Phase

This page discusses different types of phase in the context of metalens.

1. Extra Phase

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

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

, 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 are described 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.

2. Unit Cell Phase

Definition

We define the term “unit cell phase (𝜙_uc)” such that it is equal to the phase of the scattering parameter S₂₁ of the unit cell, regardless of which phase sign convention is used. In other words, unit cell phase is the phase of S₂₁.

Phase Sign Convention

Depending on the computation method, the phase of S₂₁ can have different signs for the same unit cell: +90 deg or -90 deg, for example. In the below image, the phase of S₂₁ for the red wave is +90 deg. In this sign convention, for an ideal metalens, unit cell phases at all unit cell positions must be equal to their corresponding extra phase (𝜙_uc = 𝜙_ext). In the other sign convention, for an ideal metalens, flip the sign of unit cell phase and that value must be equal to the extra phase (-𝜙_uc = 𝜙_ext) — the term "-𝜙_uc" here is also referred to as "target phase" in some literature.

In SMD PRO, users can choose the sign convention for their unit cell.

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.