preparing an optical element according to any one of the herein described embodiments comprises converting said total phase profile in a discrete phase map of a Fresnel lens or a Spatial Light Modulator (SLM) according to any one of the methods known in the field. The conversion of a phase profile according to any one of the embodiments described in the present one in a discrete map of a metasurface can be performed according to any one of the methods known to a person skilled in the art. By pure way of example, as shown in the experimental section of the present application (for example Figure 6), in order to convert the continuous phase profile according to any one of the herein illustrated embodiments into a discrete phase map of a metasurface, it is possible to apply a phase calculation algorithm in MATLAB by implementing operations generally known to a person skilled in the art, such as:

[0087] - loading as input COMSOL data relating to the phase and width of the field exiting the nanostructures of the metasurface;

[0088] - calculating the analytical phase profile and the numerical aperture (NA) for a metasurface characterized by input parameters (for example focal length, radium, period) set by the user;

[0089] - digitalizing the two-dimensional (2D) phase profile wherein the different geometries of the nanostructures are distributed on the metasurface according to the provided spatial scheme;

[0090] - generating the layout of the metasurface and calculating the phase error;

[0091] - simulating and displaying the beam propagation with relative quantitative information; for example, implementing a beam propagation method (BSM) to recover the effective profile of the beam structured by the metasurface and extracting the position of its focus, its FWHM, its focus depth and its effectiveness.

[0092] Preferably, step b) of a method for preparing an optical element according to any one of the herein described embodiments comprises manufacturing on a surface of the selected flat optical element, a metasurface comprising a matrix of meta-atoms, also called in the present description and claims as nanostructures or pillars, having sizes comparable with the wavelength of the beam of plane electromagnetic waves emitted by the light source. Preferably, the meta-atoms or nanostructures have a higher height than the wavelength of the above-mentioned beam, and smaller transverse sizes of the transverse section with respect to such wavelength.

[0093] In particular, the meta-atoms or nanostructures manufactured on the surface of the optical element have predetermined shapes and arrangement so that, when the resulting optical element is struck by a beam of plane electromagnetic waves, imparting the beam of waves incident on them a phase profile represented by according to any one of the previously illustrated variants.

[0094] The meta-atoms or nanostructures of the metasurface manufactured on the surface of the optical element can have a section having a square, rectangular, round or triangular shape or even a combination of different shapes. Typically, nanostructures implemented with dielectric materials are designed with transverse sections which keep equal along the propagation axis / height of the nanostructure.

[0095] Preferably, such nanostructures have a parallelepiped shape.

[0096] According to a preferred aspect of the invention, the nanostructures of the metasurface manufactured on the surface of the optical element comprise pillars or nanopillars having a square cross-section.

[0097] As a person skilled in the art, the height of each nanostructure of the metasurface depends upon the light length, upon the used material, as well as upon the type of approach used to monitor the phase accumulation. In the context of the metasurfaces, the phase accumulation, or better the phase delay, is as defined as the phase difference between the phase of the electromagnetic wave entering and exiting the nanostructure, that is the phase delay inserted in the phase of the light incident by the nanostructure. In case of a nano-pillar of dielectric material, the phase delay is as defined by h wherein h is the height of the pillar, is the wavelength of the light that is of said beam of plane electromagnetic waves emitted by the light source and n is the refractive index of the dielectric material used to implement the metasurface.

[0098] According to an aspect of the invention, the phase accumulation can be monitored by varying the lateral size of the nanostructures. This type of approach allows not to have constraints on the type of polarization of the input radiation which one wants to structure.

[0099] Alternatively, as a person skilled in the art knows, for applications wherein the input and output light can or must be polarized circularly, the phase accumulation can be monitored by relative rotation of the pillars (geometrical phase or Pancharatnam Berry phase). In this case, the pillar section has to be necessarily asymmetric (that is for example a rectangle).

[0100] According to an aspect, said nanostructures have a height h represented by the following formula: wherein A is the wavelength of the light that is of said beam of plane electromagnetic waves emitted by the light source and n is the refractive index of the dielectric material used to implement the metasurface. According to an aspect of the invention, as illustrated in the experimental section of the present application, said height is equal to 520 nm.

[0101] By pure way of example, for a working wavelength equal to 488 nm and for nanostructures made of SiNx on SiOzas it will be illustrated in more detail in the experimental section of the present application, it is possible to obtain a variation in the phase accumulation of the whole range from 0 to 2Pi by using nanostructures with height 520 nm and by varying their lateral side from 60 to 340 nm.

[0102] According to an additional aspect of the invention, during the manufacturing process, said nanostructures are positioned on the surface of said flat optical element with a fixed spatial period represented by the following formula: wherein is the wavelength of the light that is of said beam of plane electromagnetic waves emitted by the light source and NA is the numerical aperture of the converging lens generated in step a), that is of the converging lens having a phase profile as defined in step a) of a computer-implemented method according to any one of the embodiments described in the present one. According to an aspect of the invention, the nanostructures can be implemented on the surface of said flat optical element with a fixed spatial period therebetween, in particular equal to 400 nm.

[0103] In an embodiment according to the present invention, the active area of metasurface manufactured on the surface of the optical element has a diameter comprised between 100 - 1000 pm, adaptable to the application needs.

[0104] Materials suitable to be used for manufacturing the nanostructures of a metasurface according to any one of the variants illustrated in the present description and in claims comprise silicon nitride (SiNx), silicon (density 2.33 g / cm3), GaN (6.15 g / cm3) and TiOz (4.23 g / cm3). Silicon, GaN, and TiOz are dielectric materials which can be good candidates as core layer for implementing flat optics in the visible. Such materials, in fact, have a good transparence in the visible, good chemical stability, compatibility with CMOS technology and manufacturing processes.

[0105] According to an embodiment of the invention, said nanostructures are made of SiNx. SiNx is particularly preferred since it is a dielectric material compatible with the manufacturing processes of the type “complementary metal-oxide semiconductor” (CMOS), with good chemical and thermal stability, with high contrast index with the oxide coatings used in the manufacturing process (An~0.5) and with a wide-band transmission extended to the visible spectrum.

[0106] The manufacturing of the metasurface on the surface of the optical element according to any one of the embodiments of the present invention can be performed by means of any one of the techniques known to a person skilled in the art.

[0107] Preferably, said manufacturing is performed by electronic beam lithography. The application of such technique is particularly advantageous in case of use of the silicon nitride, since it allows to obtain a resolution up to 5 nm.

[0108] As shown by pure way of example in Figure 7, according to an aspect of the invention, the manufacturing of the metasurface on the surface of an optical element comprises a first step of chemical vapour deposition (CVD) of a film of the material selected for the metasurface, such as SiNx, on the surface of the selected optical element, such as a substrate of silicon dioxide. Preferably, the thickness of film of the material selected for the metasurface is equal to about 520 nm.

[0109] The first step is followed by a second step of sputtering of a layer of Chrome (Cr), which acts both as reference layer and as stiff mask. The so-modified substrate is coated with a layer of few tens of nm (for example 200 nm) with a material suitable to act as positive resist for the electronic beam lithography, in particular hydrogen silsesquioxane (HSQ).

[0110] The wished model of the metasurface is impressed on the photoresist after development. The wished structure of the metasurface, that is suitable for the generation of a phase profile <pT0Tdetermined by means of any one of the methods the present invention relates to, is then transferred in the layer of Cr and of the material selected for the metasurface, for example SiNx, by means of a first ion beam etching phase (I BE) followed by a reactive plasma etching phase.

[0111] At last, the residues of Cr and photoresist are removed by means of humid acid etching.

[0112] The present invention also relates to an optical element configured for generating a lattice with Bessel beams independent therebetween and arbitrary three-dimensional geometry, obtainable by a preparation method according to any one of the embodiments illustrate in the present description and claims.

[0113] According to an aspect of the present invention, said flat optical element has a thickness comprised between said h + 100 nm and 600 pm, wherein h is as previously defined in the present detailed description.

[0114] By pure way of example, said optical element configured for generating the lattice can have a thickness comprised between 526 ± 2 pm (considering the substrate + core layer), and / or a weight comprised between 0.12 ± 0.02 g.

[0115] An additional aspect of the present invention relates to an optical system for generating a lattice of independent Bessel beams, comprising (i) a light source configured for emitting a beam of plane electromagnetic waves and (ii) an optical element according to any one of the embodiments described in the present one.

[0116] According to an aspect of the invention, the optical system comprises in particular a source of plane electromagnetic waves having a wavelength comprised between 380 and 750 nm.

[0117] In an additional aspect, the invention also provides a method for generating a lattice of independent Bessel beams comprising the following steps: a) providing a light source configured for emitting a beam of plane electromagnetic waves; b) directing said beam of electromagnetic waves through an optical element according to any one of the embodiments described in the present one.

[0118] Advantageously, in a preferred embodiment, the method the present invention relates to allows to produce a structured light of arbitrary Bessel beams in 3D.

[0119] The present invention in particular provides a method for generating a lattice of Bessel beams which provides the use of un optical system comprising the following elements: a beam of incoming plane waves and a surface of a flat optical element configured so as to allow to monitor the phase of the incoming light according to a predetermined spatial distribution of the phase represented by <f>T0Taccording to any one of the previously illustrated variants. This allows to be able to model the incoming beam to make a lattice of quasi-Bessel beams at the exit of the optical system.

[0120] EXAMPLES Some embodiment examples, not limiting the method according to the present invention, are herein illustrated by way of illustration.

[0121] Example 1 - Generation of lattices of Bessel beams by one single metasurface, to record the neuronal activity in zebrafish larvae with cellular resolution

[0122] Introduction

[0123] A new solution for generating excitation profiles with several Bessel beams by means of one single flat optical element was designed. In particular, a method was developed to encode on one single optical surface all functions of the excitation target, of the apodization mask for the generation of a profile of Bessel beams and of the beam multiplexing, so as to return, after the light propagation through the optical element, a lattice with several Bessel beams to the sample. In order to test this approach, a corresponding optical element was constructed, by using the technology of metasurfaces, Meta-Surface (MS). The metasurfaces have a typical thickness comparable to the wavelength and they allow to monitor with a lower resolution than the wavelength different light properties, such as width, phase and polarization. After a phase of characterizing the optical element which confirmed the generation of the provided light models, the engineered metasurface was aligned along the excitation path of a Light Sheet Microscope (LSM). By using the developed excitation approach based upon metasurface, the neuronal activity from the brain of a zebrafish larva was recorded with a resolution sufficient to solve the single cells.

[0124] Results

[0125] Generation of multiple Bessel beams starting from one single optical element

[0126] The electromagnetic field of a Bessel beam has an intensity profile extended along the propagation direction Z with an almost uniform transverse section XY showing a central maximum surrounded by a series of side lobes with decreasing intensity. It is known that the spectrum of this field recapitulates a ring function, with wave vectors lying on the surface of a cone. Therefore, by limiting the field intensity of a Gaussian beam with plane waves to a thin ring in the source space, it was possible making a Bessel beam in the Fourier space of a converging lens or objective.

[0127] This principle, already introduced as alternative to the approaches based upon axicon, adopts an apodization filter, to select an annular portion of the original beam, and its conjugation by a so-called “4f” optical arrangement to the rear focal plane of an objective to make a Bessel beam in the sample space. Then, the possibility of collapsing this optical path and of encoding on the same optical element the functions of the ring and objective was considered, so as to construct on the light propagation the convolution of an annular intensity profile and a phase modulation of the wave front corresponding to a focusing lens (Figures 1a and b).

[0128] By assuming an incoming plane wave, a phase mask defined in the domain and characterized by focal length f, acts as a Fourier transformation of an annular mask , with external radius R and thickness Δ. The resulting intensity profile (Figure 1c) corresponds to an approximation of the intensity profile of a beam Bessel of zero order as defined by the field wherein J is the zero order of the Bessel function of the first type, is the wave vector, are the longitudinal and radial components of the wave vector k, and z and r are the longitudinal and radial spatial coordinates in a system of polar coordinates.

[0129] The resulting intensity profile in the sample (Figure 1c and d) represents a Bessel beam whose spatial properties depend upon the design parameters (Figure 1a). R and f allow to monitor the real numerical aperture NA of the beam and its longitudinal position; the thickness of the ring A, affects the equilibrium of the relative weight of the Gaussian properties of the beam with respect to the Bessel properties. In fact, a beam generated by a ring-like mask with finished opening, will have a mix of features both of the Gaussian beam and of the Bessel one. The Gaussian envelop suppresses the Bessel lateral lobes of higher order (due to the continuum of wave vectors through the ring). Besides, the removal of wave vectors inside the internal diameter of the ring guarantees a much longer focus depth than a traditional Gaussian beam. Therefore, the thinner the ring A is, the more prominent the Bessel contribution is: the central peak of the beam becomes thinner and the longitudinal size of the lighting profile becomes more extended. It is common to use Bessel beams to implement a lighting scheme based upon lattice with several beams on the sample. The optical lattices are schemes with periodical interference, of any regular geometry with equidistant beams, generated by the coherent overlapping of a finished number of plane waves. It was found that it is possible to extend the above-mentioned approach and to implement on the same plane a total phase mask capable of generating several Bessel beams and then, as shown previously, a lattice of Bessel beams not limited to a regular geometry, but with arbitrary transverse and longitudinal geometry.

[0130] Two approaches to obtain the requested phase profile were identified.

[0131] In the first case, (method-1), the total phase profile is the argument of the sum of the fields , each one corresponding to a Bessel beam (Figure 1 i-n) and each one as defined independently with the wished parameters of the ring : wherein is a real number greater than zero. and wherein are two spatial parameters which define the beam position j.

[0132] This approach allows to produce a structured light of arbitrary Bessel beams in 3D. The various Bessel beams of the lattice can be generated with different focal lengths and with transverse arbitrary positions. For example, the transverse positions can be selected so as to optimize the destructive interference between the secondary lobes of the Bessel beams and to optimize the constructive interference of the central peaks.

[0133] In the second approach (method-2), the total phase profile of the lattice 1D of Bessel beams is generated by overwriting each consecutive field (Figure 1e-g).

[0134] It is known that the period P which separates the beams can be optimized as to produce a destructive interference between the lateral lobes of adjacent beams and to maximize the contribution of the main peak of the Bessel beam. With the design parameters adopted to define the optical properties of the ring (external radius of the ring R =100 pm, internal radius R- focal length f=2000 pm, and numerical aperture relative to the internal diameter of the ring Bessel beams were obtained moved laterally by P ~ 20 pm, (Figure 1e-g). With the purpose of testing the result of the developed methods, an algorithm of the beam propagation was implemented for calculating the spatial density profiles of the generated Bessel beams (see methods). This analysis confirmed that such modulation models of the single-plane wave front return to the light propagation more Bessel beams according to the design parameters: each beam has a mid-height width of FWHMx ~ 4 pm and FWHMy ~ 7 pm along the transverse directions X and Y and FWHMz ~ 700 pm along the direction of the beam propagation.

[0135] Generation of lattices of Bessel beams by using the technology of meta-surface (MS)

[0136] In order to evaluate the real capabilities of a flat optical element with the above-described features, one proceeded to its manufacturing by adopting the technology of metasurfaces, MetaSurface (MS). This approach uses matrixes of a few-hundred-nanometre-high nanopillars to monitor the properties of the electromagnetic field with a lower lateral resolution than the wavelength. The metasurfaces can be used to monitor simultaneously the width, the phase and the polarization of a wave front of propagation light, thus by allowing the colour routing, the polarization multiplexing and focusing. A series of possible materials was considered for the manufacturing and selected silicon nitride (SiNx). SiNx is a dielectric material compatible with the CMOS manufacturing processes, with good chemical and thermal stability, with high contrast index with the oxide coating (An~0.5) and with a wide-band transmission extended to the visible spectrum. Therefore, a series of MS SiNx Huygens were manufactured and tested, insensible to polarization, devised to function at A = 478 nm for the production of several Bessel beams.

[0137] The developed metasurface design has a series of nano-pillars, constructed above a substrate of silicon dioxide, with a different lateral dimension to modulate locally the phase of the incoming wave front, then by acting as phase delay element (Figure 3a). Numerical simulations in the time domain at the finished differences (FDTD) (COMSOL Multiphysics®) were used to study the modulation of the electromagnetic field of the meta-atoms and to identify a library of 28 structures which guarantee the covering of the phase rang with a transmittance effectiveness higher than 50% (Figure 5d). These meta-atoms have a square transverse section with a variable lateral length D, comprised between 60 and 340 nm in steps of 10 nm, and a fixed height h=520 nm. The designed active area of metasurface resulted to have a diameter of about 200 pm with nanopillars distributed with a fixed spatial period (p=400 nm). Then, a MATLAB routine was developed for converting the continuous phase profile obtained with the method-2 in the discrete phase map of the metasurface. The numerical simulations confirmed that the adopted metasurface layout generates a set of ten Bessel beams in strict agreement with the theorical model (Figure 3f-on the left).

[0138] Based upon this confirmation, a series of metasurfaces were manufactured with a process based upon electronic beam lithography (see Figure 3d-e, Figure 7, Figure 8 and Methods). The acquisition of the intensity profile at the focal length of MS confirmed that a metasurface was obtained capable of generating a set of ten Bessel beams with sizes ~ 5 x 7 x 700 pm3(X, Y, Z) and that they extend on a lateral dimension X of about 200 pm, close to the values predicted by the simulations (Figure 3f-on the right).

[0139] Lattice light sheet imaging based upon the meta-surface of the neuronal activity of the zebrafish

[0140] The manufactured MS was integrated in the excitation path of a customized LSM configuration to demonstrate the performances of the proposed approach. MS was inserted between the laser source and the sample, by replacing the optics and the objective of the typical LSM excitation design, and by guaranteeing a transverse excitation area of 200 pm x 700 pm in the plane XZ. The system was tested by recording the neuronal activity of the zebrafish larvae which express the GCaMP6 genetically encoded activity indicator in most neurons of the brain. This fluorescent indicator of calcium, by modifying its fluorescence emission depending upon the activation speed of the neuron action potential, allows to reconstruct the spatial pattern of the neuronal activity at the cellular resolution. In order to record the brain activity, MS was aligned parallelly to the sagittal plane of the zebrafish brain (see Figure 4 and Methods), so as to generate the excitation with Bessel beams on a wide area of the brain along the medio-lateral direction. By means of an orthogonal detection arm provided with a CMOS camera, the neuronal activity was recorded successfully in several brain regions, placed at different depths in the brain and characterized by different amount of light diffusion, such as Medulla Oblungata, Tectum Opticum and Habenulae. The Zebrafish brain has cells with a typical diameter of 6-7 pm and a degree of neuronal packing almost without extracellular space. Even under these difficult conditions, the developed approach allowed to solve single cells situated in strict proximity and to recover their independent functional signals by means of an automated segmentation analysis pipeline commonly used for this application.

[0141] Methods

[0142] Numerical simulations of the electromagnetic response of the nano-pillars

[0143] The electromagnetic response of the single meta-atom has to be designed such that their space of the phase parameters fulfils the target phase within the selected tolerance error, fixed in the present study as equal to ±2 degrees. In order to foresee the phase accumulated by each nano-pillar and their transmittance the COMSOL Multiphysics® platform was used, widely used for numerical simulations at the finished differences in the time domain (FDTD) of optoelectronic and photonic devices.

[0144] The transmittance and phase results of each meta-atom are described in note S2 and shown in Figure 5.

[0145] Design and manufacturing of metasurfaces

[0146] A metasurface was designed and a layout was created, required for the manufacturing by implementation of a semi-automated MATLAB routing (see note S3, for further details).

[0147] In particular, metasurfaces made of SiNx were manufactured at EPFL Center of MicroNanotechnology (CMi). Over the last years, these characteristics made the SiNx production technology increasingly accessible for the micro-nanomanufacturing of optical components for imaging applications.

[0148] Firstly, a 520 nm-thick layer of silicon nitride (SiNx) was deposited on silicon dioxide (SiO2 - 525 pm di thickness) as material on which the metasurface had to be created. Secondly, a layer of of electrons during the e-beam lithography and as mask for the SiNx etching phase. Subsequently, 200 nm of hydrogen silsesquioxane (HSQ) were deposited on the substrate as positive resist for the e-beam lithography by spin coating procedure. The structure of the metasurfaces was then created on the photoresist after developing the latter and then transferred on the Cr layer and on the SiNx film by a first ionic beam etching phase (I BE) followed by an etching phase with high density inductively coupled plasma (ICP) based upon fluorine chemistry (CHF3 / SF6). At last, the Cr residue and photoresist was removed by an acid solution. The main manufacturing phases are illustrated in the note S4 and in Figure 7. The images of the manufactured chips are shown in Figure 3b-c, in Figure 6 and in Figure 7.

[0149] Characterization of the light sheet of the lattice del Bessel beam based upon the metasurface

[0150] The light sheet constituted by the lattice 1 D of Bessel beams was measured by acquiring its profile along the propagation direction. For these characterization measurements, an Olympus 20x objective was mounted on a kinematic support, in turn fixed on a one-axis motorized micropositioner (PT1 / M-Z8, Thorlabs). The objective was oriented parallelly to the propagation radius and adjusted so as to focus the chip surface which was defined as zero propagation position, z = 0. The chip of the metasurfaces was fixed in the vertical slot of a rotating support implemented for the purpose and mounted on a three-axis micropositioner to align its chip surface perpendicularly to the incoming beam. The profiles of the light sheet with Bessel beams were acquired in different z propagation positions, from z = 500 pm up to z = 3500 pm. The thickness of the lighting profile was then estimated by measuring the whole width at high height (FWHM) of the intensity profile Y at a depth comprised between 1500 pm and 2200 pm (Figure 3f).

[0151] Microscope based upon metasurfaces

[0152] The devised chip of metasurfaces was integrated in a customized vertical microscope thanks to two mechanical supports devised to adapt to the required physical constraints imposed by the chip of the metasurfaces, by the sample support and by the imaging objective (Figure 4). The support of the chip of the metasurface can be easily screwed in a micropositioner XYZ (a three-axis micropositioner, MBT616D / M, Thorlabs) and it allows to align the chip of the metasurfaces with the incoming beam. The sample support is mounted on a second micropositioner identical to the preceding one to adjust the sample sagittal plane parallel to the chip surface. The light emitted by the sample was collected by the objective lens (MPLAN APO2020x / NA0.42 Mitutoyo) and focused by a second lens (fTL = 200 mm) on the sensor of the camera CMOS (XIMEA, MQ013RG-ON). As most light sheet microscopes, even the LLS microscope based on metasurfaces shown in this work is constructed on two arms for providing an optical path of lighting independent from that of imaging.

[0153] S1. Design of the phase profile of a lattice of Bessel beams

[0154] The first approach (method-1) for the generation of a phase profile for forming a lattice of Bessel beams is based upon the following steps.

[0155] Step (1): a phase profile j of a converging lens with radius Rj and focal length fj are defined: wherein j is a natural number: j = 1, 2, .. N corresponding to the beam index and N corresponds to the total number of beams in the lattice. Pj and Qj are two spatial parameters which define the beam position j. The values of Rj and fj determine the focus position and the real numerical aperture of the beam and, then, even the maximum thickness of the beam and the minimum focus depth of the beam.

[0156] Step (2): the mask is created defined by:

[0157] Such mask allows the incoming light to pass through the ring as defined by the external radius and thickness . The radius of the ring must have the same value of the radius as defined in step 1.

[0158] The narrower the ring is, that is the smaller the narrower the central peak of the beam is and the longer its focus depth is.

[0159] Step (3): the phase profile is multiplied by the mask so as to obtain the phase profile:

[0160] Step (4): the field distribution of an electromagnetic wave is calculated, characterized by the phase profile equal to the phase profile obtained in step (3) and with width given by wherein is a real number greater than zero.

[0161] Step (5): Step (1) to Step (4) are repeated for a number N of times corresponding to the wished number of beams in the lattice. At each iteration j, the phase profile can be as defined with a different length and radius and the beam position can be translated by the wished parameters Pj and Q}-

[0162] Step (6): the total phase profile is calculated as the argument of the sum of the fields each one corresponding to a Bessel beam translated by Therefore, the final phase profile is is the number of total beams.

[0163] The second approach (method-2) for generating the phase profile of a lattice of Bessel beams is based upon the following fundamental steps:

[0164] Step (1) to Step (3) as for method- 1.

[0165] Step (4): Steps from step (1) up to step (3) are repeated N times to calculate all phase rings of the wished beams wherein N corresponds to the number of beams in the lattice. At each iteration j, the phase profile of each ring can be as defined with the wished focal length , a thickness and radius of the ring and its position can be translated by the wished parameters Step (5): the total phase profile corresponds to the phase values of the single rings wherein in all points (x,y) of the codomain belonging to more than one ring always the phase value of the ring of lowest index is assigned. The phase profile in this case is defined as: wherein j is a natural number less than or equal to N, as defined by the relation wherein are as defined as in 1) e 2), and k is a support index, natural number and as

[0166] S2. Numerical simulation for studying the response of the nano-pillars

[0167] In order to engineer the nano-pillars and to extract the transmittance and phase thereof, their electromagnetic response was studied with the COMSOL Multiphysics module (version 5.5) which solves the Maxwell equation: wherein is the relative permittivity, n and k are respectively the real portion and the imaginary one of the refractive index is the wave number of the free space. Each nano-pillar unit of silicon nitride was simulated with a periodic boundary condition along the transverse direction with respect to the light propagation, an ideal absorption layer (perfectly matched layer PML) at the extremes and boundary conditions of the input / output ports along the longitudinal direction. Then their transmission T and phase of the electric field were extracted from the scattering parameter S21of the matrix S' - parameter, measured by the expansion of the proper mode of the electromagnetic field at the output port 2 (see Figure 5a). Therefore, if are models of electric field of the fundamental modes on the port i 1 , 2, 3, ..., and by assuming that the fields are normalized with respect to the integral of the power flow through each transverse section port, the outgoing calculated field is: wherein

[0168] We analysed the geometrical parameters of the nano-pillars, until finding a set of nano-pillars with lateral lengths D, height h and period p, which could provide a phase covering of 2TT at a working wavelength of 478 nm and with an average transmittance higher than 50%. As described in Figure 5, we selected a set of nanopillars with square section having a height h = 520 nm, a period p = 400 nm and one side D going from D = 60 nm up to D = 340 nm with a lateral pitch of 10 nm.

[0169] It is worth noting that we inverted the final sign of the output phase extracted from our COMSOL simulation. The phase was inverted since in COMSOL the phase convention is based upon:

[0170] Therefore, whereas in the present design an increase in the phase value designates a delay (delay accumulation) in the phase profile, in COMSOL a phase increase introduces an anticipation of the electric field.

[0171] S3. Design of metasurface (MS) and numerical simulation of the beam propagation of MS

[0172] The design of the chips of metasurfaces (MS) is based upon a semi-automated routine written in MATLAB. The code is structured in four main phases. In the first phase, the software loads as input the COMSOL data related to the phase and width of the field outgoing from the nanopillars. Then, one calculates the analytical phase profile and the numerical aperture (NA) for a metasurface characterized by the input parameters (for example focal length, radius, period) set by the user. The two-dimensional phase profile (2D) is then digitalized (second phase) wherein the different geometries of the nanopillars are distributed on the MS surface. At this point (third passage) the software generates the metasurface layout and calculates the phase error. In the last phase, we implemented a radiation propagation method (BPM) to recover the real profile of the beam structured by the MS and to extract the position of its focus, its FWHM, its focus depth and its effectiveness.

[0173] Figure 6 shows the layouts of a metasurface designed to structure a beam as a lattice sheet of Bessel beams. Such layout is obtained starting from the geometries of nano-pillars of silicon nitride with square section described previously. The layout is obtained starting from the design of a converging lens with focal length f = 2000 pm and diameter of 200 pm, for an operating wavelength of 478 nm.

[0174] S4. Manufacturing and characterization di metasurface The manufacturing of the metasurfaces made of silicon nitride requires the electronic beam lithography technology to obtain a resolution up to 5 nm. Figure 7 and table 1 describe the main phases of the manufacturing process flow. The surface of the chip of metasurface was inspected by both optical and scan electronic microscopy, as shown in Figure 8 and in Figure 9.

[0175] The MS characterization was performed systematically by using a semi-automated procedure and the experimental setup described in Figure 10. The profile of the beam structures by MS was acquired along the axis of the beam propagation at intervals of 2 pm by using pManager1to monitor both the position z of the micropositioner and the camera. Then, the beam profile was reconstructed with a script based upon MATLAB which we wrote to measure the main parameters of the beam.

[0176] Table 1 - Process for manufacturing the metasurface Table 2 - Experimental conditions and localization results

[0177] 1. Edelstein, A., Amodaj, N., Hoover, K., Vale, R. & Stuurman, N. Computer Control of Microscopes Using pManager. Curr. Protoc. Mol. Biol. 92, 14.20.1-14.20.17 (2010).

[0178] 2. Suite2p: beyond 10,000 neurons with standard two-photon microscopy | bioRxiv. https: / / www.biorxiv.Org / content / 10.1101 / 061507v2.

Claims

CLAIMS1. A computer-implemented method for generating a phase profile of a lattice ofindependent Bessel beams, suitable for configuring an optical element, which method comprises the following steps: a) providing as input a phase profile of a converging lens with radius Rj and focallength represented by the following formula:wherein: j is a natural number ranging from 1 , 2, up to N and corresponds to the index of a single Bessel beam in the lattice;N corresponds to the total number of Bessel beams in the lattice;Pj and Qj are spatial parameters which define the position of a single Bessel beam of index j in the lattice; and the and values determine, respectively, the focus position and the numerical apertureof a single Bessel beam of index j in the lattice;b) providing as input a ring phase mask represented by the following formula:wherein: are as defined in a),represents the external radius of the ring; represents the thickness of the ring; and wherein the radius of the ring has the same value of Rj as defined in in stepa); c) multiplying the phase profile provided in step a) by the phase mask providedin b) so as to obtain a phase profile with a ring shape represented by the followingformula:wherein for each index are as defined asin a) and b); andwherein said phase profile reproduces the intensity profile of a single Bessel beamof index j; and d) determining said total phase profile lattice starting from the phase profilesobtained in c).

2. The computer-implemented method according to claim 1, wherein said step d) comprises the following steps: e) calculating the field distribution of an electromagnetic wave represented by the formula wherein corresponds to a phase profile equal to the phase profileobtained in step c) and corresponds to the amplitude represented by the followingformula:wherein is a real number greater than zero and wherein for each j index, Pj, Qj, Rj areas defined in the preceding claim; f) repeating each one of the steps a)-c) and e) for a number N of times corresponding to the number of beams of the lattice; wherein, at each iteration of the index j, said lengthand said radius Rj have different values, and wherein, at each iteration of the index j, said spatial parameters Pj and Qj have different values such that the position of each beam of index j in the lattice results to be translated with respect to that of a beam of different index j; and g) calculating said total phase profile as the argument of the sum of the fields Ej eachone corresponding to a Bessel beam translated by the values n the lattice,according to the following formula:wherein j = 1, 2, up to N and N is the number of total beams in the lattice.

3. The computer-implemented method according to claim 1, wherein said step d) comprises the following steps: e’) repeating said steps a)-c) N times to calculate a number of phase profileswherein N corresponds to the number of beams in the lattice; wherein, at each iteration of the index J, each phase profile is defined with focal length a thickness and radiusof the ring equal or different and wherein, at each iteration of the index j, said spatial parameters Pj e Qj have different values such that the position of each beam of index j in the lattice results to be translated with respect to that of a beam of different index j; and f’) determining said total phase profile as:wherein j is a natural number less than or equal to N, as defined by the relation:wherein are as defined as in a) and b) of any one of the precedingclaims, and k is a natural number less than or equal to N.

4. The computer-implemented method according to any one of claims 1 to 3, wherein said lattice comprises a number N of Bessel beams comprised between 1 and 10.

5. A computer program configured for generating a phase profile of a lattice of Besselbeams, comprising a list of instructions which, when executed on an electronic computer, implement the steps of a method according to any one of claims 1 to 4.

6. A method for preparing an optical element configured for generating a lattice of independent Bessel beams when struck by a beam of plane electromagnetic waves emitted by a light source, comprising the following steps: a) generating a total phase profile of a lattice of independent Bessel beams byperforming the steps of a computer-implemented method according to any one of claims 1to 5; b) encoding or making the total phase profile generated in a) on a surface of a flatoptical element.

7. The method according to claim 6, wherein said step b) comprises converting said total phase profile in a discrete phase map of a metasurface on said surface of the flatoptical element.

8. The method according to claims 6 or 7, wherein said step b) comprises manufacturing on said surface of the flat optical element, a metasurface comprising a matrix of nanostructures having dimensions smaller than a wavelength of the beam of plane electromagnetic waves emitted by the light source; wherein said nanostructures have predetermined shapes and arrangement in such a way as to impart to the beam of emitted waves incident on them a total phase profile represented by9. The method according to claim 8, wherein said nanostructures comprise nanopillars having a square cross-section.

10. The method according to claim 8 or 9, wherein said nanostructures have a variable lateral length D comprised between 60 and 340 nm, in steps of 10 nm.

11. The method according to any one of claims 8 to 10, wherein said nanostructures have a height h represented by the following formula:wherein A is the wavelength of said beam of plane electromagnetic waves emitted by the light source, and n is the refractive index of a dielectric material selected tomake said metasurface.

12. The method according to any one of claims 8 to 11 , wherein said nanostructures are positioned on the surface of said flat optical element with a fixed spatial period represented by the following formula:wherein A is the wavelength of said beam of plane electromagnetic waves emitted by thelight source and NA is the numerical aperture of the converging lens having a phase profile as defined in step a) of a computer-implemented method according to claim 1.

13. The method according to any one of claims 8 to 12, wherein said nanostructures are made of silicon nitride (SiNx).

14. The method according to any one of claims 8 to 13, wherein said manufacturing is carried out by electron beam lithography.

15. The method according to any one of claims 6 to 14, wherein said flat optical element comprises or consists of silicon dioxide.

16. The method according to any one of claims 6 to 15, wherein said flat optical element has a thickness comprised between said h + 100 nm and 600 pm.

17. An optical element configured for generating a lattice of independent Bessel beams, obtainable by a method according to any one of claims 6 to 16.

18. An optical system for generating a lattice of independent Bessel beams, comprising (i) a light source configured for emitting a beam of plane electromagnetic waves and (ii) an optical element according to claim 17.

19. The optical system according to claim 18, wherein said plane electromagnetic waves have a wavelength comprised between 380 and 750 nm.

20. A method for generating a lattice of independent Bessel beams comprising the following steps: a) providing a light source configured for emitting a beam of plane electromagnetic waves; b) directing said beam of electromagnetic waves through an optical element according to claim 17.

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