Coherent control of optical polarization effects in metamaterials

Processing of photonic information usually relies on electronics. Aiming to avoid the conversion between photonic and electronic signals, modulation of light with light based on optical nonlinearity has become a major research field and coherent optical effects on the nanoscale are emerging as new means of handling and distributing signals. Here we demonstrate that in slabs of linear material of sub-wavelength thickness optical manifestations of birefringence and optical activity (linear and circular birefringence and dichroism) can be controlled by a wave coherent with the wave probing the polarization effect. We demonstrate this in proof-of-principle experiments for chiral and anisotropic microwave metamaterials, where we show that the large parameter space of polarization characteristics may be accessed at will by coherent control. Such control can be exerted at arbitrarily low intensities, thus arguably allowing for fast handling of electromagnetic signals without facing thermal management and energy challenges.

The coherent interaction of electromagnetic waves on thin functional linear materials can be described by Jones transmission and scattering matrices t and s. Here we consider signal E in and control E in e iα input beams with a phase difference α that are incident on opposite sides of a thin linear material as described in the main manuscript. The signal output E s,out is formed by the superposition of the transmitted signal beam tE in and the scattered (reflected) control beam sE in e iα . Similarly, the control output E c,out is the superposition of the scattered signal and transmitted control input beams. Therefore, where right and left arrows mark Jones matrices for the directions of the incident signal an control beams, respectively. In case of planar metamaterials, transmission simply corresponds to the superposition of incident and scattered fields, i.e. t = 1 + s, with 1 being the unity matrix.

A. Coherent control of manifestations of optical anisotropy
Optical anisotropy has been studied for a planar metamaterial (t = 1 + s). Planar anisotropic metamaterials are described by the same ordinary and extraordinary scattering coefficients s x and s y for both directions of illumination and therefore equation (1) simplifies to where j = x, y. From here, it is clear that the corresponding input and output beams must be identical for α = 180 • , when e iα = −1, implying complete transparency at the magnetic node, as observed in our experiments and simulations. It also follows that the output intensities will have a sinusoidal phase dependence, which can be written in the general form I out = A cos 2 ( α−α0 2 ) + C and has been seen experimentally and numerically.

B. Coherent control of manifestations of optical activity
Optical activity is most easily characterized in terms of its circular polarization eigenstates. Simple constructive or destructive interference occurs for incident control and signal polarizations that have the same projection onto the metamaterial plane. As the handedness of polarization states is measured looking into the beam, this implies that coherent control is achieved with signal and control beams of opposite handedness. Thus equation (1) becomes with E c,in ∓ = E s,in ± e iα± , where indices "+" and "-" stand for right-handed and left-handed circular polarizations.
The polarization states of the input and output beams are defined by their azimuth Φ and ellipticity angle η ) . (5)

C. Intrinsic 3D chirality
In the simplest case, an intrinsically 3D-chiral metamaterial is characterized by a polarization-independent scattering coefficient s 0 and transmission coefficients t + and t − , which are identical for opposite directions of illumination. Simulations show that this holds for the pair of twisted crosses discussed in the main manuscript, while it becomes an approximation at resonances in our experimental case, where resonant reflectivity is slightly different for opposite sides of the sample. Based on the above, equation (3) simplifies to For linear polarization, we can always choose coordinates so that E s,in + = E s,in − and α + = α − . In this case it is apparent from equations (4) and (5) that the control output at phase α is equivalent to the signal output at phase −α with the same azimuth rotation and ellipticity. The same conclusion has been drawn from symmetry arguments in the main manuscript.

D. Extrinsic 3D chirality
Extrinsic 3D chirality has been studied for planar metamaterials (t = 1 + s). In the limit of a vanishing anisotropic response the extrinsically 3D-chiral metamaterials have circular polarization eigenstates and are characterized by scattering coefficients s + and s − for the incident signal beam. As extrinsic chirality is reversed for the direction of the control beam, ← − s + = s − and ← − s − = s + , so that equation (3) simplifies to E s,out It follows from equations (4) and (5) that the control output at phase α corresponds to the signal output at phase −α with reversed azimuth rotation and ellipticity. The same conclusion has been drawn in the main manuscript based on symmetry arguments for planar metamaterials in general. Considering E c,in ∓ = E s,in ± e iα± , the corresponding input and output beams must be identical for α ± = 180 • , when e iα± = −1, implying complete transparency at the magnetic node as observed in our experiments and simulations. It also follows that the output intensities will have a sinusoidal phase dependence, which can be written in the general form I out = A cos 2 ( α−α0 2 ) + C and has been observed experimentally and numerically.