Extra Phase

We will use the term “extra phase (𝜙)” to refer to the phase defined by:

$$𝜙(𝑥,𝑦) = {2𝜋n\over𝜆_o} \lbrace\sqrt{(x-x_f)^2 +(y-y_f)^2+f^2}-f\rbrace$$

, where n: refractive index of the transmitted medium, 𝜆_o: wavelength in vacuum, (x_f, y_f): x and y positions of focal spot, and f: focal distance. Extra phases for several unit cell positions is visualized in the figure below; where each unit cell is a source of a spherical wave. Because of extra phase, these spherical waves cannot superpose constructively at the focal spot. This indicates that unit cells must introduce different amounts of phase shift to the incident wave, a plane wave, to cancel the effect of extra phase and produce constructive interference at the focal spot.

We will use another term “unitcell phase (or unit cell phase)” to refer to the phase shift that a unit cell imposes to the incoming wave, illustrated in the figure below. The unitcell phase is equal to the phase of the unit cell's S-parameter S₂₁. For an ideal lens, unitcell phases at all unit cell positions are equal to the extra phase at that position.

Phase shift to focus light. A plane wave is incident from the right side of the lens. Diffracted waves (spherical waves) at the unit cells superpose at the same phase (90° in the figure) at the focal spot. A and A’: phases of incident and transmitted waves, respectively, at an earlier moment. B and B’: phases of incident and transmitted waves, respectively, at a later moment.