GoKawiilFourier-pixel design
Each Fourier pixel involves the generation, propagation and diffraction of guided waves (Fig. 1a). Although we focus here on SPPs, the design principles also apply to photonic waveguide modes. In our plasmonic Fourier pixels, we used sinusoidal gratings to generate SPPs. The SPPs were launched when the SPP wavevector, \({k}_{{\rm{SPP}}}\), satisfied (Extended Data Fig. 1)
$${k}_{{\rm{SPP}}}={k}_{\parallel }+n{g}_{{\rm{m}}}$$ (1)
where \({k}_{\parallel }\) is the in-plane wavevector of photons incident on the grating, \({g}_{{\rm{m}}}=2{\rm{\pi }}/\varLambda \) is the grating momentum, \(\varLambda \) is the grating period and \(n\) is the diffraction order. The photons have wavevector \({\bf{k}}\) with \(|{\bf{k}}|=k=\frac{2{\rm{\pi }}}{\lambda }\) with wavelength \(\lambda \). If the grating contains multiple spatial frequencies, it can couple photons of different wavelengths simultaneously at the same incident angle, launching SPPs with different \({k}_{{\rm{SPP}}}\).
In a Fourier pixel, the generated SPPs propagate in \(x\) across the \(x,y\) interfacial plane with transverse-magnetic polarization. We treat the SPPs as scalar reference waves of the form
$$r(x,y)={{\rm{e}}}^{{\rm{i}}{k}_{{\rm{SPP}}}x}$$ (2)
The SPPs then encounter the Fourier element that creates a desired complex-valued optical wavefront \(g(x,y)\) at a specific output plane through diffraction. For simplicity, we assumed a constant-amplitude SPP wave, neglecting propagation and outscattering-induced attenuation over the extent of the Fourier element. For higher-efficiency designs, such attenuation can be compensated by apodization of the scattering strength, as commonly used in integrated photonics, guided-wave holography and metasurface systems.
The inverse-design process for our Fourier pixels must predict the height profile \({h}_{{\rm{p}}}(x,y)\) of the Fourier element that generates the desired \(g(x,y)\). In general, diffraction of light by metallic surfaces can be treated by considering the local optical path differences introduced by the structured interface. We apply a similar diffraction model to describe the interaction of SPPs with our Fourier elements. For amplitude and phase, we use a scalar diffraction model. After the SPP reference wave \(r(x,y)\) interacts with the Fourier element, the optical wavefront at the diffractive surface, \(f(x,y)\), is described by the relation
$$f(x,y)=r(x,y)\,t(x,y)$$ (3)
where we introduced a complex-valued transparency function \(t(x,y)\). It describes how the Fourier element converts the SPPs into the desired wavefront at the diffractive surface. This wavefront then propagates to generate \(g(x,y)\). Equation (3) can be rearranged to give
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