Generalized polarization transformations with metasurfaces: supplement

Metasurfaces are arrays of sub-wavelength spaced nanostructures, which can be designed to control the many degrees-of-freedom of light on an unprecedented scale. In this work, we design meta-gratings where the diffraction orders can perform general, arbitrarily specified, polarization transformation without any reliance on conventional polarization components, such as waveplates and polarizers. We use matrix Fourier optics to design our devices and introduce a novel approach for their optimization. We implement the designs using form-birefringent metasurfaces and quantify their behavior - retardance and diattenuation. Our work is of importance in applications, such as polarization abberation correction in imaging systems, and in experiments requiring novel and compact polarization detection and control.


SATISFYING THE SYMMETRY CONDITION FOR DESIGNING WITH METASURFACES
As mentioned in the main text, each metasurface nanopillar can only do a unitary and symmetric transformation of polarization at the incident plane. The far-field matrix distribution is given by the Fourier transform of the metasurface matrix distribution, made up of arrays of such nanopillars, at the incident plane. Note that a Fourier transform or series is essentially just a sum; a summation of unitary matrices can, in general, result in totally hermitian, or partially hermitian matrices, which is why we can design diattenutators and analyzers in the farfield, using phase-only (unitary) nanopillar arrays; however, note that a sum of symmetric matrices can never result in an asymmetric matrix. This is why our designs are restricted to symmetric only matrices in the far-field, if we use this metasurface platform. To better understand the extent of our metasurface polarization optics design space, we perform the following analysis to derive 'selection rules' for our devices, by using Pauli matrices.

A. Pauli Matrices and Polar Decomposition
We can write a Jones Matrix, most generally, as a sum of Pauli matrices [1]: where α, a 1 , a 2 , a 3 are, in general, complex, and the Pauli matrices are defined as: Furthermore, while describing the evolution of an incident state, through a polarizing system, we always describe the acts of retardance and diattenuation, sequentially. (For example, in commonplace optics setups, we talk about waveplate followed by a polarizer or vice versa). Therefore, we can use the polar decomposition to describe any Jones matrix as a multiplication of a (Unitary) retarder Jones Matrix, and a (Hermitian) diattenuation Jones matrix. Matrix operations, as we know, are sequential, and the sequence is particularly important when matrices do not commute. We will use the more common 'right' polar decomposition sequence: where U is Unitary and H is Hermitian. Now we can further decompose these two matrices as a sum of pauli matrices.

B. Unitary Matrix Decomposition
The Unitary matrix U can be written as a matrix exponential involving Pauli Matrices: here − β 0 2 is the common phase (scalar) which we can ignore from our subsequent analysis because its contribution is non-polarizing. Furthermore given − → β = β β, −β is the full retardance and β is the axis of retardation. Now using the Matrix equivalent of Euler's identity, we can write: Here − → σ is a vector of the three Pauli matrices − → σ = (σ 1 , σ 2 , σ 3 ), where the retarder axis vector β can be thought of as a 3-element Stokes vector so β = (β 1 , β 2 , β 3 ) = (S 1 , S 2 , S 3 ). We can write U more explicitly as: Or, without loss of any polarization properties, we can re-write:

C. Hermitian Matrix Decomposition
Following our analysis for Unitary Matrices, we can similarly analyse the Hermitian Matrix: Ignoring the overall scaling factor e α 0 2 , we can expand, analogously to the Unitary case: Without loss of any polarization properties, we can re-write H as: Note that − → α = α α. Further expansion gives us: From the equation, we see that the maximum and minimum differential losses are 1 ± tanh( α 2 ), with the axis of transmission parallel to α. But since Jones matrices deal with electric fields and not intensities, we need to square the expression for these losses. Using the definition of diattenuation D, involving maximal and minimal intensities, we can write: Rearranging, we can get α in terms of D.
Where D is constrained to be within 0 and 1. Note that we choose the − √ 1 − D 2 root over the + √ 1 − D 2 , because the negative root maintains physicality, while the positive root diverges over certain ranges of D.

D. Symmetry Condition
Now we can construct an arbitrary Jones matrix, starting with its polar decomposed form: We are ignoring the overall phase and scaling, as it is non polarizing. We can multiply the expression above and get the coeffecients of Pauli matrices in terms r s and d s. Furthermore, we realize that for any symmetric Jones matrix, its σ 3 component needs to be zero (as σ 3 is the only non symmetric Pauli matrix), which provides the necessary and sufficient condition to ensure symmetry. Thus to ensure there is no σ 3 component in our Jones matrix expression, we have to satisfy the following relations: r 3 = 0 (S17)