Engineering the Phase Front of Light with Phase-Change Material Based Planar lenses

A novel hybrid planar lens is proposed to engineer the far-field focusing patterns. It consists of an array of slits which are filled with phase-change material Ge2Sb2Te5 (GST). By varying the crystallization level of GST from 0% to 90%, the Fabry-Pérot resonance supported inside each slit can be spectrally shifted across the working wavelength at 1.55 µm, which results in a transmitted electromagnetic phase modulation as large as 0.56π. Based on this geometrically fixed platform, different phase fronts can be constructed spatially on the lens plane by assigning the designed GST crystallization levels to the corresponding slits, achieving various far-field focusing patterns. The present work offers a promising route to realize tunable nanophotonic components, which can be used in optical circuits and imaging applications.


Supplementary Information
Section 1

The comparison of focal length error from lenses with different Fresnel number based on the analytical model
Lens span W is an important parameter when comparing the focusing capability from lenses sharing the same working wavelength λ and nominal focal length f nominal . However, since W, λ and f nominal can vary from one design to another, for example, a GHz planar lens with a span of a few centimeters and a nanoscale planar lens working in visible frequencies, a dimensionless gauge is needed to explain the optical responses of the planar lenses across different scales. The Fresnel number (FN), defined as W 2 /(λf nominal ), serves this purpose. A clear study about the effect of FN value on the focal length error is presented in the paper 'On the chromatic aberration of microlenses ' (Opt. Express, 14, 4687-4694 (2006)), demonstrating that a lens of small FN has a larger focal length error.
Here, we include a comparison of focal length error between two lenses of different spans (different FN) to show its effect on the lens performance. We use the analytical model of 'optimal' planar lens for the demonstration. The other two analytical models ('quasi-optimal' and 'realistic' planar lenses) are just the derivations of the 'optimal' case by limiting the phase coverage and adding intensity variation.
We report in Supplementary Table S1 on the parameters of the two situations considered. The original design denotes the design we show in the main manuscript with W=10 µm and the expanded design represented the case of W=32 µm. Lens span for the expanded case is chosen in this way so that the ratios among W, λ and f nominal are the same as the design reported in Appl. Phys. Lett. 103, 183507 (2013). In other words, they have the same FN values, therefore a direct comparison can be established. We keep the spacing between neighboring point sources the same at 0.5 µm, therefore there are 65 point sources in the expanded case. Nominal focal length is set to be 13.95 µm (9λ) for both situations.

Supplementary Table S1. The parameters of the two planar lenses
It can be seen that the expanded case has a focal length f at 13.67 µm (8.8λ) and the focal length error is reduced to 2.0% compared to the original desgin. This focal length error is in the same scale as show in the reference paper mentioned above. The FN values are reported in the last row of supplementary Table 1 for comparison.
For the sake of completness, we plot the normalized magnetic intensity distributions along the lens axis for the expanded and original designs, which is shown in Supplementary Fig. 1.   Supplementary Fig. S1.

The normalized magnetic intensity distributions along the lens axis
Here we also report the oberseved focal lengths and FWHMs from 'realistic' and 'quasioptimal' cases at the expanded W=32 µm. The data are shown in Supplementary Table S2. Clearly, the limitations of the phase coverage and intensities introduce a trade-off between the focal length error and FWHM. Without Fresnel or Fraunhofer diffraction theory approximations, the 2D integral representation of the scalar electric field diffraction from an aperture is given by is the transmission function, and the integral ranges the whole span of the lens, . Similarly, the electric field directly transmitted through the aperture reads If we choose the position ! = !"#$!%& , = !"#$!%& as the focal point of our lens We assume that the transmission of our lens is given by We have The equal optical path principle states that the phase accumulated by the diffracted waves must be the same as the directly transmitted waves except for a 2 factor (where n is an integer), having: In Fresnel diffraction, the transmitted fields, ≫ and