Analysis and experimental investigation of a subwavelength phased parallel-plate waveguide array for manipulation of electromagnetic waves

Phase-gradient metasurfaces can be designed to manipulate electromagnetic waves according to the generalized Snell’s law. Here, we show that a phased parallel-plate waveguide array (PPWA) can be devised to act in the same manner as a phase-gradient metasurface. We derive an analytic model that describes the wave propagation in the PPWA and calculate both the angle and amplitude distribution of the diffracted waves. The analytic model provides an intuitive understanding of the diffraction from the PPWA. We verify the (semi-)analytically calculated angle and amplitude distribution of the diffracted waves by numerical 3-D simulations and experimental measurements in a microwave goniometer.


Derivation of the equation system (Eq. (17) -(20) in the main article)
We briefly derive the equation system that describes the x-component of the electric field E x and the y-component of the magnetic field H y of the electromagnetic waves inside and outside the phased parallel-plate waveguide array (PPWA). Region I corresponds to the space of the incident and back diffracted wave, region II refers to the PPWA space and region III to the area after transmission through the PPWA, as illustrated in Fig. 1. As in the main article, E i,x and E i,z with i = I, III describe the electric field xand z-component in the i th region, E II,m,x and E II,m,z the electric field xand z-component in the m th waveguide in region II, H i,y with i = I, III the magnetic field y-component in the i th region, H II,m,y the magnetic field y-component in the m th waveguide in region II, n m the refractive index of the m th waveguide in a single cell, η i with i = I, III the wave impedance in region i, η II,m the wave impedance in the m th waveguide in region II, L the width of a cell, w the waveguide width, h the waveguide length, r l the reflection coefficient for the l th order back diffracted wave at the boundary at z = 0, t l the transmission coefficient for the l th order forward diffracted wave at the boundary at z = h, k x and k z the xand z-component of the vacuum wave vector of the incident wave, k 0 the magnitude of the vacuum wave vector of the incident wave, k x,l and k z,l the xand z-component of the vacuum wave vector of the l th order diffracted wave, sinc(x) = sin(x)/x the sinc-function, δ i j the Kronecker delta, and C + m and C − m the complex amplitudes of the forward and backward propagating waves in the m th waveguide. In the following, we consider the boundary conditions for the tangential component E x of the electric field and the tangential component H y of the magnetic field at z = 0 and z = h, see Fig. 1. At this point it must be noted that we cannot obtain an analytic solution if we strictly uphold the boundary condition of matching tangential components for each single point at the boundary of each waveguide aperture. In order to obtain an analytic solution, it is necessary to approximate the boundary conditions for the tangential component in such a way as to ensure that the average value of the tangential component of the electric and magnetic fields match along the boundary of each waveguide aperture. Although this loosened requirement does not strictly match the exact boundary conditions, we obtain good agreement between the numerical simulation of the electromagnetic waves and the analytic calculation, as shown later by comparison.
We assume a perfect electric conductor for the waveguide walls, which implies that E x must equal zero at the waveguide walls at z = 0. Along the waveguide aperture, the condition E II,m,x = E I,x must be fulfilled for continuity. Equation (5) contains the approximation step for the boundary condition by averaging the tangential component of the electric field over the waveguide aperture.
By changing the index of the diffraction order from r to l, we obtain the same nomenclature as in the primary manuscript: 2/7 H y at z=0 As above, we assume a perfect electric conductor for the waveguide walls, which implies that H y must be continuous at the waveguide walls at z = 0. Along the waveguide aperture, the condition H II,m,y = H I,y must also be fulfilled for continuity. Equation (14) contains the approximation step for the boundary condition by averaging the tangential component of the magnetic field over the waveguide aperture.
H I,y = H II,m,y for x in the m th PPW (11) dx Integration over m th PPW (13)

3/7
In analogy to the previous evaluation of the boundary conditions at z = 0, we calculate E x and H y at z = h. Here, Eqs. (23) By changing the index of the diffraction order from r to l, we obtain the same nomenclature as in the primary manuscript: dx Integration over m th PPW (30) The relaxation of the electromagnetic boundary conditions by requiring continuity only for the average value of the tangential components of the electric and magnetic field instead of point-wise compliance can be observed by comparison between the numerically calculated electromagnetic fields in the PPWA and the corresponding analytic calculation. This is shown for the example of H y in the PPWA in Fig. 2, where H y is calculated analytically and numerically for −L < z < h + L. As can be seen in the insets, which display a zoom of the magnetic field H y at the boundary z = 0, the magnetic field H y is strictly continuous in case of the numeric calculation, whereas H y is discontinuous along the waveguide boundary in the analysis due to the averaging process described above. However, since the width of the waveguides is small compared to the wavelength of the waves, the averaging error is marginal and deviations between the numerical and analytic calculation of the tangential fields at the boundaries are almost negligible. 2 Fourier Series of a 180 • phase grating and an arbitrary phase and amplitude grating As described in the main article, the impact of the PPWA on the forward wave diffraction of arbitrary waveforms at z = h can be described by a superposition of an L/2-periodic amplitude and phase grating g(x) and an L-periodic binary phase grating h(x) that induces zero phase shift in the first half cycle and a 180 • phase shift in the second half cycle. For the calculation of the transmitted diffraction orders at z = h, we develop both functions g(x) and h(x) in terms of a Fourier series and obtain the function of the diffracted waves as the product f (x) = g(x) × h(x).

Fourier series of an arbitrary amplitude and phase grating with a period of L/2
The Fourier series of an arbitrary L/2-periodic amplitude and phase grating can be readily obtained by where a k is the complex Fourier amplitude of the amplitude and phase grating.

Fourier series of the binary phase grating
The function h(x) of the binary phase grating is depicted in Fig. 3. As obvious, the binary phase grating induces a 180 • phase shift in the second half cycle of the periodic function, while the amplitude remains constant at unity. With ω 0 being the (one-dimensional) wavevector, L the width of a cell and c k the Fourier amplitude, the Fourier series of h(x) is described by ⇒ c k,k =0 = 0 for k = even = ±2, ±4, ...