Multipole-phase division multiplexing: Supplemental Document

The control of structured waves has recently opened innovative scenarios in the perspective of radiation propagation, advanced imaging, and light-matter interaction. In information and communication technology, the spatial degrees of freedom offer a wider state space to carry many channels on the same frequency or increase the dimensionality of quantum protocols. However, spatial decomposition is much more arduous than polarization or frequency multiplexing, and very few practical examples exist. Among all, beams carrying orbital angular momentum gained a preeminent role, igniting a variety of methods and techniques to generate, tailor, and measure that property. In a more general insight into structured-phase beams, we introduce here a new family of wave fields having a multipole phase. These beams are devoid of phase singularities and described by two continuous spatial parameters which can be controlled in a practical and compact way via conformal optics. The outlined framework encompasses multiplexing, propagation, and demultiplexing as a whole for the first time, describing the evolution and transformation of wave fields in terms of conformal mappings. With its potentialities, versatility, and ease of implementation, this new paradigm introduces a novel playground for space division multiplexing, suggesting unconventional solutions for light processing and free-space communications.


S1: Laplace's equation in structured phase transmission
In the paraxial approximation, the propagation of a structured field (S1) According to the stationary phase approximation [1,2] a two-dimensional integral with the form in Eq. (S1) can be approximated with its contributions around the saddle points of the total phase function  of the argument, that is: and Eq. (S1) can be rewritten in the form: where H is the Hessian determinant of  , are the saddle points of  , that is the solutions of the differential equation 0   , where the differential operator should be intended in 2D. Then, the field in z is endowed with a phase function ( ) z  given by: As a consequence of the relation in Eq. (S2), the condition 0   leads to the identity: in Cartesian coordinates. Under the existence of partial derivatives for ρ, Eq. (S5) implies the following relation: Taking the divergence in 2D of Eq. (S5) and using Eq. (S7) we obtain the following condition on the phase: It should be noted that the previous condition is still valid in the Fresnel regime, by including a focusing term and considering the propagated field at z f  .
The explicit form of Eq. (S8) in polar coordinates is given by: Solving Eq.(S9) under variable separation gives the following general solution: where m is assumed to be integer, after imposing periodic boundary conditions in 0 independent on the radial coordinate. The latter recalls the azimuthal phase term which is peculiar of orbital angular momentum (OAM) beams. Therefore, in this general approach to structured-phase division multiplexing in the paraxial regime, OAM beams are rediscovered as particular solutions of the phase patterns which can be transferred between two parties.

S2: Circular-sector transformation in the stationary phase approximation
For the benefit of the reader, we provide here a detailed calculation of the phase elements required to optically perform an n-fold circular-sector transformation. In the paraxial approximation, the propagation of an input beam U (i) at a distance z, after illuminating a phase plate with transmission function exp(iΩ) located at 0 z  , is described by the Fresnel diffraction integral: According to the stationary phase approximation [1,2] the integral can be approximated with its contributions around the saddle points of the phase function. If we consider the phase function Ω in Eq. (S11) as the sum of two contributions, i.e., the phase term Ω n imparting the desired n-fold circular-sector transformation and a quadratic focusing term: then the nullification of the gradient of the total phase inside the integral in Eq.(S11) leads to the following condition: We now consider the coordinate change: describing a mapping between the reference frame   , r  on the input plane   0 z  and the new reference frame   ,   on the destination plane at z f  . This mapping performs a rescaling of the azimuthal angle by a factor n , while the power scaling on the radial coordinate is dictated by the Cauchy-Riemann conditions [3]. After substituting Eqs. (S14) in Eq. (S13) and solving for z f  , we obtain: The integration can be done easily, remembering the definition: n n r r . We obtain: and after a straightforward integration we get the final result: Actually, a second phase plate is required in z f  in order to complete the phase transformation and account for the phase distortions due to the propagation in-between. In the reverse configuration, this element works as a circular-sector transformation with a factor equal to 1 / n , performing the inverse optical transformation: After performing analogous calculations as above, we find out the following phase pattern for the phase-corrector element:

S3: Multiplexer design
Under the condition n=-1/m, Eq. (S18) and Eq. (S20) provide the phase patterns for the sequence of optical elements, i.e. transformer and phase-corrector, required to demultiplex a superposition of multipole-phase beams of order m: When illuminated in reverse, the same configuration can be exploited to multiplex several input beams into a collimated bunch of multipole-phase beams with the same order but differing in phase strength and rotation angle on a plane perpendicular to the propagation direction. However, the first element is not exactly as the second one in the demultiplexer (Eq. (S22)). Since the inverse transformation performs the mapping of the whole pattern onto a circular sector with amplitude 2 / m  , then an |m|-fold multiplication of the input beam is needed in order to obtain a beam defined over the whole 2π range. For this reason, the required phase turns out to be the combination of m phase patterns performing n-fold circular-sector transformations with n=-m and rotated by 2 / m  with respect to each other.
The phase pattern of the first element, i.e., the multiplier, is described by the combination: while the second element of the multiplexer is given again by Eq. (S21), that is  In Fig. S1 a scheme of the optical configuration for the generation of a beam carrying a multipole-phase term with m=+2 is depicted. An input linear phase gradient is transformed into the desired phase by applying a 2-fold multiplication plus a phase inversion (n=-2). While the second element (Fig. S1(c)) is equal to the first element of the demultiplexer, in this case the first element ( Fig. S1(b)) is given by the combination of two phase elements rotated by π, as a result of Eqs. (S23) and (S24). The strength and orientation of the output phase can be controlled by acting properly on the input linear phase gradient. As shown in Fig. S2 and Fig. S4, supposing it is generated by illuminating an f-f optical system with a Gaussian beam, it is possible to tune the output phase strength α and its rotation angle ϑ 0 by changing the axial displacement of the input beam according to: and the cross-talk on the ith channel can be evaluated by: As shown in Fig. S3 and S5 for the two sets of beams generated in Fig. S2 and S4, respectively, the output beams can be assumed to be quasi-orthogonal, due to the negligible cross-talk among them. As a matter of fact, since there is no spatial superposition between the input spots illuminating the f-f system, the null overlap is expected to be maintained also during beam multiplexing and propagation, due to the unitary nature of the conformal mapping and free-space propagation.
As shown in Fig. S2 and S4, the intensity distribution exiting the multiplexer exhibits a decrease in intensity close the centre. If the multipole phase is uploaded onto a Gaussian intensity distribution, this choice increases the cross-talk, as shown in Fig. S5. However, the values below -40 dB can be still acceptable for telecom applications.

S4: Distortion of circular-sector transformation for multipole phases in input
When the stationary phase approximation is applied, it is usually assumed that the input field impinges on the first phase element with a planar wavefront. On the other hand, a multipolephase beam illuminates the optics with a non-planar phase contribution . The effect of a non-planar input wavefront has been for instance discussed in the case of log-pol optical transformation [4], when the twisted wavefront introduces a distortion in the output field distribution that is negligible as far as the input OAM is far below a threshold value. The same analysis has been performed for the circular-sector transformation applied to OAM multiplication and division [3], and an analogous limit for the input OAM has been calculated, depending on the transformation parameters and on the size of the input beam. In the following, we apply the same analysis to the case of a multipole phase in input. In the paraxial ray approximation, a ray passing through a phase mask Ω at the position (x 0 , y 0 ), placed in the plane 0 z  , is deflected at an angle