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TitleShaping light beams with dielectric metasurfaces Daniël Pieter Stellinga
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Page 1

Shaping light beams

with dielectric metasurfaces

Dani�el Pieter Stellinga

Doctor of Philosophy

University of York


May 2016

Page 2


Page 76

amplitude regions are completely disconnected, it is unlikely that a good path

will be found. Excluding these beforehand and only running the pathfinding on

likely candidates is more practical.

The final result is a thickness, polarisation and set of periods and duty cycles

that together form the basis set for the design of a metasurface.

4.1.3 Phase pro�les and grating designs.

Once a suitable basis set of grating parameters is found, the next step is to

determine the phase profile that needs to be encoded into the metasurface. As

discussed in chapter 2.2, this involves finding the difference between the phase

profile of a beam incident onto the surface and then subtracting it from the desired

phase profile of the output beam. Time reversal is still true, so the output phase

profile can be found by propagating the resulting beam in the reverse direction

using, for example, Huygen’s principle (see chapter 2.1) and again determining

its phase distribution where the metasurface intersects it.

In principle this holds for any combination of phase profiles, limited only by

the unit cell size and Nyquist’s theorem. For simplicity, assuming a plane wave

and normal incidence makes the phase profile on the incident side a constant, and

therefore permits only encoding the desired phase profile onto the metasurface.

Crucially, a phase profile is independent of transmission or reflection. In

other words, a parabolic mirror has the same phase profile as a lens. Moreover,

while the phases of transmission and reflection are not equal and the resonance

conditions are independent, they do both increase with optical path length and

therefore tend to show similar behaviours along a path. This behaviour leads to a

metasurface designed for reflection generally also creating a similar beam profile

in transmission, albeit less intense.

Two particular beam types tend to be interesting for exploring metasurface func-

tionalities: lenses (or parabolic mirrors) and vortex beam phase plates. The

former has a purely radial phase profile which is completely independent of the

angle around the origin, while the latter creates a vortex beam with a purely

azimuthal phase distribution that is independent of phase. Their orthogonal na-

ture means they can also be combined into a single metasurface that combines

both behaviours without loss of functionality. Phase profiles and example meta-

surfaces based on them are shown in figure 4.5. In the phase profiles (a), (c) and

(e), the symmetry of the different types of beam is reflected very clearly, with (a)

changing only radially, (c) only azimuthally and (e) in both those dimensions.

This symmetry is still present, though less obvious, in the resulting metasurface

designs of (b), (d) and (f).


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(a) Parabolic mirror phase profile (b) Parabolic mirror grating design

(c) Phase plate phase profile (d) Phase plate grating design

(e) Combined phase profile (f) Combined grating design

Figure 4.5 – On the left are shown the phase profiles of a focusing mirror (a), LG
generating phase plate with l = 1 (c), and a combination of lens and phase plate
with l = 2 (e). On the right corresponding grating designs.


Page 152

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