Coherent control of light-matter interactions in polarization standing waves

We experimentally demonstrate that standing waves formed by two coherent counter-propagating light waves can take a variety of forms, offering new approaches to the interrogation and control of polarization-sensitive light-matter interactions in ultrathin (subwavelength thickness) media. In contrast to familiar energy standing waves, polarization standing waves have constant electric and magnetic energy densities and a periodically varying polarization state along the wave axis. counterintuitively, anisotropic ultrathin (meta)materials can be made sensitive or insensitive to such polarization variations by adjusting their azimuthal angle.


Planar symmetries: Effect on scattering matrices and coherent absorption in polarization standing waves (PSWs)
An ultrathin sample can be categorized by its planar symmetry -whether it is or is not anisotropic and/or 2D-chiral. In the main body of this work we consider an anisotropic and 2D-achiral sample in an Figure S2. Schematic of the experimental arrangement for measurements of coherent absorption in ultrathin media. The beam from a mode-locked Ti:sapphire laser (130 fs pulse duration; 10-20 nm spectral FWHM) is modulated by a mechanical chopper and then split by a pellicle into two beams. A piezoelectric translation stage located in one beam path sets/tunes the relative time delay and thus the relative phase difference between pulses arriving at the sample from opposing sides. A half-wave plate in the other beam path controls the mutual orientation of the two beams' polarization to establish either an energy or a polarization standing wave at the sample. The two beams are focused at normal incidence from opposite sides, to a diameter ~10 μm, onto the sample using plano-convex lenses. Their average powers at the sample position are balanced using a variable neutral density filter (ND filter), and maintained below 1 mW per beam to exclude opto-thermal and nonlinear effects. The two output beams (transmitted and reflected from both sides of the sample) are directed via two non-polarizing beamsplitters (NBSs) to a pair of identical photodiodes, the signals from which were monitored using lock-in amplifiers referenced to the chopping frequency of 1.6 kHz.
3 anisotropic, 2D-chiral, and 3D-achiral PSW formed by counter-propagating orthogonally linearly polarized waves. Here, we use the scattering matrices ± to elucidate the behaviour of lossy planar media with all possible planar symmetries in such a PSW.
As discussed, an ultrathin medium illuminated at normal incidence has scattering matrices of the general form where a, b and c are complex scattering parameters for linearly polarized waves. In what follows we show how a structure's planar symmetry can limit the number of free parameters in this set.
An ultrathin medium subjected to planar operations such as rotation around a point or reflection about a line in the xy-plane, is described by a new matrix ± ′ , where is the operation matrix. For example, counter-clockwise rotation around the origin by an angle φ is described by the rotation matrix ,

= ( cos − sin sin cos )
Similarly, reflection in the x-axis is described by the reflection matrix ,

Isotropic media
Homogeneous and (chiral or achiral) structured planar media with at least 3-fold rotational symmetry present isotropic scattering properties that do not depend on the azimuthal orientation. The relation ± −1 = ± must therefore hold true for any rotation angle . In principle, this condition is satisfied by a scattering matrix with identical diagonal elements, wherein one off-diagonal element is the negative of the other.
However, as has already been established (see article main text), for a vanishingly thin medium the offdiagonal polarization conversion terms of the matrix must be identical (Eq. S1). So in this case they can only be zero: Thus, isotropic planar media cannot exhibit coherent absorption modulation in a PSW formed by counter-propagating orthogonally linearly polarized waves -the material response will show no dependence on either the phase difference θ between incident waves or on the sample's azimuthal orientation.

Anisotropic and 2D-achiral media
The L-slot metasurface discussed in the main body of this work and employed for the experimental demonstration of coherent PSW absorption modulation is an example of an anisotropic and 2D-achiral structure. Patterns of this type always have a mirror symmetry axis.
If this line of mirror symmetry is oriented parallel to the x-axis, then the structure (and its scattering matrix) must be unaffected by reflection in the x-axis, i.e. Thus, in a PSW the coherent absorption of a lossy, anisotropic and 2D-achiral sample will be a function of the mutual phase θ of the incident beams, except when its mirror symmetry axis is aligned with one of the incident linear polarization states, in which case absorption is independent of θ (as seen in the reported experiments).

Anisotropic and 2D-chiral interfaces
Low symmetry anisotropic, planar chiral patterns of rotational order 1 or 2 generally place no constraints on the allowed scattering properties for normal incidence illumination, i.e. the parameters a, b and c in Eq. S1 remain mutually independent and may all be non-zero.
For such samples, coherent absorption in a PSW will be modulated with both the azimuthal orientation of the sample and the mutual phase θ of the incident beams. There will be no azimuthal orientation for which absorption is independent of θ.