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Supplemental Document
Multi-dimensional wavefront sensing using
volumetric meta-optics: supplement
C
ONNER
B
ALLEW
,
G
REGORY
R
OBERTS
,
AND
A
NDREI
F
ARAON
Kavli Nanoscience Institute and Thomas J. Watson Sr. Laboratory of Applied Physics, California Institute
of Technology, Pasadena, California 91125, USA
faraon@caltech.edu
This supplement published with Optica Publishing Group on 14 August 2023 by The Authors
under the terms of the Creative Commons Attribution 4.0 License in the format provided by the
authors and unedited. Further distribution of this work must maintain attribution to the author(s)
and the published article’s title, journal citation, and DOI.
Supplement DOI: https://doi.org/10.6084/m9.figshare.23906151
Parent Article DOI: https://doi.org/10.1364/OE.492440
Multi-dimensional wavefront sensing using
volumetric meta-optics: supporting information
Conner Ballew, Gregory Roberts, and Andrei Faraon
Kavli Nanoscience Institute and Thomas J. Watson Sr. Laboratory of Applied Physics,
California Institute of Technology, Pasadena, California 91125, USA
E-mail: faraon@caltech.edu
S1 Illustration of design procedure
Figure S1: An overview of the design methodology described in the main manuscript. For
each iteration the process combines the results of multiple FDTD simulations, denoted for-
ward and adjoint sources, to compute the gradient of many different FoMs. The various gra-
dients are combined and used to update the structure using either a density-based method
1
or a level-set method.
2
.
1
S2 Level-set optimization
The term level-set optimization stems from treating the device boundaries as the zero-level
contour of a level-set function
φ
(
x,y
). The level-set function (LSF) is perturbed in accor-
dance with the gradient, which has the effect of perturbing the zero-level contour of the LSF
and the associated device boundaries to increase performance. We use a signed-distance
function to define the level-set function (LSF), where the value of
φ
(
x,y
) is proportional to
the signed distance from the device boundary. The gradients of the electromagnetic FoM
and a fabrication penalty function are combined and used to perturb the LSF, which has
the effect of perturbing the boundary of the device such that the electromagnetic FoM is
increased and the fabrication penalty term is decreased. The fabrication penalty term in-
cludes both a minimum radius of curvature constraint and a minimum gap size constraint of
60 nm. The LSF is then recomputed to ensure it remains a signed-distance function. This
process is repeated until the FoM has converged (recovering the performance lost from the
binarization step), and the fabrication penalty term is minimized.
It is critical that we preserve fabrication restrictions during the level-set optimization,
and we do so here using the techniques described in detail in Ref.
2
To briefly summarize that
work, a multi-dimensional fabrication penalty function is first analytically computed over the
entire device region. This function includes two types of fabrication constraints: one limits
the radius of curvature of the device boundaries, and the other limits the smallest gap size
of the device. The terms are integrated over the entire design region, yielding a real number
(the fabrication penalty term) that we wish to minimize. The gradient of this fabrication
penalty term is computed over the device region using a finite-difference approximation and
is subsequently combined with the gradient of the electromagnetic FoM that is computed
through the adjoint method. The level-set function is perturbed in the direction of this
combined gradient, resulting in a shifting of the material boundaries that co-optimizes the
electromagnetic FoM and the fabrication penalty term.
2
S3 Convergence plots
The convergence plots showing the average device figure of merit (FoM) and device bina-
rization are shown in Fig. S2. The FoM being plotted is the power transmission through
the appropriate pixel. It is worth noting that this is different from the FoM that is used to
define the adjoint source, which is the intensity at the center of the appropriate pixel.
The density-based optimization is considered converged when the binarization, which is
defined by Eq. 1 of the main manuscript, is nearly 100%. At various points the binarization
is forced to increase by passing the permittivity through a sigmoidal function and changing
the device permittivity to the output of this function. The FoM is then allowed to recover
before repeating this discrete push in binarization. At iteration 512 the optimization switches
to a level-set optimization, which recovers some of the performance that was lost during the
final phases of the density optimization. During the level-set optimization, the binarization
is recorded as approximately 90% because the boundaries of the device are smoothed out
yielding a continuous permittivity value. This can intuitively be thought of as the level-set
function passing through a simulation mesh voxel, which is modelled in the FDTD simulation
as a dielectric volume average of the two materials. Therefore the density optimization is
overly restrictive at binarization values above 90%, since it does not allow for this type
of border smoothing, instead modelling the device as a discretized grid of binary voxels.
This explains why the level-set function is able to recover substantially more performance
than what is lost at the initial conversion from density-based optimization to a level-set
optimization, recovering to approximately the same average FoM value as the density-based
optimization when it was 90% binary.
A non-intuitive aspect of this convergence plot is the FoM being permitted to decrease
during the density-phase of the optimization. The reason for this is described in detail in
Ref.
1
In short, a hard constraint is imposed that forces binarization to increase, thus forcing
the device to tend towards a binary device while either maximizing or minimally sacrificing
the FoM using the gradient information from the adjoint-retrieval process.
3