At-will chromatic dispersion by prescribing light trajectories with cascaded metasurfaces

Chromatic dispersion spatially separates white light into colours, producing rainbows and similar effects. Detrimental to imaging but essential to spectroscopy, chromatic dispersion is the result of material properties in refractive optics and is considered an inherent characteristic of diffractive devices such as gratings and flat lenses. Here, we present a fundamental relation connecting an optical system’s dispersion to the trajectories light takes through it and show that arbitrary control over dispersion may be achieved by prescribing specific trajectories, even in diffractive systems. Using cascaded metasurfaces (2D arrays of sub-micron scatterers) to direct light along predetermined trajectories, we present an achromatic twisted metalens and experimentally demonstrate beam deflectors with arbitrary dispersion. This new insight and design approach usher in a new class of optical systems with wide-ranging applications.

x (μm) x (μm) x (μm) x (μm) x (μm) y x Intensity (a.u.)     Figure S5 | Beam deflector phase profiles. Metasurface phase profiles for (a) superchromatic, (b) ordinary grating, (c) achromatic, and (d) positive dispersion beam deflectors. In each panel the green curve corresponds to the profile for the input metasurface (φ 1 ), and in panels a, c, and d, the blue curve corresponds to the profile for the output metasurface (φ 2 ). Diagrams to the right of each graph depict ray paths (not to scale).      Figure S13 | Engineering dispersion. (a) Diagram of a ray traversing the beam deflector system, defining coordinate axes x 1 , x 2 , u, and z, deflection angles θ 1 and θ 2 and indicating regions with refractive indices n 1 , n 2 , and n 3 . (b) Diagram showing a plane wave arriving at u with oblique incidence.
through the system at its design frequency ω can be parameterized by a set of position vectors {r 1 , r 2 , . . . , r M }, where r m represents the intersection of the path with the mth surface. We designate as fiducial the ray which impinges the first surface at r 1 . If frequency is changed to ω + ∆ω, the deflection of the fiducial ray at each interface is modified, altering its trajectory: its intersection with the mth surface becomes r m + ∆r m , where ∆r m = dr m /dω ∆ω. At frequencies other than ω, the fiducial ray still emanates from O but does not necessarily pass through I, so we define l M +1 as the distance the ray travels between the last surface and a reference sphere [1] centred at I with radius equal to the minimum distance between I and the M th surface (see Fig. S12). To first order in ∆ω, the accumulated phase changes by where ∇ m represents the gradient with respect to r m . According to Fermat's principle, the total phase acquired is stationary with respect to path variations (∇ m Φ = 0 for all m) [1], so the second term in parentheses vanishes, leaving ∆Φ = (∂Φ/∂ω) ∆ω. Though path variation has no effect on ∆Φ to first order, it does contribute at second and higher orders, which determine the bandwidth of an achromatic system.
The system focuses rays to point I at frequency ω + ∆ω if and only if the phase accumulated along each ray's path (Φ + ∆Φ) is the same. Because Φ is the same for each ray, this implies the system is achromatic if and only if ∆Φ or equivalently l g is the same for all rays. Equation 2 is obtained by partial differentiating Eq. 1 and multiplying by c.
Design of bilayer apochromat. We present a design for a bilayer apochromatic metalens-a lens with the same focal length at three wavelengths-in Fig. S3. Like the twisted metalens presented in the main text, the apochromat has an annular aperture (150 µm< r 1 <300 µm) and 1 mm focal length, as shown schematically in Fig. S3a. Unlike the twisted metalens, the apochromat achieves the OGL criterion at two wavelengths and only deflects rays radially. The metalens comprises two metasurfaces that are separated by a 1-mm-thick substrate (n=1.46); their radially symmetric phase profiles for are shown in Fig. S3b.
The apochromat is designed to work at visible wavelengths, and is well-corrected over the 450-seen in the ray diagrams in Fig. S3c: shorter wavelengths (450 nm) are deflected to a shallow angle, arriving at the second metasurface closer to its centre, whereas longer wavelengths (650 nm) are deflected to a steeper angle by the first metasurface, and arrive at the second metasurface farther from the optical axis. In the centre of its operating bandwidth (550 nm), the apochromat has an effective NA of 0.21. The consequences of the wavelength-dependent NA are apparent in the transverse and axial intensity profiles (Fig. S3d). The apochromat, like the twisted metalens, has a narrow field of view, as evidenced by the transverse intensity profile for a 550 nm field incident at 0.1 • .
To benchmark the performance of the bilayer apochromatic metalens, we designed a singlelayer hyperbolic metalens with the same annular aperture, shown schematically in Fig. S3e. This control lens was designed to have NA=0.21 at its design wavelength of 550 nm, matching that of the apochromat and producing a focal length of 1.4 mm. The transverse and and axial intensity profiles are shown in Fig. S3f, demonstrating chromatic behaviour qualitatively similar to the other single-layer metalenses presented in this work (see Fig. 3e and Fig. S2b).
The shift of focal length with wavelength of these two metalenses are compared in Fig. S3g.
The single-layer metalens exhibits a shift in focal length of ∆f ≈500 µm over the band considered, consistent with the expected behaviour for a non-dispersive, single-layer metalens with this By contrast, the focal length of the apochromat shifts less than 5 µm over the same range.
Designing beam deflector dispersion. Figure S13a shows the path of a ray through a system of two metasurfaces. Assume the ray is at the design frequency ω. The first metasurface lies on the x 1 axis, and deflects the normally-incident ray to an angle θ 1 . The second metasurface lies on the x 2 axis, parallel to and a distance d from x 1 , and deflects the ray to an angle θ 2 . The deflection angle for the beam deflector system is θ 2 , and thus is the same for all rays at frequency ω; θ 1 may vary among rays. The u axis makes an angle θ 2 with the x 2 axis, and the origin for each axis lies at their intersection. The medium between x axes has refractive index n 2 and the medium to the right of x 2 has refractive index n 3 . At the design frequency, the total phase accumulated from x 1 to u is constant for all rays. The group length for the ray can be expressed in these coordinates and media as l g (u) = n 2 d cos θ 1 + n 3 u tan θ 2 .
We will impose the requirement that l g (u) = au+l 0 . We can then write the phase change along u due to a frequency change ∆ω as Compare this with the phase profile for a plane wave arriving at u at oblique incidence (Fig. S13b), where ∆θ is the angle of the wavefront with respect to u. Equating first-order coefficients in Eq. S4 and Eq. S5 gives ∆θ ∆ω = − a n 3 ω , where we have used sin ∆θ ≈ ∆θ. Choosing a = tan θ 2 gives ordinary grating dispersion, and a = 0 gives the achromatic condition. Superchromatic and positive behaviour results from exceeding these bounds.
For a given value of a and a choice of l 0 we can solve Eq. S3 for θ 1 (u). Coordinates x 1 , x 2 and u are related by x 2 = u cos θ 2 (S7) and x 1 = x 2 − d tan θ 1 .
Phase surfaces are then designed to produce the necessary deflections at x 1 and x 2 .