Supplemental Document

Design of continuously variant metasurfaces forconformal transformation optics: supplementGIANLUCA RUFFATO1,2,∗ AND FILIPPO ROMANATO1,2,3

1L.i.f.e.L.a.b. Program, Consorzio per la Ricerca Sanitaria (CORIS), Veneto Region, Via Giustiniani 2,35128, Padova, Italy2Department of Physics and Astronomy ‘G. Galilei’, University of Padova, via Marzolo 8, 35131 Padova,Italy3CNR-INFM TASC IOM National Laboratory, S.S. 14 Km 163.5, Basovizza, 34012 Trieste, Italy∗gianluca.ruffato@unipd.it

This supplement published with The Optical Society on 27 October 2020 by The Authors underthe terms of the Creative Commons Attribution 4.0 License in the format provided by the authorsand unedited. Further distribution of this work must maintain attribution to the author(s) and thepublished article’s title, journal citation, and DOI.

Supplement DOI: https://doi.org/10.6084/m9.figshare.12994451

Parent Article DOI: https://doi.org/10.1364/OE.400627

Design of continuously-variant metasurfaces for conformal transformation optics: supplemental document

GIANLUCA RUFFATO,1,* AND FILIPPO ROMANATO

1,2

1 Department of Physics and Astronomy ‘G. Galilei’, University of Padova, via Marzolo 8, 35131 Padova, Italy 2 CNR-INFM TASC IOM National Laboratory, S.S. 14 Km 163.5, Basovizza, 34012 Trieste, Italy *gianluca.ruffato@unipd.it

1. Transformation optics in the paraxial approximation

For the benefit of the readers, we provide in the following the mathematical derivation of the relation between the desired transformation and the phase pattern of the optical element required to perform the task, which is expressed by Eq. (10) in the main manuscript. In the paraxial regime, the field distribution after a propagation length f from an optical element, endowed with a phase function ( ),x yΩ , is given by the following formulation of the Fresnel-Kirchhoff integral [1]:

( ) ( ) ( ),( )1, ,

xu yvik

i x yi ffU u v dxdyU x y e e

i fλ

+−Ω= , (S1)

being ( )iU the input field illuminating the optical element. The method of stationary phase approximation allows one to express the result of the integral in Eq. (S1) with the values assumed by the argument at the stationary points (x*,y*) of the total phase Ф(x,y)=Ω(x,y)-k(xu+yv)/f, that is [2]:

( ) ( ) ( )

( )

* *

* *

,( ) * *

,

1 2, , j j

j j

i x yif j j

x y

iU u v U x y e

i f H

π σλ

Φ≅ , (S2)

where H is the Hessian determinant of Ф calculated at the stationary points, and σ=sgn(∂2Ф/∂x2) when H>0, σ=-i, otherwise. The stationary condition on the total phase, i.e.

0∇Φ = , leads to the following relation between the gradient of optical element phase and the point of coordinates (u, v) on the destination plane:

[ ]ku v

f∇Ω = , (S3)

This relation can be thought as a mapping (x, y)→ (u(x, y), v(x, y)) between the input field defined on the coordinate system (x, y) and the output field on the plane (u, v) after propagation at a distance f. A particularly interesting case, which is treated in this article, is given when the mapping is conformal, and it is described by a complex anti-holomorphic function ( ) ( ) ( ), ,g z u x y iv x y= + , where z=x+iy. Under that definition, the coordinates (u(x,y), v(x,y)) on the destination plane are given by the real and imaginary parts of g, respectively, then Eq. (S3) can be rewritten as it follows:

kg g

f∇Ω = ℜ ℑ , (S4)

The condition of harmonicity for the phase function ( ),x yΩ descends from the criterion of anti-holomorphic (conformal) mapping, as shown in the main text.

For a given conformal mapping g, Eq. (S4) allows one to obtain the phase pattern of the corresponding optical element which is necessary to implement the transformation. As suggested by Eq. (S2), the output field is equal to the input one calculated in the new reference frame and rescaled by a spatially-variant factor given by |H|-1/2, being H the Hessian determinant of the total phase Ф. In addition, the transformed beam carries a phase term exp(iФ(x*, y*)). The compensation of this last term is performed by placing a second phase-element, the so-called phase-corrector, having a phase pattern equal to -Ф(x*, y*). When the sequence of two elements is illuminated in reverse, the latter element performs the inverse conformal mapping g-1, mapping a point (u, v) to the corresponding point (x, y).

2. Far field of q-plates and conformal transformations for OAM beams

We report in this section a numerical calculation of the far-field intensity and phase distributions for the optical elements considered in this work, in particular q-plates, log-pol mapping and circular-sector transformations. For the last two mappings, the visual transformation of the input azimuthal gradient of the OAM beam allows one to better understand the effect of the transformation on the input field.

In Fig. S1, the far-field is calculated for a Gaussian beam illuminating spiral phase patterns with phase exp(iℓϑ) for different values of ℓ. The output field is endowed with the same azimuthal term, and the value ℓ assumes the physical meaning of orbital angular momentum (OAM) per photon in units of h/2π. A metasurface implementation of the central zone is reported also here. The incident beam is assumed to be right-handed circularly polarized. As shown in Eq. (1) in the main text, the same metasurfaces would transfer the opposite phase terms to a left-handed circularly polarized beam.

In Fig. S2 the effect of the sequence of unwrapper and phase-corrector is shown for an input OAM beam. The input intensity ring is unwrapped and transformed into a linear intensity distribution, and the same mapping is experienced by the input phase, mapping the input azimuthal gradient to a linear one.

In Fig. S3 the effect of circular-sector transformations with different values and signs of n are shown. The azimuthal phase gradient of an input OAM beam is mapped onto a circular sector of amplitude 2π|n|. An inversion of the phase gradient is performed when n is negative. When the sequence is illuminated in reverse, the wrapping of a circular sector of amplitude 2π|n| onto the whole circle is performed, i.e. the inverse circular-sector transformation by a factor 1/n is performed.

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