Fig. 2: Long-range optical coupling in ENZ multilayers and the anti-correlated optical near-field intensities.

a Schematic of two ENZ thin films separated by a silica spacer layer. b Near-field distribution |E | / | E₀| of optical coupling between two ENZ layers with a separation l = 0.2 µm. c Oscillatory and anti-correlated optical field intensity in ENZ outer (left) and inner (right) layers on silica with an increased spacer separation. d Phase distribution plot for the ENZ/silica/ENZ multilayers at different interlayer separation l in panel c. The shaded regions suggest location of the outer and inner ENZ layers. For separation distance l = l_a, l_b, l_c, and l_d, the phase difference between the inner and outer ENZ layer (∆ϕ> = \({\phi }_{{inner}}\) - \({\phi }_{{outer}}\)) is π/2, π, - π/2, and 2π, respectively. e Angle-dependent optical fields ( | E | ²/ | E₀ | ²) of the two identical ENZ thin layers with a separation l = 0.2 µm under TM polarization. f Anti-correlated optical near fields between ENZ double layers with a spacing up to hundreds of microns. The film thickness is d = 50 nm with ε_i ≅ 0, pump wavelength λ = 1.3 µm and incident angle θ_i = 7°.