Nonseparable Polarization Wavefront Transformation

In this Letter, we investigate a new class of polarization wave front transformations which exhibit nonconventional far field interference behavior. We show that these can be realized by double-layer metasurfaces, which overcome the intrinsic limitations of single-layer metasurfaces. Holograms that encode four or more distinct patterns in nonorthogonal polarization states are theoretically demonstrated. This Letter clarifies and expands the possibilities enabled by a broad range of technologies which can spatially modulate light’s polarization state and, for metasurfaces specifically, rigorously establishes when double-layer metasurfaces are—and are not—required.

Figure 1

(a),(b) Schematics of separable and nonseparable transformations. (c)–(h) The output far field patterns for different incident polarization states. $\overrightarrow{\lambda }$ and ${\overrightarrow{\lambda }}_{\perp }$ are orthogonal. Among them, (b) and (c) are separable states; (d) and (f)–(h) are nonseparable states. In terms of far field superposition, (d) is an intensity sum of (b) and (c), whereas (h) shows a new pattern that is different from (f) or (g).

Figure 2

(a) Schematics of the retarder space. Each point represents a class of unitary Jones matrices (wave plates) with the same polarization effect, up to some overall phase. The radial distance and orientation are related to the phase retardation and the eigenpolarization states respectively. For example, points 1–3 have the same fast axis along $x$, but varying retardance, from zero to quarter wave to half wave. Points 3–5 are all half wave plates, but with different eigenpolarizations states. In particular, points in the equatorial plane correspond to linearly birefringent elements. (b)–(f) A polarization wave front transformation can be represented in the retarder space as a curve or surface or a collection of points. (b) Separable transformations are mapped to ellipses. (c) Nonseparable transformations can have arbitrary shapes. (d)–(f) Polarization transformations by single-layer, generalized single-layer, and double-layer metasurfaces are represented by the equatorial plane, ellipsoids, and arbitrary shapes or points respectively.

Figure 3

(a)–(c) Side views of single-layer, generalized single-layer, and double-layer metasurfaces. (d) The Venn diagram summarizes the relation between different types of polarization transformations (marked by fill pattern) and the possible metasurface implementations (marked by color). Generalized single-layer metasurfaces (light blue) can realize all separable transformations (line pattern) and a subset of nonseparable transformations (dot pattern). Double-layer metasurfaces (dark blue) can implement any unitary polarization transformation.

Figure 4

(a) A schematic of the device. It consists of two layers of ${\mathrm{TiO}}_2$ rectangular nanopillars fabricated on top of one another. The bottom layer is encapsulated in glass. The top layer is exposed to air. (b) The building blocks of the metasurface. $H_t=600\mathrm{nm}$, $H_b=1600\mathrm{nm}$. $U=225\mathrm{nm}$. The design wavelength is $\lambda =488\mathrm{nm}$. (c) The optimized target Jones matrix profile. For visualization purposes, only 1% of the points are shown. (d) The retarder space representation of the simulated library. (e)–(h) The simulated far field intensity pattern for horizontal, vertical, 45 degree, and left circular incident polarization state respectively. (i)–(j) The simulated far field polarization distribution for horizontal and vertical incident polarization. (k)–(l) The real and imaginary parts of ${\overrightarrow{E}}_H^{*}·{\overrightarrow{E}}_V$. Red and blue correspond to positive and negative value respectively.