WO2019207438A1

WO2019207438A1 - Optical device for demultiplexing and multiplexing modes with different orbital angular momentum

Info

Publication number: WO2019207438A1
Application number: PCT/IB2019/053233
Authority: WO
Inventors: Filippo Romanato; Gianluca Ruffato
Original assignee: SM Optics S.r.l.; Universita' Degli Studi Di Padova
Priority date: 2018-04-26
Filing date: 2019-04-18
Publication date: 2019-10-31

Abstract

It is disclosed an optical device (10-5, 10-3) for demultiplexing modes with different orbital angular momentum. The device comprises a substrate slab having two surfaces (2- 112, 2-126; 2-12, 2-26) at least partially facing each other and comprises two films arranged on the two surfaces respectively, wherein at least one of the two films includes a first diffractive optical zone (2-112a, 2-12a) and a second diffractive optical zone (2-126a, 2-12b). The first zone (2-112a, 2-12a) is configured to receive a multiplexed input free-space optical beam carrying a plurality of orbital angular momentum modes (OAM1, OAM2) with different orbital angular momentum, is configured to perform a conformal mapping of the multiplexed input free-space optical beam from a circular distribution to a linear distribution of luminous intensity, generate therefrom a plurality of unwrapping free-space optical beam, split the plurality of unwrapping free-space optical beams into corresponding groups of copies and generate therefrom a plurality of groups of copies of unwrapping free-space optical beams having a propagation direction which is tilted with respect to a propagation direction of the multiplexed input free-space optical beam. The substrate is configured to propagate the tilted plurality of group of copies of unwrapping free-space optical beams towards the second zone and generate therefrom a corresponding propagated plurality of groups of copies of unwrapped free-space optical beams. The second zone (2-126a, 2-12b) is configured to receive the propagated plurality of groups of copies of unwrapped free-space optical beams, perform a phase correction of the propagated plurality of groups of copies of unwrapped free- space optical beams and generate therefrom a plurality of phase-corrected free-space optical beams having different directions in the space depending on the orbital angular momentum of the plurality of received orbital angular momentum modes.

Description

Optical device for demultiplexing and multiplexing modes with different orbital angular momentum

DESCRIPTION

TECHNICAL FIELD OF THE INVENTION

The present invention generally relates to the field of optical communications.

More in particular, the invention concerns an optical device for demultiplexing and multiplexing modes with different orbital angular momentum and a mode division optical communication system comprising the demultiplexing and/or multiplexing optical device. PRIOR ART

During the last decade, Space Division Multiplexing (SDM) has experienced an upsurge of research interest, both in academia and industry, as a possible means to address the ever increasing worldwide demand for bandwidth.

In particular, an implementation of SDM exploiting a set of mutually orthogonal spatial modes, in the so-called Mode Division Multiplexing (MDM), has been considered both for free-space and guided propagation, in which independent channels are carried by coaxially propagating and spatially overlapping modes with the same frequency, therefore improving the spectral efficiency and information capacity of the optical link, proportionally to the number of modes transmitted.

Among all the different families of orthogonal modal basis, beams carrying orbital angular momentum (OAM) of light have been demonstrated to provide promising candidates for MDM in the optical range, both in free-space and optical fiber propagation.

Beams carrying OAM present a characteristic azimuthal phase term, exp(/ <p), being f the azimuthal coordinate on a plane orthogonal to the propagation direction, and the orbital angular momentum per photon in units of ft.

A pivotal stage of an optical link based on OAM-MDM is that of (de)multiplexing, i.e., how to form a collimated bunch of orthogonal OAM modes at the source and how to sort them according to their OAM content at the receiver after propagation.

Various techniques have been described and implemented in order to sort a set of multiplexed optical beams with different OAM values, including interferometric methods, time-division techniques, integrated silicon photonics, coherent detection, OAM-mode analysers, astigmatic-mode converters, transformation optics, and rotational Doppler effects.

Among all, one of the most effective methods is represented by transformation optics, mapping (conformally) angular momentum to linear momentum. This involves a unitary transformation converting the azimuthal phase gradients of OAM beams into linear phase gradients (tilted beams), which are then mapped to unique spatial positions by means of a Fourier lens.

The mapping is executed by two optical elements in sequence: the first performing a log-pol coordinate transformation and the second correcting the introduced phase distortion.

The log-pol geometric optical transformation is described in the article of G.C.G. Berkhout, M.P.J. Lavery, J.Courtial, M.W.Beijersbergen, M.J, Padgett,“Efficient sorting of orbital angular momentum states of lights”, in Phys. Rev. Lett. 105, 153601 -1 -4 (2010).

This method has been widely used as a sorting technique, for example, in recent telecom experiments both in the classical and quantum regimes.

In its first realization, spatial light modulators (SLMs) were exploited to implement the two elements, which were subsequently replaced by refractive optical components for efficiency reasons.

More recently, diffractive versions exhibiting a higher level of compactness and miniaturization have been realised (see the article of Ruffato G., Massari M. and Romanato F., Compact sorting of optical vortices by means of diffractive transformation optics. Opt. Let 42 (3), 551 -554 (2017) ).

The same setup has been demonstrated to perform multiplexing, with the two elements in reverse order (see the article of Ruffato G., Massari M., Parisi G. and Romanato F., Test of mode-division multiplexing and demultiplexing in free-space with diffractive transformation optics. Opt. Express 25, 7859-7868 (2017).).

A drawback of this demultiplexing technique is represented by the overlap between neighbouring modes, an unavoidable feature of the design, which is detrimental to the inter channel cross-talk of the communication system. This can be overcome by using a sparse mode space, but at the expense of discarding many channels included in the sorting bandwidth of the system.

Therefore, the measurement bandwidth of the sorter, which is proportional to the Fresnel number of the optics, should be increased in order to provide a sufficient number of modes after channel selection: this is achievable for instance by either decreasing the focal length or increasing the size of the first element performing optical transformation (see again the article of Ruffato G., Massari M. and Romanato F., Compact sorting of optical vortices by means of diffractive transformation optics. Opt. Lett. 42 (3), 551 -554 (2017) ).

On the other hand, applications in optical fibers could prescribe severe limitations to the number of supported OAM modes, i.e. the maximum value, and the selection of non- consecutive OAM values could dramatically reduce the number of available channels. An alternative solution consists in including a fan-out element, which creates multiple copies extending the phase gradient of the sorted beam.

The optical fan-out element is described in the article of Prongue D., Herzi, H. P. and Gale M. T., Optimized kinoform structures for highly efficient fan-out elements. App. Optics. 31 , 5706-571 1 (1992).

Referring to Figure 1 , it shows an optical device 1 10 for demultiplexing OAM modes according to the prior art.

The demultiplexing optical device 1 10 includes four optical elements 102-1 , 102-2, 102-3, 102-4 implementing the log-pol geometric optical transformation combined with the optical fan-out element.

In particular, the optical elements 102-1 , 102-2, 102-3, 102-4 have the following functions:

the first optical element 102-1 performs the unwrapper;

the second optical element 102-2 performs the phase correction of the unwrapper; the third optical element 102-3 performs fan-out copies;

the fourth optical element 102-4 performs the phase correction of the fan-out copies.

The fan-out element creates multiple copies extending the phase gradient of the sorted beam, which is focused as for the log-pol geometric optical transformation but with a narrower width, thus improving the separation between spots. This reduces the channel overlap without sacrificing modal density, but comes at the cost of increased size and complexity of the system.

In its first realization, the fan-out element and the corresponding phase-corrector were realised using spatial light modulators (SLMs) and placed following the two-piece log- polar optics, for a total of at least four optical elements, plus lenses in-between for the Fourier transform.

The possibility to integrate the two optical operations (i.e. unwrapper and phase correction) into a single optic has been demonstrated by means of a spatial light modulators (see the article of Wan, C., Chen, J., and Zhan, Q. Compact and high-resolution optical angular momentum sorter. APL Photonics 2, 031302-1 -6 (2017)), thus prohibiting compactness and also the manufacturing process would be very difficult and expensive to implement.

Therefore, while these solutions are satisfactory for laboratory tests, they are not suitable for industrial implementation in a real-world optical communication set-up.

SUMMARY OF THE INVENTION

The present invention relates to a demultiplexing and multiplexing optical device for demultiplexing/multiplexing modes with different orbital angular momentum as defined in the enclosed claims 1 and 10 and by their preferred embodiments disclosed in the dependent claims 2 to 9 and 1 1 to 15.

According to a first aspect of the invention, the demultiplexing optical device comprises:

- a substrate slab having two surfaces at least partially facing each other,

- two films arranged on the two surfaces respectively, wherein at least one of the two films includes a first diffractive zone and a second diffractive zone,

and wherein:

the first zone is configured to:

^■ receive a multiplexed input free-space optical beam carrying a plurality of orbital angular momentum - OAM - modes with different orbital angular momentum;

^■ perform a conformal mapping of the multiplexed input free-space optical beam from a circular distribution to a linear distribution of luminous intensity, generate therefrom a plurality of unwrapping free-space optical beam, split the plurality of unwrapping free-space optical beams into corresponding groups of copies and generate therefrom a plurality of groups of copies of unwrapping free-space optical beams having a propagation direction which is tilted with respect to a propagation direction of the multiplexed input free-space optical beam;

the substrate is configured to propagate the tilted plurality of groups of copies of unwrapping free-space optical beams towards the second zone and generate therefrom a corresponding propagated plurality of groups of copies of unwrapped free-space optical beams;

the second zone is configured to receive the propagated plurality of groups of copies of unwrapped free-space optical beams, perform a phase correction of the propagated plurality of groups of copies of unwrapped free-space optical beams and generate therefrom a plurality of phase-corrected free-space optical beams having different directions in the space depending on the orbital angular momentum of the plurality of received OAM modes.

According to a second aspect of the invention, the demultiplexing optical device comprises a substrate slab having a first and a second surface at least partially facing each other, the substrate slab comprising a first and a second film arranged on the first and second surfaces respectively, wherein the first film includes a first diffractive zone and the second film includes a second diffractive zone, and wherein:

the first zone on the first film is configured to:

^■ perform a conformal mapping of the multiplexed input free-space optical beam from a circular distribution to a linear distribution of luminous intensity, generate therefrom a plurality of unwrapping tree-space optical beam, split the plurality of unwrapping tree-space optical beams into corresponding groups of copies and generate therefrom a plurality of groups of copies of unwrapping tree-space optical beams having a propagation direction which is tilted with respect to a propagation direction of the multiplexed input free-space optical beam;

the substrate is configured to propagate the tilted plurality of groups of copies of unwrapping free-space optical beams towards the second zone on the second film and to generate therefrom a corresponding propagated plurality of groups of copies of unwrapped free-space optical beams;

the second zone on the second film is configured to receive the propagated plurality of groups of copies of unwrapped free-space optical beams, perform a phase correction of the propagated plurality of groups of copies of unwrapped free-space optical beams and generate therefrom a plurality of phase-corrected free-space optical beams having different directions in the space depending on the orbital angular momentum of the plurality of received OAM modes.

According to a third aspect of the invention, the optical device for demultiplexing modes with different orbital angular momentum comprises a substrate slab having a first and a second surface at least partially facing each other, the substrate slab comprising a first and a second film arranged on the first and second surfaces respectively,

wherein the first film includes a first diffractive zone and a second diffractive zone which are arranged side-by-side on the first film on the first surface,

wherein the second film includes an optical reflecting zone,

wherein the first zone is configured to:

^■ receive a multiplexed input free-space optical beam carrying a plurality of orbital angular momentum - OAM -modes with different orbital angular momentum;

wherein the substrate is configured to propagate the tilted plurality of groups of copies of unwrapping free-space optical beams towards the optical reflecting zone of the film on the second surface and generate therefrom a corresponding plurality of propagated groups of copies of unwrapping free-space optical beams,

wherein the optical reflecting zone is configured to receive the plurality of propagated groups of copies of unwrapping free-space optical beams, reflect them off-axis towards the second zone on the first film and generate therefrom a plurality of reflected groups of copies of unwrapping free-space optical beams,

wherein the substrate is configured to propagate the plurality of reflected groups of copies of unwrapping free-space optical beams and generate therefrom a corresponding plurality of reflected groups of copies of unwrapped free-space optical beams,

wherein the second zone on the first film is configured to receive the plurality of reflected groups of copies of unwrapped free-space optical beams, perform a phase correction of the plurality of reflected groups of copies of unwrapped free-space optical beams and generate therefrom a plurality of phase-corrected free-space optical beams having different directions in the space depending on the orbital angular momentum of the plurality of received OAM modes.

According to a fourth aspect of the invention, the optical device for demultiplexing modes with different orbital angular momentum comprises a substrate slab having two surfaces at least partially facing each other, wherein at least one of the two surfaces includes a first diffractive zone and a second diffractive zone,

and wherein:

the first zone is configured to:

The Applicant has perceived that the demultiplexing/multiplexing optical device according to the present invention has the following advantages:

it increases the compactness and the degree of miniaturization;

it simplifies the alignment process, which is self-aligning;

it has a high-resolution;

it reduces the manufacturing time and the costs of production;

it simplifies the manufacturing process.

A further object of the present invention is a mode division optical communication system as defined in the enclosed claim 16 and by its preferred embodiments disclosed in the dependent claim 17.

It is also an object of the present invention a process for manufacturing an optical device for demultiplexing modes with different orbital angular momentum according to the first, second, third and fourth aspect of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

Further characteristics and advantages of the invention will emerge from the following description of a preferred embodiment and variants thereof, said description being provided by way of example with reference to the enclosed drawings, wherein:

Figure 1 shows an optical device for demultiplexing OAM modes according to the prior art;

Figure 2A shows an optical device for demultiplexing OAM modes according to an embodiment of the invention;

Figure 2B shows a first specific implementation of the demultiplexing optical device of a first embodiment of the invention of Figure 2A;

Figure 2C shows a second specific implementation of the demultiplexing optical device of the first embodiment of the invention of Figure 2A;

Figure 3 shows a comparison of the output intensity between the demultiplexing optical device of the embodiment of the invention and the demultiplexing optical device of the prior art;

Figure 4A-4B show a prospective view of a third specific implementation of the demultiplexing optical device of the first embodiment of the invention performing demultiplexing of guided OAM modes with a different orbital angular momentum; Figure 4C shows a top view of the third implementation of the demultiplexing optical device of the first embodiment of the invention;

Figures 4D-4F show a variant of the third specific implementation of the demultiplexing optical device according to the first embodiment of the invention performing demultiplexing of guided OAM modes with a different orbital angular momentum and a different state of polarization;

Figure 5A-5B show a third specific implementation of a multiplexing optical device according to the first embodiment of the invention performing multiplexing of guided OAM modes with different orbital angular momentum;

Figure 5C-5E show a variant of the third specific implementation of a multiplexing optical device according to the first embodiment of the invention performing multiplexing of guided OAM modes with a different orbital angular momentum and a different state of polarization;

Figure 6A shows eight levels surface of the substrate of a diffractive optical element and Figure 6B shows a 3D surface-relief pattern of the diffractive optical element for performing dynamic phase control;

Figure 7A shows eight spatial rotations of a Pancharatnam-Berry optical element and Figure 7B shows a binary pattern of spatially rotated sub-wavelength gratings for geometric phase generation;

Figure 8 shows an optical element performing a focusing fan-out unwrapper used in the second and third specific implementation of the demultiplexing optical device according to the embodiment of the invention;

Figures 9A-9B show a prospective view of a specific implementation of the demultiplexing optical device of a second embodiment of the invention performing demultiplexing of guided OAM modes with a different orbital angular momentum;

Figures 9C-9D shows a variant of the specific implementation of the demultiplexing optical device of the second embodiment of the invention performing demultiplexing of guided OAM modes with a different orbital angular momentum and a different state of polarization;

Figure 10A-10B shows a specific implementation of the multiplexing optical device according to the second embodiment of the invention performing multiplexing of guided OAM modes with different orbital angular momentum;

Figure 10C-10D shows a variant of the specific implementation of the multiplexing optical device according to the second embodiment of the invention performing multiplexing of guided OAM modes with a different orbital angular momentum and a different state of polarization; Figure 1 1 shows a substrate slab with alignment markers of the implementation of the demultiplexing optical device of the second embodiment of the invention.

DETAILED DESCRIPTION OF THE INVENTION

It should be noted that in the description that follows identical or similar blocks, components or modules have the same reference numerals, regardless of whether they are shown in different embodiments of the invention

International patent application of the same inventors having filing number PCT/IB2017/055096 filed on August 24, 2017 is incorporated by reference.

PCT/IB2017/055096 discloses (see Figures 4A and 4B) an optical demultiplexing device 2 (and 102) performing the demultiplexing of guided OAM modes with different orbital angular momentum (that is, with different values h, h, of the angular index I).

In particular, the optical demultiplexing device 2 (and 102) implements the log-pol optical transformation with a single diffractive optical element 2-12 (and 2-13) and it further includes a mirror 2-6.

The diffractive optical element 2-12 comprises an outer annular region 2-2a and an inner annular region 2-1 a concentric (i.e. coaxial) with the outer annular region 2-2a, wherein the outer annular region 2-2a of the transmitting type implements the unwrapper of the log- pol geometric optical transformation (i.e. a conformal mapping from a circular distribution to a linear distribution of luminous intensity) and wherein the inner annular region 2-1 a of the transmitting type performs the phase corrector of the log-pol geometric optical transformation.

Figure 4B of PCT/IB2017/055096 differs from Figure 4A in that the internal circular zone 2-1 b of the diffractive optical element 2-13 is of the reflecting type, instead of the transmitting type.

The use of the single diffractive optical element 2-12 (or 2-13) as disclosed in PCT/IB2017/055096 has the advantage to increase the compactness and the degree of miniaturization, reducing the manufacturing time and the costs of production.

Moreover, the use of a single diffractive optical element 2-12 (or 2-13) has the further advantage to simplify the alignment process, because the two optical elements of the log-pol geometric optical transformation are coplanar and aligned structurally.

The optical demultiplexing device 2 (and 102) of PCT/IB2017/055096 has the disadvantage of the overlapping between neighbouring modes, which causes inter-channel cross-talk: this can be solved including a fan-out element performing multiple copies of the sorted beam as explained above, the multiple copies being focused with a narrower width thus providing a larger separation between spots. This reduces the channel overlap without sacrificing density of the OAM modes, but it increases the complexity and size of the system. Referring to Figure 2A, it shows the optical scheme of an optical device 10 for demultiplexing OAM modes according to an embodiment of the invention.

The demultiplexing optical device 10 (also referred to as OAM sorter or demuxer or demultiplexer) comprises a first diffractive optical element 2-1 followed by a second diffractive optical element 2-2.

The first diffractive optical element 2-1 is customized to perform multiple operations at once, performing a log-pol optical transformation, fan-out copying and beam focusing,

The second diffractive optical element 2-2 is a double phase-corrector which adjusts the phase distortions introduced by the unwrapping and fan-out copying operations implemented by the first diffractive optical element 2-1 .

Therefore, high-resolution OAM sorting is performed with two diffractive optical elements: the first 2-1 encoding both the optical operations of unwrapping and fan-out copying, and the second 2-2 performing two phase-corrections.

In other words, the optical layout of the demultiplexing optical device 10 is constituted of a sequence of two optical elements, the fan-out unwrapper 2-1 and the double phase- corrector 2-2.

The known unwrapper performs a conformal mapping of a point (x, y) in the input plane to a point (u, v) in the output plane, where v=a arctan(y/x) and u= -a In (rib), being r=(x²+ ²)^1/2, a and b design parameters, while the corresponding phase-corrector performs correction of the resultant (distorted) phase by taking into account the optical path differences at each point, thus completing the conversion of the input azimuthal phase gradient into a linear one.

After applying the coordinate transformation, the field can be described as a truncated tilted plane wave:

(1 )

where rect{x)= 1 for |x|<1/2, =0 otherwise.

Therefore, by passing this field through a lens with focal length f_T and observing the back focal plane, the tilted plane waves angles are converted into lateral shifts As* proportional to the OAM amount f according to:

As a result, optical beams carrying different amount of OAM are focused at different positions, i.e. they are separated in space; however, owing to the non-null width of these spots, there is an overlap between the neighbouring modes in the output intensity pattern, which can be detrimental when detecting OAM states.

The first diffractive optical element 2-1 implements additional optical operations with respect to the known unwrapper, i.e. fan-out copying and beam focusing.

Therefore, the transmission function of the first diffractive optical element 2-1 is as follows:

and results from the combination of three optical elements.

The first term performs the log-pol optical transformation and is given, according to G.C.G. Berkhout et al., Phys. Rev. Lett. 105, 153601 (2010), by

2 pa ( LL _Ί (

L JW - y arctan — — xin

f L J \

where the two parameters a and b are related to the optical transformation and control the size and the location of the transformed optical beam, respectively.

The parameter a assumes the value Z./2TT, SO that the azimuthal phase gradient is mapped over a length L on the second element in the y-direction, while the position of the unwrapped optical beam in the x-direction is controlled by the parameter b.

The second contribution encodes the fan-out term and is given by:

This term splits the unwrapped optical beam into N copies and locates the several copies of the optical beam side by side on the second optical element 2-2; this is achieved by choosing the spatial frequency carriers according to y_m=mLklf.

The parameters (c_m, <5_m) are optimized for an equal distribution of the input energy over the several copies.

After phase-correction, the field results in:

which after focusing produces an elongated spot located at the same spacing as the known sorter, see for instance in G.C.G. Berkhout et al., Phys. Rev. Lett. 105, 153601 (2010), but exhibiting a width scaled as 11N, therefore reducing the overlap between adjacent spots as shown in Figure 3.

Finally, the lens term provides the focusing of the /V-copies of the unwrapped optical beam on the second optical element 2-2. The second optical element 2-2 performs a phase correction of the field: this element is a double phase-corrector performing the correction of both the log-pol optical transformation and the fan-out operation at the same time.

An analytical formulation of the log-pol phase-corrector exists in the paraxial regime (see G.C.G. Berkhout et al., Phys. Rev. Lett. 105, 153601 (2010)), however the particular choices of focal lengths and beam size in this study require a more precise calculation of the phase patterns beyond the Fresnel regime.

Based on angular spectrum diffraction theory, the rigorous solution of the diffracted field U can be expressed in the convolution algorithm form:

where FT and FT¹ are the Fourier transform and the inverse Fourier transform, respectively, HAS is the angular-spectrum transfer function:

^HAs { fx y ) = ^C P fc- (8)

Then the required phase profile for the phase-correcting term is given by

W₂ (W, V) = 2^ - arctan [lm (t/ ) / Re (t/ )] (9)

This can be calculated numerically for Uo as an input Gaussian mode with a planar phase front and a beam waist properly chosen in order to illuminate the zone of interest of the first optical element 2-1.

In addition, the second optical element 2-2 performing the phase-correction can be endowed with a tilt term to prevent the optical beam from overlapping with a possible zero- order contribution.

OAM sorters are difficult to align and the dual phase-corrector 2-2 of the embodiment of Figure 2A requires a more precise alignment, making it difficult to obtain output optical beams of good quality unless the two optical elements 2-1 , 2-2 are perfectly planar, coaxial and aligned one to each other.

In order to simplify the alignment process, the two optical elements 2-1 , 2-2 are incorporated into a single optical element 2-1.2 as shown in the demultiplexing optical device 10-1 of Figure 2B, which is a first specific implementation (i.e., a mechanical layout) of a first embodiment of the demultiplexing optical device 10 of Figure 2A.

The term“implementation” (referred to the demultiplexing optical device 10-1 ) means the definition not only of the optical layout for performing OAM demultiplexing (as shown in Figure 2A above), but also the specific mechanical layout implementing the corresponding optical layout for OAM demultiplexing. The demultiplexing optical device 10-1 includes the single optical element 2-1.2 and a reflecting optical element 2-6 (e.g. a mirror).

The two elements 2-1.2, 2-6 can be either physically separated as shown in Figure 2B; alternatively, the optical element 2-1 .2 and the reflecting optical element 2-6 can constitute two opposite surfaces of a single substrate slab.

A multiplexed input OAM optical beam carrying two OAM modes OAM1 , OAM2 impinges on the outer annular region 2-1 2a of the single diffractive optical element 2-1 .2 that encodes the focusing fan-out unwrapper.

Subsequently, the reflecting optical element 2-6 reflects in-axis the optical beam towards the diffractive optical element 2-1.2 again through its inner zone 2-1.2b, wherein double phase-correction is performed.

It is worth noting that the reflecting optical element 2-6 reflects the impinging optical beam in-axis, i.e. the reflected optical beam illuminates a first area of the optical element 2- 1 .2 which is included (or mainly overlapping) into a second area of the optical element 2-1.2 illuminated by the multiplexed optical beam impinging directly on the optical element 2-1 .2, so that there is overlap between the optical beam impinging directly on the optical element 2- 1.2 and the optical beam impinging on the optical element 2-1.2 after reflection from the reflecting optical element 2-6.

A beam-splitter 2-4 is used to separate input and output beams.

A focusing lens 2-5 performing a Fourier-transform completes the demultiplexing.

The integration hinges on the fact that OAM optical beams have a doughnut-like intensity profile around a central null: since the optical element 2-1 .2 acts basically on the zone with non-zero input field, the first transformation leaves unexploited the inner region 2- 1 2b of the optics.

Therefore, said central zone 2-1.2b can be selected for integrating the second element 2-2 performing phase-correction.

The optical element 2-1.2 is illuminated twice: after crossing the outer unwrapping zone, the optical beam is back- reflected with a mirror and impinges on the inner central region providing the phase-corrector (see Figure 2B).

Therefore, the total diffractive phase pattern QDOE of the optical element 2-1 .2 turns out to be the composition of the two phase functions in eq. (3) and eq. (9):

being p^ the outer radius of the optics, p_å the radius of the central part, Q the Fleaviside function (Q(c)=1 for x>0, Q(c)=0 otherwise), providing the condition p2>/Vna is fulfilled.

The first implementation of Fig.2B makes the alignment operation significantly easier, since the two zones 2-1.2a, 2-1.2b are by-design aligned, parallel and coaxial one to each other; however, this imposes limitations on the input-beam radius and on the lateral extension of the phase-corrector, i.e. the number of copies N.

Moreover, the first implementation of Figure 2B has a higher resolution than a traditional sorter, meaning that it increases the capability of the optical system to spatially separate OAM modes. This is crucial for the design and implementation of optical devices performing OAM-mode division multiplexing with low values of cross-talk. Due to the inherent high superposition between nearest neighbouring OAM channels, the traditional sorter exhibits a too-high inter-channel cross-talk. In order to reduce this effect, a sparse mode set of modes must be chosen, that is considering OAM modes separated by an amount of OAM greater than 2 (as shown in Ruffato G., Massari M. and Romanato F., Compact sorting of optical vortices by means of diffractive transformation optics. Opt. Lett. 42 (3), 551 -554 (2017)). The new design of the first implementation reduces the channel overlap and therefore the cross-talk without sacrificing the density of the OAM modes.

In order to overcome the above indicated limitations of the first implementation of Figure 2B, the two optical elements 2-1 and 2-2 of the single diffractive optical element 2-1.2 are arranged side-by-side on a same surface of the substrate, as shown in the demultiplexing optical device 10-2 of Figure 2C, which is a second specific implementation (i.e., a mechanical layout) of the first embodiment of the demultiplexing optical device 10 of Figure 2A.

The term“implementation” (referred to the demultiplexing optical device 10-2) means the definition not only of the optical layout for performing OAM demultiplexing (as shown in Figure 2A above), but also the specific mechanical layout implementing the corresponding optical layout for OAM demultiplexing.

The demultiplexing optical device 10-2 is implemented with a substrate slab comprising a surface 2-20.

The substrate slab of the demultiplexing optical device 10-2 is made for example of a glass material (i.e. a glass slide) transparent for the visible wavelengths or silicon material transparent for the infra-red wavelengths.

The demultiplexing optical device 10-2 further comprises a reflecting optical element 2-7 (e.g. a mirror) and a focusing lens 2-25, which can be separate with respect to the surface 2-20 (as shown in Figure 2C) or it can be embedded thereon.

More in particular, the substrate slab of the demultiplexing optical device 10-2 comprises a first zone 2-20a and a second zone 2-20b which are arranged side-by-side on the surface 2-20, wherein:

the first zone 2-20a performs the log-pol optical transformation (i.e. unwrapping operation) and the fan-out copy operation, thus it is referred as“fan-out unwrapper”; the second zone 2-20b performs the double phase correction, i.e. performing phase correction of both the log-pol optical transformation and the fan-out copy.

The use of a same substrate for the first and second zones 2-20a, 2-20b provides the advantage that a single manufacturing technique can be used, thus simplifying the manufacturing process, reducing the manufacturing time and the costs of production, said manufacturing technique being for example Electron beam lithography of a resist layer, or Reactive Ion Etching of a silicon substrate using an EBL fabricated mask, or a replica process of a pre-fabricated master, which will be explained better afterwards.

The operation of the demultiplexing optical device 10-2 is the following:

the tip of a multimode optical fibre generates an output optical signal carried by a multiplexed guided OAM mode carrying a first OAM mode (OAM1 ) having a first angular index h and a second OAM mode (OAM2) having a second angular index /?;

a multiplexed free-space OAM optical beam is generated by the output optical signal and propagates in free space, said multiplexed free-space OAM optical beam carrying a first free-space optical beam (OAM1 ) having a first angular index h and a second free-space optical beam (OAM2) having a second angular index /?;

the multiplexed free-space OAM optical beam impinges on the first zone 2-20a of the surface 2-20 encoding the fan-out unwrapper, then the first zone 2-20a transmits two groups of a plurality N of unwrapping free-space optical beams towards the reflecting optical element 2-7, i.e. the first group of N-unwrapping free-space optical beams associated to the first angular index h and the second group of N-unwrapping free-space optical beams associated to the second angular index /?;

during propagation towards the reflecting optical element 2-7, the two groups of N- unwrapping free-space optical beams unwrap;

the two groups of N-unwrapping free-space optical beams (generated by the first zone 2-20a of the surface 2-20) impinge on the reflecting optical element 2-7, which reflects off-axis the two groups of N-unwrapping free-space optical beams towards the second zone 2-20b of the surface 2-20;

the two groups of reflected N-unwrapped free-space optical beams impinge on the second zone 2-20b of the surface 2-20, which performs phase-correction and transmits two phase-corrected free-space optical beams having linear intensity distribution,

the two phase-corrected free-space optical beams starts to separate along a transverse direction according to their OAM content, i.e. a first phase-corrected free-space optical beam starts to have a first direction in the space depending on the first angular index h, while a second phase-corrected free-space optical beam starts to have a second direction (different from the first direction) in the space depending on the second angular index . The separate focusing lens 2-25 performs a Fourier-transform of the two phase- corrected free-space optical beams (having different directions) into two parallel axially- shifted optical beams, thus completing the demultiplexing of the two input OAM modes OAM 1 , OAM2.

Alternatively, the focusing lens 2-25 can be realized in a diffractive form and it can be embedded into the second zone 2-20b of the surface 2-20.

Preferably, the demultiplexing optical device 10-2 further comprises an asymmetric focusing element (e.g. a cylindrical lens) performing a reshape of the two phase-corrected free-space optical beams in order to generate two different luminous spots with a more circular symmetry on two respective points of a photo-detector.

It is worth noting that the reflecting optical element 2-7 reflects the impinging optical beams off-axis, i.e. the reflected two groups of N-unwrapped free-space optical beams (reflected by the reflecting optical element 2-7) illuminate the area 2-20b which is different (i.e. not overlapping) with respect to the area 2-20a illuminated by the multiplexed free-space OAM optical beam impinging directly on the surface 2-20, so that there is no overlap between the multiplexed free-space OAM optical beam impinging directly on the first zone 2- 20a and the two groups of N-unwrapped free-space optical beams impinging on the second zone 2-20b after reflection from the reflecting optical element 2-7.

The above indicated second zone 2-20b of the surface 2-20 shown in Figure 2C is of the transmitting type, but it can alternatively be implemented as a reflecting type.

The second implementation of Figure 2C is compact and it allows a simple alignment of the first zone 2-20a with respect to the second zone 2-20b, because they belong to a same substrate, thus the first zone 2-20a and the second zone 2-20b are coplanar and aligned structurally.

Moreover, the second implementation has a higher resolution than a traditional OAM sorter, meaning that it increases the number of exploitable OAM modes, because the channel overlap between neighbouring OAM modes is reduced without sacrificing the density of the OAM modes.

The second zone 2-20b of the surface 2-20 can be implemented with a multi-level diffractive optical element made for example of silicon material or Poly(methyl methacrylate) (PMMA), also known as acrylic or acrylic glass, and it is realized with a holographic mask having the structure of a multi-level surface, which is composed of a plurality of pixels (that is, a matrix of pixels), each pixel having discrete phase and/or amplitude values, as shown in Figures 6A-6B; in this case the above indicated focusing element 2-25 can be integrated into the multi-level diffractive optical element, by adding a specific term in the phase pattern of the multi-level diffractive optical element of the second zone 2-20b. Alternatively, the second zone 2-20b of the surface 2-20 can be implemented with Pancharatnam-Berry optical elements (PBOE) made of silicon material, as shown in Figures 7A-7B; in this case the above indicated focusing element can be either integrated into the second zone 2-20b or it can be a lens 2-25 which is external to the second zone 2-20b (as shown in Figure 2C), or the focusing element is a lens 2-25 which is superimposed to the second zone 2-20b and in contact thereon in the same way as shown in Figure 8 for the lens 2-3 superimposed to the zone 2-12a.

The first zone 2-20a can be implemented with Pancharatnam-Berry optical elements (PBOE) made of silicon (as shown in Figures 7A-7B), which allows to perform both mode division demultiplexing and polarization division demultiplexing (PDM= Polarization Division Multiplexing).

The second implementation strictly requires the addition of a tilt term to the first zone 2-20a of the surface 2-20, in order to transmit the optical beam off-axis: this also contributes to prevent the optical beam transmitted from the first zone 2-20a from overlapping with zero order optical beam.

Therefore, the optical beam transmitted from the first zone 2-20a of the surface 2-20 has a propagation direction which is tilted (i.e. an angular tilt) with respect to the propagation direction of the optical beam impinging on the first zone 2-20a; in other words, the optical beam transmitted from the first zone 2-20a is tilted with respect to a perpendicular axis of the first zone 2-20a.

For example, the angle comprised between the propagation direction of the optical beam transmitted from the first zone 2-20a and the propagation direction of the optical beam impinging on the first zone 2-20a (i.e. the angle comprised between the propagation direction of the optical beam transmitted from the first zone 2-20a and the propagation direction of the optical beam impinging on the first zone 2-20a) is comprised between 5 degrees and 55 degrees, in particular between 25 degrees and 55 degrees. This constraint, in addition to the compact size of the sorter, set the optical conditions far beyond the paraxial approximation and the traditional form of the unwrapper phase function in equation (4) is no longer valid.

Therefore, a new and more rigorous formulation of the unwrapper term encoded in the first zone 2-20a is required.

To this aim, it is applied the stationary phase approximation to solve the Rayleigh- Sommerfeld diffraction formula, describing the propagation in the non-paraxial regime:

wherein: (x,y) and {u,v) are the orthonormal coordinate systems on the first zone 2-20a and on the second zone 2-20b of the surface 2-20, respectively;

z is twice the distance between the surface 2-20 and the reflecting optical element 2- 7;

- Q(x,y) is the phase function of the first zone 2-20a to be determined;

U¹ is the field of the input wave.

For the sake of simplicity, let’s consider an input plane wave illuminating the optics. According to the stationary phase approximation, the integral solution reduces to find the saddle points of the phase function F of the argument, that is:

(u, v) can be expressed in terms of (x,y) by substituting the log-pol transformation relations:

where a lateral shift c was considered for the side-by-side configuration.

Previous differential equations for Q(x,y) can be solved numerically in order to obtain the phase pattern of the unwrapper term encoded in the first zone 2-20a of the surface 2-20.

The fan-out unwrapper (i.e. performing sorting and /V-copying) encoded in the first zone 2-20a of the surface 2-20 can be considered as the superposition of N unwrappers {Q_m} and it is calculated as it follows:

W = arctan

where the m-th unwrapper Q_m performs the optical transformation: m

As shown above, the phase-corrector of the second zone 2-20b of the surface 2-20 can be calculated numerically with equations (7) and (9) for a Gaussian beam illuminating the first zone 2-20a of the surface 2-20 and applying the convolution algorithm in the angular spectrum method formulation.

A custom MATLAB code was implemented, based on the convolution algorithm in the angular spectrum regime, in order to compute the propagation of a Gaussian beam impinging on the fan-out unwrapper encoded in the first zone 2-20a of the surface 2-20 and thus calculate the phase pattern of the corresponding dual phase-corrector implemented in the second zone 2-20b of the surface 2-20 for phase-distortion correction.

The same code was used to calculate the numerical output of the designed diffractive elements and estimate the OAM bandwidth of the sorter on the basis of the design parameters, e.g. log-pol transformation parameters (a, b) and focal length of the system.

Referring to Figures 4A-4C, they show a third specific implementation 10-3 of a first embodiment of the demultiplexing optical device 10 performing demultiplexing of a guided OAM mode carrying a plurality of OAM modes with a different orbital angular.

The term“implementation” (referred to the demultiplexing optical device 10-3 and 10- 5 explained afterwards) means the definition not only of the optical layout for performing OAM demultiplexing (as shown in Figure 2A above and Figure 9A explained afterwards), but also the specific mechanical layout implementing the corresponding optical layout for OAM demultiplexing.

For the sake of simplicity two OAM modes OAM1 , OAM2 are considered, but more in general the demultiplexing optical device 10-3 (or 10-5) performs demultiplexing of a plurality M of OAM modes, where N is an integer number greater than or equal to 2.

The demultiplexing optical device 10-3 comprises a fiber connector 5; moreover, Figure 4A also shows the end portion of a multimode optical fiber 4 carrying a plurality of guided OAM modes with a different orbital angular momentum, wherein the optical fiber 4 can be considered external to the demultiplexing optical device 10-3.

In particular, the demultiplexing optical device 10-3 is implemented with a first substrate slab having a first surface 2-12 and a second surface 2-26, wherein the first surface 2-12 and the second surface 2-26 are at least partially facing each other.

More in particular, the first and second surfaces 2-12, 2-26 are planar surfaces and are substantially parallel each other. The first substrate slab of the demultiplexing optical device 10-3 is for example made of a glass material (i.e. a glass slide) transparent for the visible wavelengths or silicon material transparent for the infra-red wavelengths.

The first substrate slab of the demultiplexing optical device 10-3 can be a single block; alternatively, the first substrate slab of the demultiplexing optical device 10-3 can be constituted by multiple layers made of two or more transparent slabs arranged in order to form a solid unique block.

More in particular, a first film is arranged on the first surface 2-12 and a second film is arranged on the second surface 2-26.

For example, the first film is arranged on the first surface 2-12 by means of a deposition process and the first film has a thickness comprised in a range of 0,2 pm to 3 pm, in particular about 0,5 micrometres (pm), thus it is a thin film.

The first film includes both a first diffractive optical zone 2-12a and a second diffractive optical zone 2-12b, wherein the first zone 2-12a and the second zone 2-12b are arranged side-by-side on the first film on the first surface 2-12, so that the first zone 2-12a and the second zone 2-12b are coplanar and structurally aligned.

The first film is patterned in the first zone 2-12a and in the second zone 2-12b, as explained more in detail afterwards in the fabrication techniques paragraph.

The second film includes an optical reflecting zone 2-26a, for example made of chrome material or aluminum or nickel.

Preferably, the second film can also be patterned in the area determined by the optical reflecting zone 2-26a, as explained more in detail afterwards in the fabrication techniques paragraph.

A focusing element 2-3 (e.g. a Fresnel lens) is optically coupled with the first zone 2- 12a (for example, arranged in contact with the film on the surface of the first zone 2-12a) and it performs beam collimation and focusing on the second zone 2-12b.

More in particular:

the first zone 2-12a encodes the log-pol optical transformation (i.e. unwrapping operation) and the fan-out copy operation, thus it is also referred as“fan-out unwrapper”; the focusing element 2-3 performs beam collimation and focusing on the second zone

2-12b;

the second zone 2-12b is of the diffractive type and it performs the double phase correction, i.e. performing phase correction of both the log-pol optical transformation and the fan-out copy operation.

The first substrate slab of the demultiplexing optical device 10-3 has for example the shape of a solid figure such as a parallelepiped as shown in Figures 4A and 4B, wherein the first surface 2-12 and the second surface 2-26 are opposite faces (surfaces) of the parallelepiped and wherein the first zone 2-12a and the second zone 2-12b are arranged on the same face of the parallelepiped (single first substrate slab or unique block composed of multiple layers) and wherein the first zone 2-12a has a substantially circular shape and the second zone 2-12b has a rectangular shape.

The third implementation requires the addition of a tilt term to the first zone 2-12a of the first film on the first surface 2-12, in order to transmit the optical beam off-axis: this also contributes to prevent the optical beam transmitted from the first zone 2-12a from overlapping with zero order optical beam.

Therefore, the optical beam transmitted from the first zone 2-12a of the first film on the first surface 2-12 has a propagation direction which is tilted (i.e. an angular tilt) with respect to a propagation direction of the optical beam impinging on the first zone 2-12a; in other words, the optical beam transmitted from the first zone 2-12a has a propagation direction which is tilted with respect to a perpendicular axis of the first zone 2-12a. For example, the angle comprised between the propagation direction of the transmitted optical beam and the propagation direction of the impinging optical beam is comprised between 5 degrees and 55 degrees, in particular between 25 degrees and 45 degrees. This constraint, in addition to the compact size of the sorter, set the optical conditions far beyond the paraxial approximation, as above explained for the second implementation.

The use of a film on the same substrate surface 2-12 for the first zone 2-12a and the second zone 2-12b provides the advantage that a single manufacturing technique with a single lithographic step of pattern writing can be used for producing the demultiplexing optical device 10-3, thus keeping the designed geometrical relationships among the first and second zones 2-12a, 2-12b necessary for preserving the light propagation path, simplifying the manufacturing process, reducing the manufacturing time and the costs of production, said manufacturing technique being for example Electron beam lithography of a resist layer, or Reactive Ion Etching of a silicon substrate using an EBL fabricated mask, or a replica process of a pre-fabricated master, which will be explained better afterwards.

The relative distances and the geometrical relationships between the zones 2-12a e 2-12b and 2-26a allows to keep fixed the designed alignment procedures.

Preferably, a single lithographic process of writing the zones 2-12a and 2-12b on the same surface 2-12 allows to enhance the precision of relative alignment.

The second zone 2-12b of the first film on the first surface 2-12 can be implemented with a multi-level diffractive optical element made for example of polymeric material or PMMA and it is realized with a holographic mask having the structure of a multi-level surface, which is composed of a plurality of pixels (that is, a matrix of pixels), each pixel having discrete phase and/or amplitude values, as shown in Figures 6A-6B.

Alternatively, the second zone 2-12b of the first film on the first surface 2-12 can be implemented with Pancharatnam-Berry optical elements (PBOE) made of silicon material, as shown in Figures 7A-7B.

The first zone 2-12a of the first film can be implemented with Pancharatnam-Berry optical elements (PBOE) made of silicon, as shown in Figures 7A-7B.

The focusing element 2-3 is for example a Fresnel lens and it is made for example of PMMA material.

The optical reflecting zone 2-26a performs a reflection of a plurality of unwrapping free-space optical beams (generated by the fan-out copy operation) towards the second zone 2-12b of the first film on the first surface 2-12.

The reflecting zone 2-26a is for example made of a metal material.

As above explained, the first zone 2-12a and the second zone 2-12b are etched on the first film arranged on the first surface 2-12 of the first substrate slab of the demultiplexing optical device 10-3, in particular when the first substrate slab is made of glass and the first film is made of another material such as silicon or polymer; alternatively, the first zone 2-12a and the second zone 2-12b are etched directly on the first surface 2-12 of the first substrate slab of the demultiplexing optical device 10-3 (i.e. without the first film), in particular when the first substrate slab is made of only one material such as silicon.

Preferably, the demultiplexing optical device 10-3 further comprises a second substrate slab which is optically coupled to the first substrate slab, wherein the second zone 2-12b of the first film on the first surface 2-12 is included into a reflective zone 2-12c configured to reflect the plurality of phase-corrected free-space optical beams (generated by the second zone 2-12b) towards the second substrate slab.

The second substrate slab of the demultiplexing optical device 10-3 is for example made of a glass material (i.e. a glass slide) transparent for the visible wavelengths or silicon material transparent for the infra-red wavelengths.

The second substrate slab of the demultiplexing optical device 10-3 can be a single block; alternatively, the second substrate slab of the demultiplexing optical device 10-3 can be constituted by multiple layers made of two or more transparent slabs arranged in order to form a solid unique block.

The second substrate slab of the demultiplexing optical device 10-3 has for example the shape of a parallelepiped, as shown in Figures 4A and 4B.

The second substrate slab of the demultiplexing optical device 10-3 includes an output diffractive optical element having a surface 2-5 comprising an optical diffractive zone 2-5a having the function of reshaping the plurality of phase-corrected free-space optical beams in order to generate at the output of the demultiplexing optical device 10-3 a respective plurality of more symmetric circular spots.

The surface 2-5 of the output diffractive optical element is for example made of PMMA material and it can implement the function of a cylindrical lens with phase pattern:

being /_CL the focal length.

The above indicated zone 2-12c of the first surface 2-12 shown in Figure 4A is of the reflecting type, but it can alternatively be implemented as a transmitting type.

The use of the zone 2-12c of the reflecting type has the advantage (with respect to the use of the zone 2-12c of the transmitting type) that the direction of the optical beam transmitted at the output of the demultiplexing optical device 10-3 is the same as the direction of the optical beam received at the input of the demultiplexing optical device 10-3 (i.e. the direction from left to right in the drawing of Fig.4A): this simplifies the insertion of the demultiplexing optical device 10-3 into a rack housing multiple printed circuit boards performing the reception of the optical signal, processing of the optical signal (possibly after conversion into electrical signals) and transmission of the optical signal.

Figure 4B shows the possible values of the dimensions of the demultiplexing optical device 10-3 and of the fiber connector 5, wherein the first and second substrate slabs have the shape of a parallelepiped and wherein the first zone 2-12a has a circular shape.

The parallelepiped of the first substrate slab of the demultiplexing optical device 10-3 has a width equal to about 15 millimetres (mm), a height equal to about 5 mm and a depth equal to about 7 mm; in particular, the distance between a centre of the first circular zone 2- 12a and the geometric center of the second rectangular zone is about 5 mm.

The parallelepiped of the second substrate slab of the demultiplexing optical device 10-3 has a length comprised between 3 mm and 8 mm, a width equal to about 7 mm and a height equal to about 5 mm.

The fiber connector 5 has a circular section and a length comprised between 10 mm and 15 mm.

The distance between the surface 2-5 and the photo-detector with luminous spots on two points P1 , P2 is about 10 mm.

Referring to Figure 4C, it shows the operation of the demultiplexing optical device 10- 3 according to the third implementation 10-2 of the first embodiment of the demultiplexing optical device 10: the tip of the multimode optical fiber 4 generates an output optical signal carried by a multiplexed guided OAM mode carrying a first OAM mode (0AM1 ) having a first angular index h and a second OAM mode (0AM2) having a second angular index h (see letter a) in Figure 4C);

a multiplexed free-space OAM optical beam is generated by the output optical signal and propagates in free-space in the fiber connector 5, said multiplexed free-space OAM optical beam carrying a first free-space optical beam (OAM1 ) having a first angular index h and a second free-space optical beam (OAM2) having a second angular index /?( see letter b) in Figure 4C);

the multiplexed free-space OAM optical beam illuminate the focusing element 2-3 fabricated on the first film on the surface of the first zone 2-12a (see again letter c) in Figure 4C) and the focusing element 2-3 performs beam collimation and focusing over the second zone 2-12b;

the collimated multiplexed free-space OAM optical beam impinges on the first zone 2- 12a of the first film on the first surface 2-12 encoding the fan-out unwrapper (i.e. performing log-pol optical transformation and unwrap of the annular OAM profile and thus generating multiple N copies of the unwrapping optical beams, see again letter c) in Figure 4C), then the first zone 2-12a transmits two groups of a plurality N of unwrapping free-space optical beams having a propagation direction towards the optical reflecting zone 2-26a by means of the tilt term, i.e. the first group of N-unwrapping free-space optical beams associated to the first angular index h and the second group of N-unwrapping free-space optical beams associated to the second angular index /?;

during propagation inside the first substrate slab towards the optical reflecting zone 2- 26a, the two groups of focused N-unwrapping free-space optical beams unwrap (see letter d) in Figure 4C);

the two groups of focused N-unwrapping free-space optical beams (generated by the first zone 2-12a of the first film on the first surface 2-12) impinge on the optical reflecting zone 2-26a of the second film on the second surface 2-26 (see letter e) in Figure 4C), which back-reflects off-axis the two groups of N-unwrapping free-space optical beams towards the second zone 2-12b of the first film on the first surface 2-12;

the two groups of reflected N-unwrapping free-space optical beams propagate in the first substrate slab towards the first surface 2-12 and impinge on the second zone 2-12b of the first film on the first surface 2-12 (see letter f) in Figure 4C), which performs phase- correction by retaining the linear phase gradient and transmits two phase-corrected free- space optical beams having linear intensity distribution, the two phase-corrected tree-space optical beams having linear intensity distribution propagate in the second substrate slab towards the surface 2-5 (see letter g) in Figure 4C) and start to separate along a transverse direction according to their OAM content, i.e. a first phase-corrected tree-space optical beams starts to have a first direction in the space depending on the first angular index h, while a second phase-corrected tree-space optical beams starts to have a second direction (different from the first direction) in the space depending on the second angular index /?;

the two tree-space optical beams impinge on the optical diffractive zone 2-5a (see letter h) in Figure 4C), which performs a reshape of the two phase-corrected tree-space optical beams in order to generate two different luminous spots with a more circular symmetry on two respective points P1 , P2 of a photo-detector;

the photo-detector performs an opto-electrical conversion of the two received optical beams at points P1 , P2 into two respective electrical signals (see letter i) in Figure 4C), thus completing the demultiplexing of the input OAM modes OAM1 , OAM2.

Advantageously, the first zone 2-12a of the first film on the first surface 2-12 is implemented with Pancharatnam-Berry optical elements (PBOE), that allow to perform both mode division demultiplexing and polarization division demultiplexing (PDM= Polarization Division Multiplexing), as shown in Figures 7A-7B.

Referring to Figures 4D-4F, they show a demultiplexing optical device 10-4 for performing demultiplexing of guided OAM modes with a different orbital angular momentum and a different state of polarization, according to a variant of the third implementation of the first embodiment of the invention.

The demultiplexing optical device 10-4 differs from the demultiplexing optical device 10-3 in that it includes two optical paths (instead of only one) for processing the two different circular polarization states (left and right) of the received multiplexed input free-space optical beam carrying two or more OAM modes, in that it includes two second substrate slabs (instead of only one) which are optically coupled to the first substrate slab; preferably, the demultiplexing optical device 10-4 further includes a prismatic interferometer 10-4a (see Figure 4F).

The prismatic interferometer 10-4a includes a retarder 10-4a1 , a 50:50 beam splitter 10-4a2, a half-wave plate 10-4a3 and a reflective surface 10-4a4.

Therefore, the demultiplexing optical device 10-4 includes on the surface 2-12 the first diffractive optical zone 2-12a (encoding the fan-out unwrapper) and two second diffractive optical zones 2-12b1 , 2-12b2 (each one performing the double phase corrector), wherein (see Figure 4D): one second diffractive optical zones 2-12b1 is arranged on one side with respect to the first diffractive optical zone 2-12a; and

the other second diffractive optical zones 2-12b2 is arranged on the other side with respect to the first diffractive optical zone 2-12a.

The first diffractive optical zone 2-12a (fabricated in the form of a Pancharatnam-

Berry optical element) encodes a tilt contribution in the phase pattern which is experienced with opposite signs, as explained below, by orthogonal circular polarizations. Therefore the tilt term has two opposite values, one for a first circular polarization state (for example, left) and the other for a second circular polarization state (for example, right): the first diffractive optical zone 2-12a transmits two optical beams having two different propagation directions tilted with respect to the propagation direction of the impinging optical beam, wherein one transmitted optical beam is carrying the first polarization state (for example, left) and the other transmitted optical beam is carrying the second polarization state (for example, right).

One of the two tilted optical beams is reflected back by the optical reflecting zone 2- 26a and illuminates the second diffractive optical zone 2-12b1 , while the other of the two tilted optical beams is reflected back by the optical reflecting zone 2-26a and illuminates the second diffractive optical zone 2-12b2.

Instead of scalar OAM modes (i.e. OAM modes with a definite polarization), vector modes could be considered and exploited. Vector modes are built as non-separable combination of OAM modes and polarization states.

For any given OAM value

a vector modes basis of 4 elements can be defined as it follows:

where R and L stand for right-handed and left-handed circular polarization states.

Advantageously, in order to spatially separate those beams, an additional optical element, e.g. a prismatic interferometer 10-4a, is required after the demultiplexing optical device 10-4. As above indicated, first the demultiplexing optical device 10-4 acts as a polarizing beam splitter, separating the left- and right-circularly polarized contributions along two different paths:

For any OAM value f, the OAM contributions are spatially separated in / according to: yi^~ ikl

Ίp a

Therefore, for any vector mode, the two constituent contributions are located at the same transversal position y. The two light paths interfere in correspondence of a prismatic interferometer 10-4a (see Figure 4F).

The first polarization, after passing through a retarder 10-4a1 on the input facet, is reflected by a 45-degrees reflective surface, e.g. metal; as a consequence of reflection, its polarization is switched.

The second polarization enters the interferometer 10-4a through a half-wave plate 10- 4a3 for polarization conversion.

Then, the two light paths interfere in correspondence of a 50:50 beam-splitter 10-4a2.

The output optical beam proceeds either in the direction of the first or second optical beam, depending on their relative phase.

The prismatic interferometer 10-4a can be made in transparent material, e.g. glass or

PMMA.

Referring to Figure 8, it shows a possible implementation of the first zone 2-12a (fan out unwrapper) of the first film on the first surface 2-12 of the first substrate slab combined with the focusing element 2-3 (e.g. a Fresnel lens) of the demultiplexing optical device 10-3 of the third implementation of Figure 4A, thus it is also referred as “focusing fan-out unwrapper”.

The first zone 2-12a is implemented with Pancharatnam-Berry optical elements (PBOE) made of Silicon. The focusing element 2-3 is a diffractive optical element and it is made for example of PMMA material.

For example, the focusing element 2-3 is a Fresnel lens having a circular shape and composed of a plurality of concentric circular annuli, wherein said plurality of circular annuli have different radial thicknesses decreasing as a function of the increasing value of the radius.

In particular, the focusing lens 2-3 is superimposed on the first zone 2-12a and it is placed in contact with the surface of the first zone 2-12a so that the focusing lens 2-3 is optically coupled to the first zone 2-12a.

The fabrication of the focusing fan-out unwrapper is explained more in detail afterwards.

The optical element of Figure 8 can also implement the focusing fan-out unwrapper of the zone 2-20a of the second implementation of Figure 2C.

Pancharatnam-Berrv optical elements and the fan-out unwrapper

It will be described hereinafter the details of the Pancharatnam-Berry optical elements (PBOE) composing the first zone 2-12a (focusing fan-out unwrapper) of the first film on the first surface 2-12 of the first substrate slab of the demultiplexing optical device 10-3 of Figure 4A or composing the zone 2-20a of the surface 2-20 of the substrate slab of the demultiplexing optical device 10-2 of Figure 2C.

The Pancharatnam-Berry phase is a geometric phase associated with the polarization of light.

Unlike traditional diffractive and refractive elements, where the phase delay is introduced through optical path differences, in Pancharatnam-Berry optical elements (PBOE) the phase delay results from the geometric phase that accompanies space-variant polarization manipulation.

These elements can be realized in the form of birefringent media whose fast-axis orientation is spatially varying.

By locally controlling the axis orientation, it is possible to control the polarization conversion and the related Pancharatnam-Berry phase point-by-point.

PBOEs can be fabricated by structuring the surface of the sample with subwavelength features in order to introduce form birefringence.

For instance, a grating with subwavelength period (i.e. a period smaller than the wavelength) is experienced by an impinging wave as a birefringent effective medium, with its fast axis perpendicular to the grating grooves.

By pixelating the surface with spatially-variant subwavelength gratings, it is possible to shape the wavefront of the incident wave by locally changing its polarization state. In Jones matrix notation, the matrix T{x,y) of the single pixel at the position (x,y) is given by:

being Q ( x,y ) the orientation of the grating vector (orthogonal to the grating grooves), and <5 the phase delay between the two orthogonal linear polarizations.

Considering the basis constituted by right-handed (R) and left-handed (L) circular polarizations, the subwavelength grating acts as a polarization converter:

G[L] = cos ( d / 2) R - i sin ( d / 2 ) e^+2,t>L

T[L] = cos (c> / 2) - / sin (c> / 2) e ^2WR

If the phase delay <5 is equal to p, complete polarization conversion is achieved:

T[R] = -i - e^+2U>L

T[L] = -i e ^2WR

By correctly designing the grating in terms of period, duty-cycle and profile, in order to get a 77 delay between the two orthogonal linear polarizations, it is possible to totally convert the circular polarization from right to left, and viceversa, with the appearance of a phase term equal to twice the grating orientation. In addition, orthogonal polarization states experience phase patterns with opposite signs.

Therefore, a given phase pattern Q(x,y) can be materialized into a PBOE by fabricating a pixelated surface of subwavelength gratings whose local orientation is given by q(c,g)= Q(x,y)/2.

Each subwavelength grating should be properly designed and fabricated in order to assure phase delay <5=77.

Usually, the pixel lateral size is at least 3-5 wavelengths and the phase pattern results to be spatially discretized over a mesh which does not allow reproducing phase features below the micrometric scale; this could be detrimental for the PBOE realization of complex patterns such as the unwrapper and phase-corrector of the log-pol geometric optical transformation and the fan-out unwrapper encoded in the first zone 2-20a of the surface 2-20 and in the first zone 2-12a of the first film on the first surface 2-12.

A solution is the design of continuously-variant subwavelength gratings that guarantee the continuity of polarization conversion and phase manipulation.

The grating vector is locally perpendicular to the grating stripes and defined as:

K_g = K_g ( , y ) (cos q(c, y), sin #(x, y))

where K_g(x,y)=2n/A(x,y), being A(x,y) the local grating period at the point (x,y). In order for such a grating to be physically realizable in a continuous way, the grating vector should be a conserving vector, that is:

VxK_g = 0

The aforesaid vectorial condition leads to the following system of differential equations in the components of the grating vector unknown:

dK 30

= -K

dx ^s dy

¾ 30

= K

dy dx

Once the grating vector, for the given grating orientation 6(x,y)= Q(x,y)/2, is determined, the grating potential y₉(c,g) can be found by integrating along any arbitrary path on the PBOE plane.

Therefore, a corresponding binary phase-mask can be derived as:

where Q is the Heaviside function and q is the duty-cycle.

As a consequence of the grating continuity constrain, the period of the grating is no longer fixed and it becomes spatially-variant.

In principle, it could assume any value either beyond the subwavelength regime, limited by the structural cut-off A_c = /V[max(/ii,/i2)+/ii sin( 6|_p)]^1/2, being r?i and n₂ the refractive indices of the grating and surrounding medium, respectively, and 0_n the incidence angle, or far below the subwavelength size, beyond the lithographic limits. Concurrently, since the TT- delay is related to the grating profile, a variation in the grating period could determine a change in its value.

Those limitations can be overcome imposing the following restrictions:

lower and upper limits to the period variation;

spatially-varying duty-cycle in order to maintain phase delay <5=TT.

The following steps are carried out:

upper value A_c and lower value A_m for the subwavelength grating period are defined, wherein the upper value is given by the grating structural cut-off.

For instance, in case of silicon gratings, wavelength equal to 1310 nm and normal incidence, the structural cut-off is around 700 nm.

The lower value A_m is imposed by the lithographic limitations; in the specific, it is given by the minimum grating-width obtainable by etching the required thickness (a reasonable value for A_m can be around 150 nm). the whole phase pattern Q(x,y) is divided into a matrix of zones; inside each zone the grating potential is calculated, as described above, allowing the period to vary between A_m and A_c.

for the given grating profile, e.g. digital, numerical simulations are performed in order to obtain a mathematical relation q{ A) for the duty-cycle as a function of the grating period, providing the condition phase delay <5=77.

using the relation q{ A), a binarization of the grating potential is done applying the following condition inside each zone:

the phase pattern is fabricated with a lithographic process, e.g. RIE of a silicon substrate using an EBL fabricated mask.

The whole optical element (i.e. the first zone 2-12a of the first film on the first surface 2-12 of the first substrate slab of the demultiplexing optical device 10-3) results to be the composition of several zones.

Each zone is a Pancharatnam-Berry optical element in which the period of the subwavelength grating varies continuously between the lower value A_m and the upper value A_c, and the duty-cycle changes accordingly in order to keep the condition phase delay <5= p satisfied.

At the boundaries of two neighbouring zones, the grating orientation is continuous, while there is a discontinuity in the grating period.

Fabrication techniques

It will be described hereinafter the fabrication technique of the focusing fan-out unwrapper of Figure 8, after discussing the main aspects of electron beam lithography, soft- lithography and Ion Coupled Plasma-Reactive Ion Etching.

Electron beam lithography

Electron beam lithography (EBL) is the ideal technique to transfer the computational patterns from a digital stored format to an imaging layer with high-resolution profiles.

It is a mask-less process: a high-powered focused electron-beam draws high- resolution pattern directly on an electron-sensitive polymer (resist). A thin resist layer is spun over the substrate, for instance a glass slide.

Thus, each point (x, y) of the sample is exposed to a different controlled electron- dose in order to change its dissolution rate and obtain, after development, a residual thickness t x, y) proportional to the local phase delay W(c, y), according to the following formula:

wherein n is the refractive index of the resist at the working wavelength l, no is the refractive index of the surrounding medium.

After the lithographic step, the exposed sample is developed into a specific solvent and the three-dimensional surface structuring is obtained.

EBL is a lithographic technique that is characterized by an high precision and accuracy in the writing on high resolution patterns, down to 10-20 nm; this characteristic can be exploited for the generation of patterns of any morphological profile and with precise and accurate geometrical control and relative distances.

EBL techniques can generate markers patterns, e.g. specific micro structures that allow the alignment of patterns realized in different steps of process; typical alignment markers 2-1 12m and 2-126m are represented in Figure 1 1 for the first substrate slab of the demultiplexing optical device 10-5 of the implementation of the second embodiment of Figure 9A.

Soft-lithography

Electron-beam lithography is precise and provides high-resolution, but it is expensive in terms of time and machine-cost.

Soft-lithography, such as nano-imprinting, is the suitable technique for the production of master sample that should be later replicated with faster, cheaper and high-throughput.

The soft lithography process can be arranged in two steps: fabrication of the elastomeric elements and use of these elements to pattern features in geometries defined by the element's relief structure.

Many elastomeric elements can be generated from a single master, and each element can be used several times to replicate the initial master.

Elastomeric elements are generated by casting a light- or heat-curable prepolymer against the master.

After exposure to heat/UV-light for the proper time, the polymer cures and reproduces the stamp profile with nanometric resolution.

With optimized materials and chemistries, this fabrication sequence has remarkably high fidelity in the surface relief replica.

Ion Coupled Plasma-Reactive Ion Etching

Mask-less EBL is the ideal technique for the direct lithography of diffractive optics with high-resolution. Flowever, it requires an electron-sensitive substrate.

The transfer of the pattern to other materials, e.g. silicon, requires the direct milling of the surface. Silicon pattering is usually a two-step process. At first, the sample is coated with a resist layer on which the pattern is fabricated via a proper lithographic technique, e.g. electron-beam lithography. Afterwards, the pattern is used as a mask to etch the silicon surface.

One solution is offered by Ion Coupled Plasma-Reactive Ion Etching (ICP-RIE) process.

RIE is a dry etching technique that allows the transfer of a precise micro/nano-pattern to the substrate. The etching is allowed for the presence of a plasma composed by both active neutral species and ions. The formers are responsible for chemical reactions with the substrate, and the latters are responsible for physical sputtering when accelerated by a bias towards the substrate. For this reason, the etching results highly directional. Depending on the ratio between the etching rates of mask and silicon substrate, the mask thickness and etching time must be properly calibrated.

Mass production systems

EBL technique is able to pattern with high precision and accuracy of 10-20 nm the design pattern of the optics e.g of the unwrapper and of the phase corrector, as well as of the other optical elements.

However, EBL is not a lithographic technique suitable for industrial application because of its low throughput and high production costs.

For industrial applications, these patterns must be replicated on large area with the same precision and accuracy by means of lithographic techniques such as UV lithography or imprinting lithography that are industrial techniques used for replica in massive volumes of the designed patterns.

Typically, EBL lithography is used for the generation of masters for imprinting lithography or masks for UV lithography.

The masks and the masters can be provided with alignments markers that allow the aligned replica of the optics.

The parameter that measure the quality of the alignment between different structures is called“overlay” and nowadays techniques guarantee overlay of the order of 50 nm -100 nm.

Fabrication of the patterns and of the optics

The fabrication of the patterns on the substrate slab also comprising the diffractive zones, the alignments markers and the focusing optics can be performed with a sequence of lithographic and nanofabrication processes including the use of EBL, UV and imprinting lithographies or any other suitable lithography.

The patterns can be realized on single surface side (see surface 2-12 in Fig.4A and surface 202-1 in Fig.5A for the third implementation of the demultiplexing/multiplexing optical device according to the first embodiment) or can be realized on both surfaces sides of the substrate slab (see surfaces 2-1 12, 2-126 in Fig.9A and surfaces 251 -1 , 252-1 in Fig.l OA of the implementation of the demultiplexing/multiplexing optical device according to the second embodiment); in all the cases, the geometrical relationships between the patterns must be a precise and accurate reproduction of the optical design.

In the case of fabrication of the patterns on a single surface side (see again surface 2-12 in Fig.4A and surface 202-1 in Fig.5A) the pattern can be generated by EBL lithography and further processed to fix permanently the pattern on the selected surface; advantageously, the pattern can be realized by UV or imprinting lithography; in all the cases, geometrical spatial relationships between different zones of the pattern are defined precisely and accurately.

Advantageously, the alignment markers 2-1 12m and 2-126m can also be used for performing the alignment of the second slab with respect to the first slab.

In the case of lithographic process of replica on both the sides of the substrate slab (see again surfaces 2-1 12, 2-126 in Fig.9A and surfaces 251 -1 , 252-1 in Fig.l OA), the lithographic and fabrication processes can be of the same type of the first case but they require a procedure of alignment between the patterns realized on the two surface sides.

The alignment markers 2-1 12m and 2-126m can be directly patterned using EBL lithography or by using UV and imprinting lithographies together with the patterns of the optics. This procedure defines the geometrical relationships among the alignment markers 2- 1 12m and 2-126m and the optics, e.g the unwrapper 2-1 12a+2-103 and the phase corrector 2-126a.

After the first lithographic process on one of the sides the patterns are reproduced together with alignment markers.

During the second process of replica on the other surface, a procedure of alignment with respect to the markers patterned during the first process is performed exploiting the transparency of the substrate slab, such as a glass.

Advantageously, in case the substrate is silicon, the transparency of this material in the range of infrared wavelengths can be exploited and radiation in this regime for the alignment of the masks and markers can be used.

The markers have for example the shape of crosses and double crosses, whose vision must match perfectly during the lithographic process.

The alignment of the patterns fabricated on both sides of the substrate slab can be of the order of 1 micrometre or less.

According to another procedure of fabrication, the alignment markers 2-1 12m can be patterned together with unwrapper 2-1 12a+2-103 on a first substrate slab and the markers 2- 126m can be patterned together with the phase corrector 2-126a on asecond substrate slab. The two slabs can be arranged in order to form a unique multilayer block using the alignment markers 2-1 12m of the first slab and the markers 2-126m of the second slab in order to perform the alignment between unwrapper 2-1 12a+2-103 and the phase corrector 2-126a. Fabrication of the focusing fan-out unwrapper

The fabrication of the focusing fan-out unwrapper of Figure 8 combines two fabrication approaches.

At first, RIE is exploited in order to pattern the fan-out unwrapper on the silicon substrate of the first zone 2-12a in the form of a Pancharatnam-Berry optical element.

Since right-handed and left-handed polarizations experience the same phase pattern with opposite signs, the focusing term cannot be included at this step, otherwise only one polarization would be focused on the second element, while the other one would result divergent.

The solution consists in decoupling the focusing term from the fan-out unwrapper pattern and fabricating it in the form of a focusing element 2-3 (e.g. a Fresnel lens) to be placed in contact with the silicon surface of the first zone 2-12a, as shown in Figure 8.

The Fresnel lens 2-3 can be either directly fabricated with electron-beam lithography on a resist layer spun over the silicon PBOE, or patterned with soft-lithographic techniques using a pre-fabricated Fresnel lens master.

In case the lithographic material spun over the PBOE is expected to percolate inside the PBOE subwavelength relieves, the PBOE must be designed properly considering the refractive index of the DOE instead of air as surrounding medium.

Referring to Figures 5A-5B, they show a multiplexing optical device 202 according to the third implementation of the first embodiment of the invention.

The multiplexing optical device 202 has the function of performing the multiplexing of guided OAM modes with a different orbital angular momentum (that is, with different values h, h, , ... of the angular index I) and insertion into the multi-mode optical fiber 4.

The multiplexing optical device 202 has a complementary function with respect to that of the demultiplexing optical device 10-3 of the third implementation of the first embodiment of Figures 4A-4C; it is described by analogy to the demultiplexing optical device 10-3, considering reciprocity by virtue of the symmetry linked to the time reversal invariance between the demultiplexing and multiplexing processes.

In other words, the multiplexing optical device 202 is similar to the demultiplexing optical device 10-3 because it comprises a reverse path for the optical beams based on the time invariance of Maxwell’s equations; the minimal differences are identifiable in the different architecture for generating the optical signal with respect to that for receiving the optical signal. In particular, the multiplexing optical device 202 comprises:

an array (i.e. a plurality) 203 of single-mode optical fibers or waveguides;

a reshaping optical element 206;

a first substrate slab;

a second substrate slab optically coupled to the first substrate slab;

a fiber connector 205.

The first substrate slab and the second substrate slab of the multiplexing optical device 202 are made for example of a glass material (i.e. a glass slide) transparent for the visible wavelengths or silicon material transparent for the infra-red wavelengths.

The first substrate slab of the multiplexing optical device 202 has a first surface 202-1 and a second surface 202-2, wherein the first surface 202-1 and the second surface 202-2 are at least partially facing each other.

More in particular, the first and second surfaces 202-1 , 202-2 are planar surfaces and are substantially parallel each other.

The reshaping optical element 206 is optically coupled at a first side with the outputs of the array 203 and at a second side with the first substrate slab and it includes an air cavity comprised between the two sides.

The reshaping optical element 206 is for example a third substrate slab having an output connected to the input of the first substrate slab.

The reshaping optical element 206 includes at the first side an input facet 206a1 which is optically coupled with the array 203 of single-mode optical fibers/waveguides, the input facet 206a1 including an array of diffractive optical elements.

In particular, the first substrate slab comprises:

an input facet 202c which is optically coupled with the output of the reshaping optical element 206, the input facet 202c including an array of diffractive optical elements

the first surface 202-1 ;

the second surface 202-2 facing at least partially the first surface 202-1.

More in particular, a first film is arranged on the first surface 202-1 and a second film is arranged on the second surface 202-2.

For example, the first film is arranged on the first surface 202-1 by means of a deposition process and the first film has a thickness comprised in a range of 0,2 pm to 3 pm, in particular about 0,5 micrometres (pm), thus it is a thin film.

The first film on the first surface 202-1 includes both a first diffractive zone 202-1 a and a second diffractive zone 202-1 b, wherein the first zone 202-1 a and the second zone 202-1 b are arranged side-by-side on the first film on the first surface 202-1 , so that the first zone 202-1 a and the second zone 202-1 b are coplanar and structurally aligned. The first film is patterned in the first zone 202-1 a and in the second zone 202-1 b.

The second film includes an optical reflecting zone 202-2a, for example made of chrome material or aluminum or nickel.

Preferably, the second film can be also patterned in the area determined by the optical reflecting zone 202-2a.

Claims

1 . Optical device (10-5, 10-3) for demultiplexing modes with different orbital angular momentum, the device comprising:

- a substrate slab having two surfaces (2-1 12, 2-126; 2-12, 2-26) at least partially facing each other;

- two films arranged on the two surfaces respectively, wherein at least one of the two films includes a first diffractive optical zone (2-1 12a, 2-12a) and a second diffractive optical zone (2-126a, 2-12b);

and wherein:

the first zone (2-1 12a, 2-12a) is configured to:

^■ receive a multiplexed input free-space optical beam carrying a plurality of orbital angular momentum - OAM - modes (OAM1 , OAM2) with different orbital angular momentum;

^■ perform a conformal mapping of the multiplexed input free-space optical beam from a circular distribution to a linear distribution of luminous intensity, generate therefrom a plurality of unwrapping free-space optical beams, split the plurality of unwrapping free-space optical beams into corresponding groups of copies and generate therefrom a plurality of groups of copies of unwrapping free-space optical beams having a propagation direction which is tilted with respect to a propagation direction of the multiplexed input free-space optical beam;

the second zone (2-126a, 2-12b) is configured to receive the propagated plurality of groups of copies of unwrapped free-space optical beams, perform a phase correction of the propagated plurality of groups of copies of unwrapped free-space optical beams and generate therefrom a plurality of phase-corrected free-space optical beams having different directions in the space depending on the orbital angular momentum of the plurality of received OAM modes.

2. Demultiplexing optical device according to claim 1 , wherein a first film out of the two films is arranged on a first surface (2-1 12) out of the two surfaces and the first film includes the first zone (2-1 12a),

wherein a second film out of the two films is arranged on a second surface (2-126) out of the two surfaces and the second film includes the second zone (2-126a),

wherein the two surfaces are substantially parallel each other,

wherein the first zone (2-1 12a) is configured to transmit said plurality of groups of copies of unwrapping free-space optical beams having the tilted propagation direction directed towards the second zone (2-126a) of the second film on the second surface,

wherein the substrate slab is, alternatively, a single block or a unique block composed of multiple layers,

and wherein the first surface (2-1 12) and the second surface (2-126) include respective alignment markers (2-1 12m, 2-126m) for performing alignment between the first (2-1 12a) and the second (2-126a) zones realized in different steps of a manufacturing process of the demultiplexing optical device.

3. Demultiplexing optical device (10-3) according to claim 1 , wherein a first film out of the two films is arranged on a first surface out of the two surfaces and the first film includes the first zone (2-12a) and the second zone (2-12b), wherein the first zone and the second zone are arranged side-by-side on the first film on the first surface,

wherein a second film out of the two films is arranged on a second surface (2-26) out of the two surfaces and the second film includes an optical reflecting zone (2-26a),

wherein the two surfaces are substantially parallel each other,

wherein the optical reflecting zone (2-26a) is configured to receive the tilted plurality of groups of copies of unwrapping free-space optical beams, reflect them off-axis towards the second zone (2-12b) and generate therefrom a plurality of reflected groups of copies of unwrapping free-space optical beams,

wherein the substrate is configured to propagate the plurality of reflected groups of copies of unwrapping free-space optical beams towards the second zone (2-12b) and generate therefrom a corresponding plurality of propagated groups of copies of unwrapped free-space optical beams;

wherein the second zone (2-12b) is configured to receive said plurality of reflected groups of copies of unwrapped free-space optical beams, perform the phase correction of the plurality of reflected groups of copies of unwrapped free-space optical beams and generate therefrom said plurality of phase-corrected free-space optical beams having different directions in the space depending on the orbital angular momentum of the plurality of received OAM modes, and wherein the substrate slab is, alternatively, a single block or a unique block composed of multiple layers.

4. Demultiplexing optical device according to claim 4, wherein the first film further comprises a reflective zone (2-12c) including the second zone (2-12b), said reflecting zone being configured to receive the plurality of groups of copies of unwrapped free-space optical beams and reflect the plurality of phase-corrected free-space optical beams,

and wherein .

5. Demultiplexing optical device according to any of the previous claims, the demultiplexing optical device further comprising a focusing element (2-103, 2-3) arranged in contact with the first zone (2-1 12a, 2-12a) on the first film and configured to perform collimation and focusing of the plurality of groups of copies of unwrapping free-space optical beams over the second zone (2-126a, 2-12b).

6. Demultiplexing optical device according to any of the previous claims, wherein an angle comprised between the propagation direction of the plurality of groups of copies of unwrapping free-space optical beams and the propagation direction of the multiplexed input free-space optical beam has a value comprised between 5 degrees and 55 degrees, in particular between 25 degrees and 45 degrees,

and wherein the substrate slab has a shape of a parallelepiped, wherein the first surface (2- 1 12; 2-12) and the second surface (2-126; 2-26) are opposite faces of the parallelepiped.

7. Demultiplexing optical device according to any of the previous claims, comprising a further substrate slab optically coupled to the substrate slab, the further substrate slab including a surface (2-105, 2-5) comprising an optical diffractive zone (2-105a, 2-5a) configured to reshape the plurality of phase-corrected free-space optical beams,

wherein the second zone (2-126a) on the second film or the reflective zone (2-12c) of the first film are configured to transmit or reflect respectively the plurality of phase-corrected free- space optical beams towards the optical diffractive zone of the further substrate slab.

8. Demultiplexing optical device according to any of the previous claims, wherein the first zone (2-1 12a, 2-12a) of the first film on the first surface (2-1 12, 2-12) of the substrate slab is implemented with a plurality of Pancharatnam-Berry optical elements on silicon metasurfaces and having respective spatially-variant subwavelength gratings, each grating having a subwavelength period which varies continuously between a lower value and an upper value equal to a grating structural cut-off,

wherein at the boundaries of two neighbouring Pancharatnam-Berry optical elements the grating orientation is continuous, while the grating period is discontinuous.

9. Demultiplexing optical device (10-6, 10-4) according to any of the previous claims, wherein the first zone (2-1 12a, 2-12a) is further configured to:

- receive a multiplexed input free-space optical beam carrying a plurality of OAM modes (OAM1 , OAM2) with different orbital angular momentum and different states of polarization;

- generate a first plurality of groups of copies of unwrapping free-space optical beams having a first state of polarization;

- generate a second plurality of groups of copies of unwrapping free-space optical beams having a second state of polarization orthogonal with respect to the first polarization state; wherein the substrate includes a further second zone,

wherein the substrate is further configured to:

- propagate a tilted first plurality of groups of copies of unwrapping free-space optical beams towards the second zone (2-126a1 , 2-12b1 ) and generate a corresponding tilted first plurality of groups of copies of unwrapped free-space optical beams;

- propagate a tilted second plurality of groups of copies of unwrapping free-space optical beams towards the further second zone (2-126a2, 2-12b2) and generate a corresponding tilted second plurality of groups of copies of unwrapped free-space optical beams;

and wherein:

- the second zone (2-126a1 , 2-12b1 ) is configured to receive the propagated first plurality of groups of copies of unwrapped free-space optical beams, perform a phase correction of the propagated first plurality of groups of copies of unwrapped free-space optical beams and generate therefrom a first plurality of phase-corrected free-space optical beams having different directions in the space depending on the orbital angular momentum of the plurality of received OAM modes;

- the further second zone (2-126a2, 2-12b2) is configured to receive the propagated second plurality of groups of copies of unwrapped free-space optical beams, perform a phase correction of the propagated second plurality of groups of copies of unwrapped free-space optical beams and generate therefrom a second plurality of phase-corrected free-space optical beams having different directions in the space depending on the orbital angular momentum of the plurality of received OAM modes.

10. Optical device (302, 202) for multiplexing a plurality of modes with different orbital angular momentum, the device comprising:

- a substrate slab having two surfaces (251 -1 , 252-1 ; 202-1 , 202-2) at least partially facing each other,

- two films arranged on the two surfaces respectively, wherein at least one of the two films includes a first diffractive zone (251 -1 a; 202-1 a) and a second diffractive zone (252-1 a; 202- l b),

and wherein:

the first zone (251 -1 a, 202-1 a) is configured to:

^■ receive a plurality of free-space optical beams having a linear distribution of the luminous intensity and different linear phase gradients;

^■ perform a wrapping of the distribution of the luminous intensity of the received plurality of free-space optical beams;

^■ generate a plurality of wrapping free-space optical beams having a partial annular distribution of the luminous intensity and having a propagation direction which is tilted with respect to a propagation direction of the plurality of received free-space optical beams;

the substrate is configured to propagate the plurality of tilted wrapping free-space optical beams towards the second zone and generate therefrom a corresponding plurality of propagated wrapped free-space optical beams; the second zone (252-1 a; 202-1 b) is configured to:

^■ receive the plurality of propagated wrapped free-space optical beams having an annular distribution of the luminous intensity;

^■ perform a phase correction of the plurality of propagated wrapped free-space optical beams with annular distribution of the luminous intensity and generate therefrom a multiplexed free-space optical beam carrying a plurality of modes with different orbital angular momentum.

1 1 . Multiplexing optical device (302) according to claim 10, wherein a first film out of the two films is arranged on a first surface (251 -1 ) out of the two surfaces and the first film includes the first zone (251 -1 a),

wherein a second film out of the two films is arranged on a second surface (252-1 ) out of the two surfaces and the second film includes the second zone (252-1 a),

wherein the two surfaces are substantially parallel to each other,

wherein the first zone (251 -1 a) is configured to transmit said plurality of wrapping free-space optical beams having the tilted propagation direction directed towards the second zone (252- l a) of the second film on the second surface (252-1 ),

and wherein the first surface (251 -1 ) and the second surface (252-1 ) include respective alignment markers for performing alignment between the first (251 -1 a) and the second (252- l a) zones realized in different steps of a manufacturing process of the multiplexing optical device.

12. Multiplexing optical device (202) according to claim 10, wherein a first film is arranged on a first surface out of the two surfaces and it includes the first zone (202-1 a) and the second zone (202-1 b),

wherein the first zone and the second zone are arranged side-by-side on the first film on the first surface,

wherein a second film is arranged on a second surface (202-2) out of the two surfaces and it includes an optical reflecting zone (202-2a),

wherein the two surfaces are substantially parallel to each other,

wherein the optical reflecting zone (202-2a) is configured to receive the plurality of wrapping free-space optical beams having the partial annular distribution of the luminous intensity, reflect them off-axis towards the second zone (202-1 b) and generate therefrom a plurality of reflected wrapping free-space optical beams,

wherein the substrate is configured to propagate the plurality of reflected wrapping free- space optical beams towards the second zone and generate therefrom a corresponding plurality of propagated wrapped free-space optical beams, wherein the second zone (202-1 b) is configured to receive said plurality of propagated wrapped free-space optical beams having the annular distribution of the luminous intensity, perform the phase correction of the received plurality of propagated wrapped free-space optical beams with annular distribution of the luminous intensity and generate therefrom a multiplexed free-space optical beam carrying a plurality of modes with different orbital angular momentum,

and wherein the substrate slab is, alternatively, a single block or a unique block composed of multiple layers.

13. Multiplexing optical device according to any of claims 10 to 12, wherein an angle comprised between the propagation direction of the plurality of wrapped free-space optical beams and the propagation direction of the multiplexed free-space optical beam has a value comprised between 5 degrees and 55 degrees, in particular between 25 degrees and 45 degrees, and wherein the substrate slab has a shape of a parallelepiped, wherein the first surface (251 -1 , 202-1 ) and the second surface (252-1 ; 202-2) are opposite faces of the parallelepiped.

14. Multiplexing optical device according to any of claims 10-13, further comprising a reshaping optical element (206) optically connected to the substrate slab, the reshaping optical element having:

an array of diffractive optical elements configured to receive a plurality of guided modes, perform a reshaping of the received plurality of guided modes and generate therefrom a plurality of reshaped free-space optical beams having a rectangular distribution of the luminous intensity;

an air cavity configured to propagate the plurality of reshaped free-space optical beams towards the substrate slab;

wherein the substrate slab further includes an array of diffractive optical elements configured to perform a phase correction of the received plurality of reshaped free-space optical beams and to convert therefrom a lateral displacement into an angular tilt, thus generating said plurality of free-space optical beams with a linear distribution of the luminous intensity and a plurality of linear phase gradients.

15. Multiplexing optical device according to any of claims 10-14, wherein at least one of the two films includes a further first diffractive zone,

wherein the first zone (251 -1 a1 , 202-1 a1 ) is configured to:

^■ receive a first plurality of free-space optical beams having a linear distribution of the luminous intensity, different linear phase gradients, and a first state of polarization;

^■ perform a wrapping of the distribution of the luminous intensity of the received first plurality of free-space optical beams; ^■ generate a first plurality of wrapping free-space optical beams having a partial annular distribution of the luminous intensity and having a propagation direction which is tilted with respect to a propagation direction of the first plurality of received free-space optical beams;

wherein the further first zone (251 -1 a2, 202-1 a2) is configured to:

^■ receive a second plurality of free-space optical beams having a linear distribution of the luminous intensity, different linear phase gradients, and a second state of polarization orthogonal with respect to the first polarization state;

^■ perform a wrapping of the distribution of the luminous intensity of the received second plurality of free-space optical beams;

^■ generate a second plurality of wrapping free-space optical beams having a partial annular distribution of the luminous intensity and having a propagation direction which is tilted with respect to a propagation direction of the second plurality of received free-space optical beams;

wherein the substrate is configured to propagate the first and second plurality of tilted wrapping free-space optical beams towards the second zone (252-1 a, 202-1 b) and generate therefrom a corresponding first and second plurality of tilted wrapped free-space optical beams;

and wherein the second zone (252-1 a, 202-1 b) is configured to:

^■ receive the first plurality of propagated wrapped free-space optical beams having an annular distribution of the luminous intensity, perform a phase correction of the first plurality of propagated wrapped free-space optical beams with annular distribution of the luminous intensity;

^■ receive the second plurality of propagated wrapped free-space optical beams having an annular distribution of the luminous intensity, perform a phase correction of the second plurality of propagated wrapped free-space optical beams with annular distribution of the luminous intensity;

^■ generate a multiplexed free-space optical beam carrying a plurality of modes with different orbital angular momentum and different polarizations.

16. Mode division optical communication system for demultiplexing modes with different orbital angular momentum, the system including a multimode optical fiber (4), a fiber connector (5) and a demultiplexing optical device (10-2, 10-3) according to any of claims from 1 to 9.

17. Mode division optical communication system according to claim 16, further including a multiplexing optical device according to any of claims from 10 to 15.