# A full degree-of-freedom photonic crystal spatial light modulator

Christopher L. Panuski<EMAIL_ADDRESS>Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Ian R.
Christen  Research Laboratory of Electronics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Momchil Minkov  Flexcompute, Inc., 130
Trapelo Rd., Belmont, MA 02478, USA  Cole J. Brabec  Research Laboratory of
Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Sivan Trajtenberg-Mills  Research Laboratory of Electronics, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  Alexander D. Griffiths
Institute of Photonics, Dept. of Physics, University of Strathclyde,
Technology and Innovation Centre, Glasgow G1 1RD, UK  Jonathan J.D. McKendry
Institute of Photonics, Dept. of Physics, University of Strathclyde,
Technology and Innovation Centre, Glasgow G1 1RD, UK  Gerald L. Leake  State
University of New York Polytechnic Institute, Albany, NY 12203, USA  Daniel J.
Coleman  State University of New York Polytechnic Institute, Albany, NY 12203,
USA  Cung Tran  State University of New York Polytechnic Institute, Albany, NY
12203, USA  Jeffrey St Louis  State University of New York Polytechnic
Institute, Albany, NY 12203, USA  John Mucci  State University of New York
Polytechnic Institute, Albany, NY 12203, USA  Cameron Horvath  Applied
Nanotools, Inc., Edmonton, AB T6G2M9, CA  Jocelyn N. Westwood-Bachman  Applied
Nanotools, Inc., Edmonton, AB T6G2M9, CA  Stefan F. Preble  Microsystems
Engineering, Rochester Institute of Technology, Rochester, NY 14623, USA
Martin D. Dawson  Institute of Photonics, Dept. of Physics, University of
Strathclyde, Technology and Innovation Centre, Glasgow G1 1RD, UK  Michael J.
Strain  Institute of Photonics, Dept. of Physics, University of Strathclyde,
Technology and Innovation Centre, Glasgow G1 1RD, UK  Michael L. Fanto  Air
Force Research Laboratory, Information Directorate, Rome, NY, 13441, USA  Dirk
R. Englund<EMAIL_ADDRESS>Research Laboratory of Electronics, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA

###### Abstract

Harnessing the full complexity of optical fields requires complete control of
all degrees-of-freedom within a region of space and time — an open goal for
present-day spatial light modulators (SLMs), active metasurfaces, and optical
phased arrays. Here, we solve this challenge with a programmable photonic
crystal cavity array enabled by four key advances: (i) near-unity vertical
coupling to high-finesse microcavities through inverse design, (ii) scalable
fabrication by optimized, 300 mm full-wafer processing, (iii) picometer-
precision resonance alignment using automated, closed-loop “holographic
trimming”, and (iv) out-of-plane cavity control via a high-speed $\upmu$LED
array. Combining each, we demonstrate near-complete spatiotemporal control of
a 64-resonator, two-dimensional SLM with nanosecond- and femtojoule-order
switching. Simultaneously operating wavelength-scale modes near the space- and
time-bandwidth limits, this work opens a new regime of programmability at the
fundamental limits of multimode optical control.

##  I Introduction

Programmable optical transformations are of fundamental importance across
science and engineering, from adaptive optics in astronomy [1] and
neuroscience [2, 3, 4], to dynamic matrix operations in machine learning
accelerators [5] and quantum computing [6, 7, 8]. Despite this importance,
fast, energy-efficient, and compact manipulation of multimode optical systems
— the core objective of spatial light modulators (SLMs) — remains an open
challenge [9, 10]. Specifically, the limited modulation bandwidth and/or pixel
density of liquid crystal (LC) SLMs, digital micromirror displays, and other
two-dimensional (2D) modulator arrays prevents complete control over the
optical fields they tune.

Figure 1a illustrates these limitations for a typical SLM comprised of a two-
dimensional (2D), $\Lambda$-pitch array of tunable pixels (subscript $p$)
emitting at wavelength $\lambda$ into the solid angle $\Omega_{p}$ with a
system (subscript $s$) modulation bandwidth $\omega_{s}$. Given these
parameters, each “spatiotemporal” degree-of-freedom (DoF) simultaneously
satisfying the minimum-uncertainty space- and time-bandwidth relations
($\delta A/\lambda^{2}\cdot\delta\Omega=1$ and $\delta t\cdot\delta\omega=1$,
respectively) can be illustrated as a real-space voxel with area
$\lambda^{2}/\Omega_{p}$ and time duration $1/\omega_{p}$ for pixel bandwidth
$\omega_{p}$. The optical-delay-limited pixel bandwidth
$\omega_{p}\approx(\Delta\epsilon_{p}/\epsilon)ck$ can be approximated as a
function of the achievable permittivity swing $\Delta\epsilon_{p}$ (for the
speed of light $c$) using first-order perturbation theory [11] or similarly
derived from linear scattering theorems [12].

Figure 1: Full degree-of-freedom (DoF) spatiotemporal optical programming.
Present-day SLMs (a) feature a 2D array of $\Lambda$-pitch pixels within an
aperture area $A$. Each pixel radiates at wavenumber $k=2\pi/\lambda$ into the
solid angle $\Omega_{p}$ and can be switched (blue $\leftrightarrow$ red color
change indicates a $\pi$ phase change of the emitted field) over the timescale
$T=1/\omega_{s}$ (given a modulation bandwidth $\omega_{s}$) with a large but
slow fractional permittivity perturbation $\Delta\epsilon_{p}/\epsilon$ (e.g.
liquid crystal rotation). The shaded volume indicates the smallest
controllable near-field spatiotemporal mode. In the far-field (right), the
corresponding shaded spatiotemporal bandwidth
$\nu=\Omega_{s}\omega_{s}=(\lambda/\Lambda)^{2}\omega_{s}$ counts the
controllable DoF per unit area and time in a single diffraction order. Trade-
offs between $\omega_{s}$ and $\Omega_{s}$ in liquid crystal- (LC) [13, 3,
14], thermal- [15, 16, 17], micro-electro-mechanical system- (MEMS) [18, 19,
20, 21, 22], and electro-optic-driven (EO) [23, 24, 25, 26] SLMs (b) limit
$\nu\ll\Omega_{p}\omega_{p}$, the accessible pixel bandwidth given the delay-
limited bandwidth $\omega_{p}\sim ck\Delta\epsilon_{p}/\epsilon$.
Spatiotemporal control is thus limited and scattering into undesired
diffraction orders (grey $\times$s) reduces diffraction efficiency.
Alternatively, a fully-filled array of wavelength-scale resonant apertures (c)
emitting into the solid angle $\Omega_{p}^{\prime}$ can enhance the effect of
fast (modulation frequency $\omega_{s}^{\prime}$), low-energy perturbations
$\Delta\epsilon_{p}^{\prime}\ll\epsilon$ to simultaneously achieve space- and
time-bandwidth limits (C1 and C2, respectively), yielding near-complete
spatiotemporal control with
$\nu^{\prime}\approx\Omega_{p}^{\prime}\omega_{p}^{\prime}$.

Integrating over the switching interval $T=1/\omega_{s}$ and aperture area $A$
then gives the total DoF count [27] 111Note that the exact coefficient of
proportionality in Eqn. 1 depends on the number of polarizations, the complex
amplitude and phase controllability of each mode, and the exact definition of
distinguishability when defining the Fourier uncertainty relations. For
simplicity, we have omitted these $\mathcal{O}(1)$ coefficients.

$F=\int_{A,\Omega_{p}}\frac{{\rm d}A}{\lambda^{2}}\cdot{\rm
d}\Omega\int_{T,\omega_{p}}{\rm d}t\cdot{\rm d}\omega.$ (1)

By comparison, the same switching period contains $N=A/\Lambda^{2}\leq F$
controllable modes, each confined to the pixel area $\Lambda^{2}$ and time
window $T$ (shaded box in Fig. 1a). Complete spatiotemporal control with $N=F$
is only achieved under the following criteria: (C1) emitters fully “fill” the
near-field aperture such that $\Omega_{p}$ matches the field-of-view
$\Omega_{s}=(\lambda/\Lambda)^{2}$ of a single array diffraction order; and
(C2) $\omega_{s}=\omega_{p}$. In the Fourier domain, the system’s
“spatiotemporal bandwidth” $\nu=\Omega_{s}\omega_{s}$ counts the controllable
DoF per unit area and time within a single far-field diffraction order. As
illustrated by the shaded pillbox in Fig. 1a, (C1) and (C2) are both satisfied
when $\nu$ matches the accessible pixel bandwidth $\Omega_{p}\omega_{p}$.

Practical constraints have prevented present-day SLM technology from achieving
this bound. In general, commercial devices approximate (C1) without achieving
(C2). Specifically, they offer excellent near-field fill-factor across
megapixel-scale apertures but use large $\mathcal{O}(\epsilon)$, slow index
perturbations. Liquid crystal SLMs, for example, are limited to
$\omega_{s}\sim 2\pi\times 10^{3}\text{ Hz}\ll\omega_{p}$ by the slow rotation
of viscous, anisotropic molecules that modulate the medium’s phase delay [29,
30]. Digital micromirror-based SLMs offer moderately faster
($\mathord{\sim}10^{5}$ Hz) binary amplitude modulation by displacing a
mechanical reflector, but at the expense of diffraction efficiency [31].
Mechanical phase shifters [32, 33, 19, 20, 34] improve this efficiency but
still require design trade-offs between pixel size and response time.

Recent research has focused on surmounting the speed limitations of commercial
SLMs with integrated photonic phased arrays [16, 35, 36, 37] and active
metasurfaces comprised of thermally [38, 39, 40], mechanically [41, 20], or
electrically [25, 42, 43, 44, 26] actuated elements. These devices, however,
do not satisfy (C1) (Table A2). Silicon photonics in particular has attracted
significant interest due to its fabrication scalability; however, the
combination of standard routing waveguides, high-power ($\mathord{\sim}$
mW/$\pi$ phase shift) thermal phase shifters, and vertical grating couplers in
each pixel reduces the fill-factor of emitters, yielding
$\Omega_{p}\gg\Omega_{s}$ [16]. Scattering into the numerous diffraction
orders within $\Omega_{p}$ then reduces the achievable zero-order and overall
diffraction efficiencies ($\eta_{0}$ and $\eta$, respectively). For this
reason, $\eta_{0}$ is a useful measure of near-field fill.

Various workarounds, including 1D phased arrays with transverse wavelength
tunability [35, 45, 46], sparse antenna arrays [15], and switched arrays [47,
22] improve steering performance but restrict the spatiotemporal basis (i.e.
limit $F$). Alternative nanophotonics-based approaches, often limited to 1D
modulation, have their drawbacks as well: phase change materials [38, 39, 40]
have slow crystallization rates and large switching energies, while electro-
optic devices [48, 23, 25, 43, 44, 26, 49], to date, have primarily relied on
large-area grating-based resonators to achieve appreciable modulation.

Figure 2: The photonic crystal spatial light modulator (PhC-SLM). Complete
spatiotemporal control is achieved by modulating an array of high-quality-
factor ($Q>10^{5}$), small-mode-volume ($V<0.1\lambda^{3}$) silicon PhC
cavities with a high-speed incoherent $\upmu$LED array (a). Absorbed
$\upmu$LED pulses control the detuning $\Delta$ of resonant pixels via free
carrier dispersion, which varies the amplitude and phase (illustrated by the
length and color, respectively, of emission arrows at each cell) of the
pixel’s complex reflection coefficient $r(\Delta)$. Despite sub-wavelength
near-field confinement (b, inset simulated mode profile overlaid on a SEM
micrograph of an $L4/3$-type cavity [50]), each pixel is designed for
directional ($\Omega_{p}\approx\Omega_{s}=\lambda^{2}/\Lambda_{x}\Lambda_{y}$)
far-field scattering $S(\vec{k})$ into the zeroth diffraction order (marked by
$\times$s) to satisfy (C1). Combining the reflection from each resonant
“antenna” in a large-scale aperture fabricated via optimized 300 mm wafer-
scale processing (b, inset photograph) enables near-ideal SLM performance per
the design criteria (C1-5) (c).

Figure 1b compares the performance of these and other experimentally-
demonstrated, active, 2D SLMs as a function spatiotemporal bandwidth’s two
components: modulation bandwidth $\omega_{s}$ and field-of-view $\Omega_{s}$.
Controllability aside, the evident trade-off between these parameters
illustrates the difficulty of creating fast, compact modulator arrays with
high $\nu$. Thus, in addition to satisfying the complete control criteria (C1)
and (C2), an “ideal” SLM would (Fig. 2c): (C3) maximize $\nu$ by combining
wavelength-scale pitches (for full-field $\Omega_{s}\rightarrow 2\pi$
beamforming) with gigahertz (GHz)-order bandwidths $\omega_{s}$ competitive
with electronic processors; (C4) support femtojoule (fJ)-order switching
energies as desired for information processing applications [51]; and (C5)
have scalability to state-of-the-art megapixel-scale apertures.

These criteria motivate the resonant architecture in Fig. 1c. Here, (C3) and
(C4) are achieved by switching a fully-filled array of wavelength-scale
resonant optical antennas with fast, fJ-order perturbations
$\Delta\epsilon_{p}/\epsilon\ll 1$. Each resonator’s far-field scattering and
quality factor $Q$ can then be tuned to achieve (C1) and (C2), respectively.
Combined, this resonant SLM architecture enables complete, efficient control
of the large spatiotemporal bandwidth supported by its constituent pixels.

Figure 2 illustrates our specific implementation of this full-DoF resonant
SLM: the photonic crystal spatial light modulator (PhC-SLM) [52]. Coherent
signal light is reflected off a semiconductor slab (permittivity $\epsilon$)
hosting a 2D array of semiconductor PhC cavities with instantaneous resonant
frequency $\omega_{0}+\Delta_{mn}(t)$. A short-wavelength incoherent control
plane imaged onto the cavity array controls each resonator’s detuning
$\Delta_{mn}(t)\approx-\Delta\epsilon(t)/2\epsilon$ via the permittivity
change $\Delta\epsilon_{p}(t)$ induced by photo-excited free carriers [53,
54]. We optimize the resonator bandwidth $\Gamma\approx\omega_{s}\approx
2\pi\times\mathrm{GHz}$ (corresponding to a quality factor
$Q=\omega_{0}/\Gamma\sim 10^{5}$) to maximize the linewidth-normalized
detuning $\Delta/\Gamma$ without significantly attenuating the cavity’s
response at the carrier lifetime ($\tau$)-limited modulation rate
$\omega_{s}=1/\tau$. Under these conditions, free carrier dispersion
efficiently modulates the complex cavity reflectivity $r(\Delta)$ to enable
fast ($>100$ MHz given a $\mathord{\sim}$ns free carrier lifetime [55]), low-
energy (fJ-order) conversion of incoherent control light into a dense array of
coherent, modulated signal modes (Appendix A).

This out-of-plane, all-optical switching approach is motivated by the recent
development of high-speed, high-brightness $\upmu$LED arrays [56, 57]
integrated with complementary metal-oxide-semiconductor (CMOS) drive
electronics for consumer displays [58, 59] and high-speed visible light
communication [60, 61]. In particular, gallium nitride $\upmu$LED arrays with
GHz-order modulation bandwidths [61, 62], sub-micron pixel pitches [63], and
large pixel counts [64] have been demonstrated within the past few years.
Applying these arrays for reconfigurable, “wireless” all-optical cavity
control eliminates electronic tuning elements at each pixel to avoid optical
loss, pixel pitch limitations, and interconnect bottlenecks for planar
architectures (as aperture area $A$ grows, $\mathcal{O}(A)$ pixel controls
eventually cannot be routed through the $\mathcal{O}(\sqrt{A})$ perimeter)
[65].

Free of these constraints, we designed high-finesse, vertically-coupled
microcavities offering coupling efficiencies $>90\%$, phase-dominant
reflection spectra [17, 66], and directional emission
$\Omega_{p}\approx\Omega_{s}$ for high-efficiency beamforming (Section II).
Bespoke, wafer-scale processing allows us to fabricate these “resonant
antennas” in arrays with mean quality factors $\langle Q\rangle>10^{6}$ and
sub-nm resonant wavelength standard deviation (Section III). For fine tuning,
we developed a parallel laser-assisted thermal oxidation [67, 68] protocol to
then trim $8\times 8$ cavity arrays to picometer-order uniformity [67, 68]
(Section IV), enabling high-speed spatial light modulation with fJ-order
switching energies and $\omega_{s}>2\pi\times 100$ MHz (Section V). Compared
to the previous devices surveyed in Fig. 1b, our PhC-SLM offers near-complete
control over an order-of-magnitude larger spatiotemporal bandwidth.

##  II Inverse-Designed Resonant Pixels

The sub-wavelength (i.e. normalized volume $\tilde{V}=V/(\lambda/n)^{3}<1$
relative to a cubic wavelength in the confining dielectric of refractive index
$n$), high-$Q$ (up to $\mathord{\sim}10^{7}$) [69, 70] modes of 2D PhC
cavities enable (C4) [71], but at the expense of (C1) since $Q$ optimization
(via hole displacements as in Fig. 3a [72]) cancels radiative leakage. The
exact displacement parameters are typically numerically optimized in
computationally expensive finite-difference time-domain (FDTD) simulations,
which ultimately limits the number of free parameters. Compared to the ideal
apertures in Fig. 1c, the optimized cavity unit cell confines a spatially
complex mode with $\Omega_{p}\gg\Omega_{s}$ (Fig. 4b, background), violating
(C1) and limiting the zero-order diffraction efficiency to $\eta_{0}\approx
0.04$. The result is poor beamforming performance as exemplified by the
distorted, low-efficiency far-field pattern emitted by a $64\times 64$ cavity
array with optimized detunings (derived with the algorithm in Appendix F) to
match a target far-field image (MIT logo).

Figure 3: Optimized holography with inverse-designed, vertically-coupled
microcavity arrays. (a) Silicon $L3$ slab defect cavity design (hexagonal
lattice constant $a=0.4~{}\upmu$m; hole radius $r/a=0.25$; slab thickness
$t=220$ nm) with overlaid midplane magnetic field profile $H_{z}$ after $Q$
optimization by displacing ($\delta x_{i},~{}\delta y_{i}$) and resizing
($\delta r_{i}$) the shaded holes in the $16a\times 16(\sqrt{3}/2)a$ periodic
unit cell. Hole shifts are magnified by $3\times$ for visualization. The
confined cavity mode radiates into the broad far-field profile in (b,
background), violating (C1) and yielding a zero-order diffraction efficiency
$\eta_{0}\ll 1$. As a result, simulated trial holograms (c) from a $64\times
64$ cavity array with optimized detunings (Appendix F) have minimal overall
diffraction efficiency $\eta$. Inverse design (b) solves these problems.
Guided mode expansion (GME) approximates the mode’s $Q$ and far-field profile
by sampling the losses $\\{c\\}$ at the array’s diffraction orders (white
$\times$s) displaced by Bloch boundary conditions $\vec{k}_{i}$ (i.e. at the
colored dots). An objective function $f$ that maximizes $Q$, confines
$\vec{H}$, and minimizes $\\{c\\}$ at any non-zero diffraction order can then
be efficiently optimized with respect to all hole parameters using reverse-
mode automatic differentiation (b). The resulting devices with high-$Q$,
efficient coupling, and directional emission enable high-performance
($\eta\sim 1$) resonant holography (d). Figure 4: Experimental comparison of
existing (a-c) and inverse-designed PhC cavities with high-$Q$ and near-
diffraction-limited vertical beaming. Superimposing a grating perturbation (a,
green; $\delta r_{i}$ magnified by $20\times$ for visualization) on the
$Q$-optimized design of Fig. 3a improves vertical coupling at the expense of
reduced $Q$, yielding the simulated far-field intensity profile in (b, left)
with $\eta_{0}=0.18$. Our measured far-field profile (b, right), collected
from a grating-coupled cavity using a cross-polarized imaging setup (Appendix
D), confirms the broad emission relative to the array field-of-view (dashed
white line) $\Omega_{s}$. This mismatch explains the low effective “fill
factor” and poor coupling observed in our resonant imaging (c, inset) and
near-field reflection spectra (c, blue), respectively. An input Gaussian beam
(with waist matched to the unit cell dimensions) is undercoupled and exhibits
an amplitude-dominant power reflectivity $R=|r|^{2}$ modulation (c, solid
green) with low phase variation $\Delta\phi$ (c, dashed green). Our inverse
designed cavities (d) overcome these issues by optimizing every hole in the
unit cell to vertically scatter cavity leakage for any target $Q$, producing
“ideal” resonant SLM pixels satisfying (C1). Specifically, they support near-
diffraction-limited emission (e) with $\eta_{0}\sim 1$ due to fully-filled
near-field resonant scattering (f, inset), a $\mathord{\sim}5\times$
experimental resonance contrast enhancement (f), and $>94\%$ single-sided
(i.e. assuming an ideal back-reflector described in Appendix G) coupling to an
input Gaussian beam for phase-dominant modulation (f, green).

Fortunately, these limitations are not fundamental: the effective scattering
aperture $A_{0}=\lambda^{2}/\Omega_{p}$ of a resonant mode can extend beyond
its $1/e$ decay area $A_{e}$. This apparent space-bandwidth violation
$(A_{e}/\lambda^{2})\cdot\Omega_{p}=A_{e}/A_{0}<1$ is enabled by resonant
scattering from the mode’s evanescent field, which raises the basic question:
how should scatterers be arranged to produce a desired far-field emission
pattern?

One established approach is a harmonic $2a$-period grating perturbation (Fig.
4a) that “folds” energy concentrated at the band-edge $k_{x}=\pi/a$ back to
$k_{||}=0$, yielding vertical radiation at the expense of reduced $Q$ [73, 74,
75, 76]. In the perturbative regime, the far-field scattering profile is an
image of the broad band-edge mode. Thus, once the grating-induced loss becomes
dominant, further magnifying the perturbation reduces $Q$ without
significantly improving directivity. Fig. 4b shows the narrowed far-field
profile produced by a $\delta r_{i}/r\approx 0.02$ grating perturbation, which
balances the reduced $Q\approx 8\times 10^{5}$ and a modest diffraction
efficiency improvement ($\eta_{0}=0.18$).

By contrast, our design strategy (Fig. 3b) combines semi-analytic guided mode
expansion (GME) simulations with automatic differentiation to maximize
$\eta_{0}$ (and thereby the effective near-field fill factor) for a given
target $Q$ using all of the hole parameters. In each iteration, GME
approximates the cavity eigenmode and radiative loss rates $c_{mn}^{(i)}$ at
each of the array’s reciprocal lattice vectors (i.e. diffraction orders)
offset by the Bloch periodic boundary conditions $\vec{k}_{i}$ [77]. These
coupling coefficients coarsely sample the cavity’s approximate far-field
emission (Appendix C). Scanning $\vec{k}_{i}$ over the irreducible Brillouin
zone of the rectangular cavity array improves the sampling resolution, and an
overall $Q$ can be estimated by averaging the total loss rates
$\Gamma^{(i)}=\sum_{mn}c_{mn}^{(i)}$ in each simulation. Reverse-mode
automatic differentiation then allows us to efficiently optimize an objective
function

$f=\frac{1}{N}\sum_{i=1}^{N}\frac{c_{00}^{(i)}}{\Gamma^{(i)}}\arctan\left(\frac{Q}{Q_{0}}\right)|E_{0}|^{2}$
(2)

targeting three main goals: 1) increase $Q$ to a design value $Q_{0}$; 2)
force the associated radiative loss into the array’s zeroth diffraction order
for efficient vertical coupling; and 3) minimize $V$ by maximizing $|E_{0}|$,
the electric-field magnitude at the center of the unit cell. The resulting
designs support tunable-$Q$ resonances with near-diffraction-limited
($\Omega_{p}\approx\Omega_{p}$) vertical beaming comparable to the ideal
planar apertures of Fig. 1c. The example design of Fig. 4d, for instance,
maintains $Q\approx 8\times 10^{5}$ with $\eta_{0}=0.86$ based on the
simulated far-field profile in Fig. 4e.

We prototyped each design at a commercial electron beam lithography (EBL)
foundry 222Applied Nanotools, Inc. https://www.appliednt.com/ before
transitioning to the wafer-scale foundry process described in Sec. III. The
near- and far-field reflection characteristics of the fabricated devices were
measured with the cross-polarized microscopy setup detailed in Appendix D.
Fig. 4b-c and Fig. 4e-f show the results for the grating coupled and inverse-
designed cavities, respectively. The optimal grating-coupled cavities offer
$Q\sim 4\times 10^{5}$ at $\lambda\approx 1553\text{ nm}$ with a near-field
resonant scattering profile well-centered on the cavity defect (Fig. 4c,
inset). The mode mismatch between this wavelength-scale PhC mode and the wide-
field input beam (Gaussian beam with $\mathord{\sim}150~{}\upmu\text{m}$ waist
diameter for array-level excitation) is further evidenced by the small
normalized reflection amplitude (relative to that of the inverse designed
cavities) on resonance as well as the broad far-field profile (Fig. 4b) with
$\eta_{0}=0.24$.

By comparison, inverse design non-perturbatively modifies the cavity mode
(Fig. 4d) to produce the near-ideal measured far-field profile in Fig. 4e
satisfying (C1) with $\eta_{0}=0.98$ while simultaneously increasing $Q$ to
$5.7\times 10^{5}$. We attribute the slight increase in zero-order diffraction
efficiency over the simulated value $\eta_{0}=0.86$ to the substrate-dependent
effects described in Appendix G. The fully-filled near-field resonant
scattering image in Fig. 4f explains the close resemblance between this
measured $S(\vec{k})$ and that of an ideal uniform aperture [79]. In addition,
the narrowed emission profile $S(\vec{k})$ yields a $\mathord{\sim}5\times$
increase in cross-polarized reflection and the phase-dominant simulated direct
reflection spectrum in Fig. 4f. The latter is achieved by 94% one-sided
coupling to a Gaussian beam with optimized waist diameter (Appendix E).

Combined, these results break the traditional coupling–$Q$ tradeoff (offering
an order-of-magnitude improvement in the figure-of-merit $\eta_{0}\cdot Q$ for
the prototype devices in Fig. 4) to enable high-performance beamforming at the
space-bandwidth limit (C1). These results are supported by the simulated
hologram in Fig. 3d: an array of optimally detuned, inverse-designed cavities
forms a clear far-field image with a several order-of-magnitude improvement in
overall diffraction efficiency ($\eta=0.83$) over existing designs.

##  III Foundry-Fabricated High-Finesse Microcavity Arrays

While EBL enables fabrication of few-pixel prototypes with state-of-the-art
resolution and accuracy, serial direct-write techniques do not satisfy (C5).
Field stitching issues and sample preparation aside, a single cm2, megapixel-
scale sample would require a full day of EBL write time alone. We therefore
developed a full-wafer deep-ultraviolet photolithography process specifically
optimized for wavelength-pitch arrays of high-$Q/V$ PhC microcavities in a
commercial foundry [80].

A central goal was to create vertical etch side-walls. The transmission
electron microscope (TEM) cross-section in Fig. 5i shows that the default
fabrication process (optimized for isolated waveguides) yielded an oblique
($100^{\circ}$), incomplete etch through the silicon device layer for the
target PhC lattice parameters. Both nonidealities erase the membrane’s
vertical reflection symmetry, leading to coupling between even- and odd-
symmetry (about the slab midplane) modes that ultimately limits the achievable
$Q$ of bandgap-confined resonances [81]. By contrast, our revised fabrication
process achieves near-vertical $91^{\circ}$ sidewall angles (Fig. 5ii),
yielding high-quality PhC lattices for a range of hole diameters between the
$\mathord{\sim}100$ nm critical-dimension and $2r\approx a$ (Fig. 5c). Using
TEM cross-sectioning and automated optical metrology as feedback over multiple
300 mm wafer runs in the AIM Photonics foundry’s 193 nm DUV water-immersion
lithography line, this new process relies on a combination of dose-optimized
reverse (positive) tone lithography, high-accuracy laser written masks, and
optimized etch termination. Following fabrication and dicing, we post-
processed individual die with a backside silicon nitride anti-reflection
coating and, as required, suspend the PhC membranes with a timed wet etch.

Figure 5: Full-wafer photonic crystal fabrication in an optimized 300 mm
foundry process. A wafer (a) contains 64 complete reticles (b) each comprising
millions of inverse designed PhC cavities. The before (i) and after (ii)
false-color (blue: metal fill; red: silicon; yellow: silicon dioxide; green:
etch mask) transmission electron microscope cross-sections show how process
optimization enables high-quality PhC lattices (c) that support $Lm$-type
cavity arrays with $\langle Q\rangle>10^{6}$ and sub-nanometer wavelength
standard deviation (d).

The resulting die contain isolated and arrayed PhC cavities with swept
dimensions to offset systematic fabrication biases. We chose $Lm$-type cavity
designs — formed by removing $m$ holes from the PhC lattice as demonstrated by
the $L3$ unit cells in Fig. 4 — to host tunable-volume (via variable $m$),
high-$Q$ resonant modes with even reflection symmetry (about the unit cell
axes) as required for vertical emission [82]. The highest-performance isolated
devices feature $Q>10^{6}$ with normalized volumes $\tilde{V}\approx 0.3$.
With a joint spectral- and spatial-confinement (quantified by the figure-of-
merit $Q/\tilde{V})\approx 4\times 10^{6}$, these devices are among the
highest-finesse optical cavities ever fabricated in a foundry process.

Our optimized foundry processing extends this exceptional single-device
performance (rivaling record EBL-fabricated devices) to large-scale cavity
arrays. We developed a fully-automated measurement system (Appendix D) to
locate and characterize hundreds of cavities per second via parallel camera
readout. The resulting data, extracted from over $10^{5}$ devices measured
across the wafer, allow us to statistically analyze resonator performance and
fabrication variability at the die, reticle, and wafer level. Fig. 5d, for
example, shows resonant wavelength and $Q$ variations within $8\times 8$
arrays of four different cavity designs. Using camera readout of the reflected
wide-field excitation, each data set is extracted from a single wavelength
scan of a tunable laser. Besides the expected correlation between uniformity
and mode volume [83], the data demonstrates — for the first time, to our
knowledge — the ability to fabricate sub-wavelength ($\tilde{V}<1$)
microcavity arrays with $\langle Q\rangle>10^{6}$ and sub-nanometer resonant
wavelength standard deviation ($\sigma_{\lambda}\approx 0.6$ nm). Critically
for beamforming, this uniformity also extends to the far-field: Appendix H
shows that each cavity in an $8\times 8$ array emits vertically with
$\eta_{0}=0.86\pm 0.07$, in quantitative agreement with the simulated result
in Fig. 4.

##  IV Holographic Trimming

Figure 6: Parallel, fully-automated, and low-loss microcavity trimming via
structured laser oxidation. In each iteration of the trimming loop (a), a
weighted GS algorithm distributes a visible trimming laser with power $P_{0}$
to powers $\\{P_{i}\\}$ at desired cavities based on the measured resonant
wavelength $\lambda_{i}$ of each cavity. A few nm-thick layer of thermal oxide
grows at each optical focus (photographed spots in the inset cavity array
image), reducing the as-fabricated standard deviation in hole parameters
$\sigma\sim\text{nm}$ (b) and permanently shifting the targeted resonances.
The initial and final near-field hyperspectral reflection images (c, color-
coded by each device’s wavelength normalized detuning $\Delta/\Gamma$) show
the $>100\times$ reduction in resonant-wavelength standard deviation to
$\sigma_{\lambda}=2.5~{}\mathrm{pm}$ without affecting the mean quality factor
$Q>10^{5}$. The effective dimensions of the final array (d) are thus
homogenized to $\sigma^{\prime}\sim\text{pm}$ length-scales by oxidation.
Regions of local oxide growth in helium ion microscopy images of the trimmed
PhC-SLM (d) appear as bright areas. The device summary (e) shows the
wavelength shift, $Q$ variation (quantified by dot area), and aligned
reflection spectra of each device.

In addition to these overlapping far-field emission profiles, programmable
multimode interference requires each cavity to operate near a common resonant
wavelength $\lambda_{0}$. For sufficiently high-$Q$ resonators, this tolerance
cannot be solely achieved through optimized fabrication since
$\mathcal{O}(\text{nm})$ fabrication fluctuations translate to
$\mathcal{O}(\text{nm})$ resonant wavelength variations [84, 85]. Our
prototype $8\times 8$ arrays of $L3$ cavities (chosen to optimally balance
requirements on $Q$, $V$, directive emission, and fabrication tolerance)
typically span a $\mathord{\sim}3~{}\text{nm}$ peak-to-peak wavelength
variation (given $\sigma_{\lambda}\approx 0.6$ nm), corresponding to hundreds
of linewidths for the target $Q\sim 10^{5}$.

To correct this nonuniformity, we developed an automated, low-loss, and
picometer-precision trimming procedure based on laser-assisted thermal
oxidation (Fig. 6). Two features of our approach resolve the speed and
controllability limitations of prior single-device implementations [67, 68]:
1) accelerated oxidation in a high-pressure chamber with in-situ
characterization; and 2) holographic fanout of the trimming laser to
simultaneously address multiple devices. In each iteration of the automated
trimming loop (Fig. 6a), the resonant wavelengths $\\{\lambda_{i}\\}$ are
measured and a subset $T$ containing $N$ devices is selected to maximize the
total trimming distance $N(\min_{T}\\{\lambda_{i}\\}-\lambda_{t})$ to a target
wavelength $\lambda_{t}$. Each cavity in $T$ is then targeted by a visible
laser distributed by the liquid crystal SLM setup described in Appendix D. To
generate the required phase masks, we developed an open-source, GPU-
accelerated experimental holography software package that implements fixed-
phase, weighted Gerchberg-Saxton (GS) phase retrieval algorithms 333slm-suite.
https://github.mit.edu/cpanuski/qp-slm. Using camera feedback, the algorithm
can generate thousands of near-diffraction-limited foci with
$\mathord{\sim}1\%$ peak-to-peak power uniformity and single-camera-pixel-
order location accuracy within a few iterations (Appendix I).

The holographically-targeted pixels are then laser-heated with a computed
exposure power and duration (based on the current trimming rates, resonance
locations, and other array characteristics) to grow thermal oxide at the
membrane surface. For thin oxide layers, the consumption of silicon during the
reaction with ambient oxygen permanently blueshifts the cavity resonance in
proportion to the oxide thickness $t_{\text{SiO}_{2}}$ (Fig. 6b) [68]. Per the
Deal-Grove model, the rate-limiting diffusion of oxygen through the grown
oxide accelerates with increasing oxygen pressure — a well-known technique in
microelectronics fabrication [87]. We therefore oxidize our samples in pure
oxygen with partial pressure $P_{\text{O}_{2}}=5~{}\text{bar}$, enabling ${\rm
d}\lambda_{0}/{\rm d}t\approx 0.1~{}\text{nm/s}$ resonance trimming rates over
$\Delta\lambda_{0}>20~{}\text{nm}$ wavelength ranges. After each trimming
exposure, we remeasure the resonance statistics and recycle the loop until all
devices are aligned within a set tolerance about $\lambda_{t}$. The trimming
algorithm also accounts for long-term moisture adsorption to the membrane
surface, thermal cross-talk, and trimming rate variations (Appendix J).

Fig. 6 demonstrates the results of this trimming procedure applied to our
prototype $8\times 8$ pixel PhC-SLM. Prior to trimming, the hyperspectral
near-field reflection image in Fig. 6c shows the large ($>200$ linewidths for
the mean quality factor $\langle Q\rangle=1.6\times 10^{5}$) resonant
wavelength variation between the otherwise spatially uniform and high-fill
resonant modes. Holographic trimming reduces the wavelength standard deviation
and peak-to-peak spread by $>100\times$ to $\sigma_{\lambda}=2.5~{}\text{pm}$
and $\Delta\lambda_{0}^{\text{p-p}}=1.3\Gamma=13~{}\text{pm}$, respectively,
enabling all 64 devices (imaged in Fig. 6d) to be resonantly excited at a
common operating wavelength (Fig. 6e). Since $\sigma_{\lambda}$ is directly
related to the corresponding hole radius and placement variability
($\sigma_{r}$ and $\sigma_{h}$, respectively) with an $\mathcal{O}(1)$ design-
dependent constant of proportionality, the thermal oxide homogenizes the
effective dimensions of each microcavity to the picometer scale. The mean
quality factor and near-field reflection profile of the array remain largely
unmodified throughout the process as evidenced by Fig. 6c and Fig. 6e.

To our knowledge, these results are the first demonstration of parallel, in
situ, non-volatile microcavity trimming. The achievable scale is currently
limited by environmental factors that could be overcome with stricter process
control (Appendix J). Even without these improvements, the current uniformity,
scale, and induced loss outperform the corresponding metrics of the previous
techniques reviewed in Appendix J, paving the way towards scalable integrated
photonics with high-$Q$ resonators.

##  V All-Optical Spatial Light Modulation

Once trimmed to within a linewidth, each resonator reflects an incident
coherent field $E_{\text{i}}(\vec{r},t)$, producing a far-field output [88]

$E_{\text{r}}(\vec{k},t)=S(\vec{k})\sum_{m,n}r\\{\Delta_{mn}(t)\\}E_{\text{i}}(\vec{r}_{mn},t)e^{j\vec{k}\cdot\vec{r}_{mn}}$
(3)

that can be dynamically controlled within $S(\vec{k})$ by setting the detuning
$\Delta_{mn}(t)$ (and therefore the near-field reflection coefficient $r$) of
each resonator. Experimentally, we measure the intensity pattern
$|E_{r}(\vec{k})|^{2}$ on the back focal plane of a microscope objective above
the PhC-SLM (as with the single-device characterization in Sec. II) and
optically program $\Delta_{mn}(t)$ via photo-excited free carriers. The
corresponding setups are detailed in Appendix D.

Figure 7: Nanosecond switching (a-c) and spatial light modulation (d-f). (a)
Peak phase shift $\Delta\phi$ and half-maximum switching interval
$T_{\text{switch}}$ produced by pulses from a CMOS-integrated $\upmu$LED array
imaged onto the cavity array (inset: cavities illuminated to form the letters
‘S’, ‘L’, and ‘M’) as a function of trigger duration $T_{\text{CMOS}}$ and
pulse energy density $E_{\upmu\text{LED}}$. (b) Complex reflectivity
($r=\sqrt{R}e^{j\phi}$) modulation with femtojoule-order pulse energies
$E_{\text{laser}}$ from a focused visible laser. (c) Output probe to input
visible (pump) power transfer function $T(\omega)$ fit to a second-order
response function, yielding $\omega_{s}=2\pi\times 135$ MHz limited by the
free carrier lifetime $\tau\approx 1.1$ ns and cavity bandwidth $\Gamma\approx
1$ GHz. (d) Far-field intensity profile, half-maximum beam widths, and zero-
order diffraction efficiency $\eta_{0}$ (integrated within the dashed white
box) of the trimmed array at time $t_{0}$ with horizontal and vertical cross-
sectional profiles (blue traces) compared to those of an $8\times 8$ array of
planar apertures with 80% linear fill (black, dashed). (e/f) Analogous results
for the switched array with an optically-patterned horizontal (vertical)
amplitude grating at the maximum extinction time $t_{0}+6$ ns, producing $\pm
1^{\text{st}}$-order diffraction peaks over a $10.6^{\circ}$ ($14.5^{\circ}$)
field-of-view and diffraction efficiency $\eta_{x}=0.22$ ($\eta_{y}=0.20$).

Absent a control input ($\Delta_{mn}\approx 0$), Fig. 7d shows the static far-
field intensity pattern $|E_{r}(\vec{k})|^{2}$ of a wide-field-illuminated
(i.e. $E_{i}(\vec{r})\approx E_{i}$) $8\times 8$ trimmed array with
$\\{Q\\}=1.85\times 10^{5}$ and $\sigma_{\lambda}=5$ pm at $\lambda=1562$ nm.
The inverse-designed cavity unit cells minimize scattering into undesired
diffraction orders, producing a high-efficiency ($\eta_{0}=0.66$) zero-order
beam with the expected $1.3^{\circ}$ and $1.6^{\circ}$ horizontal and vertical
beamwidths given the $42.0\lambda\times 36.4\lambda$ aperture size. The cross-
sectional beam profiles are well matched to the simulated emission profile of
uniform apertures with width $w=0.8\lambda$, suggesting an 80% effective
linear fill of the array. This extracted value agrees with the observed zero-
order efficiency and the array’s physical design (each $16a\times 16a$ cavity
offering near-unity fill was padded to $20a\times 20a$ to limit coupling to
adjacent cells).

After confirming the static performance of the array, we conducted optical
switching experiments with two sources: an incoherent $\upmu$LED array and a
pulsed visible laser. The $\upmu$LED array contains $16\times 16$
individually-addressable gallium nitride $\upmu$LEDs with $>$150 MHz small-
signal bandwidth and $\mathord{\sim}10^{6}$ cd/m2 peak luminances (at 450 nm)
flip-chip bonded to high-efficiency CMOS drivers [89, 60]. Using the setup in
Appendix D, we imaged the 100 $\upmu$m-pitch array with variable
demagnification and rotation onto the PhC cavity array. Digitally triggering
the CMOS drivers then enables reconfigurable, binary optical addressing as
illustrated by the imaged projections of three letters on the PhC-SLM (Fig.
7a). We measured the resulting pixel reflection amplitude and phase using
locked, shot-noise-limited balanced homodyne detection (Appendix D). Fig. 7a
depicts the maximum phase shift $\Delta\phi$ as a function of CMOS trigger
duration $T_{\text{CMOS}}$ and imaged pump energy density
$E_{\upmu\text{LED}}$. Single-cavity switching is possible with energy
densities below $10$ fJ/$\upmu$m2 (corresponding to $\mathord{\sim}100$ fJ
total energy for our chosen demagnification) and a minimum trigger duration
$T_{\text{CMOS}}\approx 5$ ns. Shorter trigger pulses produce relatively
constant-width pulses (due to the $\upmu$LED fall time) with insufficient
energy for high-contrast switching.

Confining visible pump pulses in space and time to the silicon free-carrier
diffusion length ($\sim 1~{}\upmu$m) and lifetime ($\tau\approx 1$ ns),
respectively, would reduce the required switching energy and maximize
bandwidth bandwidth. While either metric is achievable with existing
$\upmu$LED arrays [90, 63] and optimization to achieve both simultaneously is
ongoing [91], we demonstrated the expected performance enhancement with a
pulsed visible ($\lambda=515$ nm) laser. Fig. 7b shows that 3 dB power
reflectivity changes and high-contrast phase modulation are feasible for 5 fJ
pump pulses over a switching interval $T_{\text{switch}}\approx 1$ ns, thereby
satisfying (C4). Free-carrier dispersion is the dominant switching mechanism
for these isolated, ns-order switching events (Appendix A). While repeated
switching over $\upmu$s-order timescales leads to a slowly-varying thermo-
optic detuning [92], various optical communications techniques (constant-duty
line codes, for example [93]) can maintain average device temperature during
high-speed free carrier modulation. To demonstrate this decoupling of
switching mechanisms, we measured the normalized small-signal transfer
function $T(\omega)$ between a harmonic pump power (produced by a network
analyzer-driven amplitude electro-optic modulator) and the phase-locked
homodyne response. When aligned to the thermally-detuned resonance, the
results (Fig. 7c) match the expected second-order response
$T(\omega)=1/\\{\left[1+(\omega\tau)^{2}\right]\left[1+(\omega/\pi\Gamma)^{2}\right]\\}$
set by carrier- and cavity-lifetime-limited bandwidths ($1/\tau$ for a fitted
carrier lifetime $\tau=1.1$ ns and the measured $\Gamma=1.0$ GHz,
respectively). While satisfying (C2) therefore requires higher-$Q$ resonators,
the current regime of operation enables near-complete control over a larger
bandwidth $\omega_{s}=2\pi\times 135~{}\text{MHz}\approx 1/\tau$, i.e. without
significantly degrading the carrier-lifetime-limited modulation bandwidth.

Combining this optimized switching with the space-bandwidth-limited vertical
beaming of each resonator enables multimode programmable optics approaching
the fundamental limits of spatiotemporal control. We currently probe the PhC-
SLM in a wide-field, cross-polarized setup that produces amplitude-dominant
Lorentzian reflection profiles $r(\Delta)\propto 1/(1+j\Delta)$ regardless of
the resonator coupling regime (cavity emission is isolated from specular
reflection). For simplicity, we therefore conducted proof-of-concept
demonstrations using the PhC-SLM as an array of high-speed binary amplitude
modulators. In this modality, a nanosecond-class pulsed visible laser is
passively fanned out to the desired devices. Devices targeted by pump light
are detuned far from resonance ($\Delta\gg\Gamma$) and effectively
extinguished, whereas unactuated cavities retain their high $\Delta\approx 0$
reflectivity.

We used pump-probe spectroscopy for wide-field imaging of these few-nanosecond
switching events (Appendix D). Short infrared probe pulses were carved with a
electro-optic amplitude modulator (DC biased to an intensity null) and
variably delayed to coincide with the arrival of visible pump light at the PhC
membrane, gating probe field transmission to the IR camera. We then measured
the near- and far-field reflection as a function of the probe delay to
reconstruct switching events with sub-nanosecond resolution. Fig. 7e-f plots
the resulting far-field intensity profiles $|E_{r}|^{2}$ for horizontal and
vertical on-off gratings. For a $5$ ns probe pulse width, the maximum near-
field extinction of targeted cavities (7.4 dB and 9.8 dB for horizontal and
vertical gratings, respectively) occurs within a $\mathord{\sim}6$ ns delay;
i.e. just after the pump and probe pulses completely overlap. This minimum
probe pulse width is limited by the requirement for high imaging contrast
between probe pulses and leakage (due to the imperfect probe modulator
extinction) given the instrument-limited trigger repetition rate
($\mathord{\sim}$MHz) and camera integration time.

As expected, the input field is primarily scattered into first-order
diffraction peaks within the (greater than $10^{\circ}$) 2D field-of-view of
$S(\vec{k})$. The illustrated cross sectional beam profiles again agree with
analytic results for a 80% filled linear array of uniform apertures (black
dashed lines). For the horizontal grating Fig. 7e, the fit is scaled by a
factor of $\mathord{\sim}2$ to account for the increased reflectivity of
unactuated cavities during switching events, which we attribute to residual
coupling between adjacent cavities. In both cases, the pattern diffraction
efficiencies — measured as the fraction of integrated power within the
outlined regions in Fig. 7 — $(\eta_{x},\eta_{y})=(0.22,0.20)$ compare
favorably to the efficiency of the fitted uniform aperture array. Even with
amplitude-dominant modulation, these metrics exceed the efficiencies of
previous resonator-based experiments due to our high-directivity PhC antenna
array [17].

##  VI Summary and Outlook

These proof-of-concept experiments demonstrate near-complete spatiotemporal
control of a narrow-band optical field filtered in space and time by an array
of wavelength-scale, high-speed resonant modulators. While the general
resonant architecture (Fig. 1c) is applicable to a range of microcavity
geometries and modulation schemes, the combination of our high-$Q$,
vertically-coupled PhC cavities with efficient, all-optical free-carrier
modulation achieves (C1-5) with an ultrahigh per-pixel spatiotemporal
bandwidth $\nu\approx 5.6~{}\text{MHz}\cdot\text{sr}$. This MHz-order
modulation bandwidth per aperture-limited spatial mode corresponds to a more
than ten-fold improvement over the 2D spatial light modulators reviewed in Fig
1b. Our wafer-scale fabrication and parallel trimming offer a direct route
towards scaling this performance to spectrally-multiplexed,
$\mathcal{O}(\text{cm}^{2})$ apertures for exascale interconnects beyond the
reach of current electronic systems, thus motivating the continued development
of optical addressing and control techniques.

The PhC-SLM opens the door to a number of applications and opportunities,
including: high-definition, high-frame-rate holographic displays by the
integration of a back-reflector (see Appendices E-G) for one-sided, phase-
only, and full-DoF spatiotemporal modulation; compact device integration via
direct transfer printing of our cavity arrays onto a high-bandwidth $\upmu$LED
array [94]; three-dimensional optical addressing and imaging by combining on-
demand $\upmu$LED control with statically trimmed detuning profiles that
continuously steer pre-programmed patterns [95]; large-scale programmable
unitary transformations for universal linear optics processors [96]; focal
plane array sensors for high-spatial-resolution readout of refractive index
perturbations in imaging applications from endoscopy to bolometry and quantum-
limited superresolution [97, 98, 99]; optical neural network acceleration via
low-power, high-density unitary transformation of free-space optical inputs
[5, 100]; and high-speed adaptive optics enabling free-space compressive
sensing, deep-brain neural stimulation, and real-time scattering matrix
inversion in complex media [101, 102]. Moreover, whereas we have so far
considered only mode transformations, the PhC-SLM’s high-$Q/V$ resonant
enhancement suggests the possibility of programming the quantum optical
excitations/fields of these modes for applications ranging from multimode
squeezed light generation [103], to multiplexed single photon sources for
linear optics quantum computing [6, 104] or deterministic photonic logic [105,
106].

###### Acknowledgements.

The authors thank Flexcompute, Inc. for supporting FDTD simulations, the
MIT.nano staff for fabrication assistance, and M. ElKabbash (MIT) for useful
discussions. C.P. was supported by the Hertz Foundation Elizabeth and Stephen
Fantone Family Fellowship. S.T.M. is funded by the Schmidt Postdoctoral Award
and the Israeli Vatat Scholarship. Experiments were supported in part by Army
Research Office grant W911NF-20-1-0084, supervised by M. Gerhold, the
Engineering and Physical Sciences Research Council (EP/M01326X/1,
EP/T00097X/1), and the Royal Academy of Engineering (Research Chairs and
Senior Research Fellowships). This material is based on research sponsored by
Air Force Research Laboratory under agreement number FA8650-21-2-1000. The
U.S. Government is authorized to reproduce and distribute reprints for
Governmental purposes notwithstanding any copyright notation thereon. The
views and conclusions contained herein are those of the authors and should not
be interpreted as necessarily representing the official policies or
endorsements, either expressed or implied, of the United States Air Force, the
Air Force Research Laboratory or the U.S. Government. C.P. and D.E. conceived
the idea, developed the theory, and led the research. M.M. and C.P. developed
the far-field optimization technique and designs. C.B. developed the optimized
resonator detuning theory. C.P. conducted the experiments with assistance from
I.C. (trimming experiments), S.T.M. (holography software), and A.G.
($\upmu$LED measurements). J.J.M., M.D., and M.S. contributed the $\upmu$LED
arrays and guided the incoherent switching experiments. C.H. and J.W.B.
fabricated the initial samples for evaluation prior to foundry process
development by C.T., J.S.L., J.M., and G.L. S.P. assisted with wafer post-
processing. M.F. coordinated and led the foundry fabrication with assistance
from G.L. and D.C. C.P. wrote the manuscript with input from all authors.

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##  Appendix A Analytic Model for Slab Switching

Here, we develop an analytic model for all-optical switching in slab-type PhC
cavities to estimate the required tuning energy. An absorbed control pulse
produces a refractive index change

$\delta n(\vec{r},t)=-\alpha_{c}N(\vec{r},t)+\alpha_{t}T(\vec{r},t)$ (4)

proportional to the photo-excited carrier density $N$ and induced temperature
change $T$ through the plasma dispersion and thermo-refractive effects,
respectively. The thermo-refractive coefficient $\alpha_{t}={\rm d}n/{\rm d}T$
and empirical free-carrier “scattering volume” $\alpha_{c}=-{\rm d}n/{\rm d}N$
are typically both positive such that the two effects counteract one another.
The evolution of $\delta T$ and $N$ are governed by the diffusion equations

$\displaystyle\frac{\partial N(\vec{r},t)}{\partial t}$
$\displaystyle=\nabla\cdot(D_{c}\nabla N)-\frac{N}{\tau}+g(\vec{r},t)$ (5a)
$\displaystyle\frac{\partial T(\vec{r},t)}{\partial t}$
$\displaystyle=\nabla\cdot(D_{t}\nabla T)+q(\vec{r},t)$ (5b)

given the thermal diffusivity $D_{t}$ and assuming ambipolar diffusion of
carriers with lifetime $\tau$ and diffusivity $D_{c}$. Over relevant
timescales $t>w^{2}/D_{c}$ in a $w$-thick uniform slab, vertical diffusion can
be neglected to yield solutions

$\displaystyle N(\vec{r},t)$
$\displaystyle=g(\vec{r},t)*_{\vec{r},t}G(\vec{r},2D_{c}t)e^{-t/\tau}$ (6a)
$\displaystyle T(\vec{r},t)$
$\displaystyle=q(\vec{r},t)*_{\vec{r},t}G(\vec{r},2D_{t}t)$ (6b)

expressed as convolutions ($*$) of the inhomogeneous sources $g(\vec{r},t)$
and $q(\vec{r},t)$ with the two-dimensional Green’s function

$G(\vec{r},\sigma^{2})=\frac{1}{2\pi
w\sigma^{2}}\exp{-\frac{|\vec{r}|^{2}}{2\sigma^{2}}}.$ (7)

All variables are considered uniform along the vertical axis; $\vec{r}$ in our
notation thus corresponds only to transverse coordinates in the slab plane. We
specifically consider solutions to Eqns. 6 in response to a focused, square-
wave Gaussian control pulse with beam waist $2\sigma_{p}$, pulse-width $T$,
and pulse (photon) energy $E$ ($E_{0}$) absorbed into the cavity with
efficiency $\eta_{\text{abs}}$. The results can be considerably simplified
with the conservative (i.e. underestimating plasma dispersion at short
timescales $t\lesssim\tau$), albeit crude, assumption of instantaneous carrier
diffusion to the diffusion length $\sqrt{D_{c}\tau}$. This method decouples
carrier decay and diffusion to yield the carrier density

$N(\vec{r},t)=N_{0}(t)G(\vec{r},2D_{c}\tau+\sigma_{p}^{2})$ (8)

with time-dependent total population

$N_{0}(t)=\eta_{\text{abs}}\frac{\tau}{T}\frac{E}{E_{0}}\begin{cases}(1-e^{-t/\tau}),&t\leq
T\\\ e^{-t/\tau}(e^{T/\tau}-1),&t>T.\end{cases}$ (9)

The recombination of each carrier pair releases the bandgap energy $E_{g}$
back into the slab with volumetric heat capacity $c_{v}$, yielding the source

$q(\vec{r},t)=-\left(\frac{\partial N}{\partial
t}\right)_{\text{decay}}\frac{E_{g}}{c_{v}}=\frac{N(\vec{r},t)}{\tau}\frac{E_{g}}{c_{v}}$
(10)

that produces the temperature profile

$T(\vec{r},t)=\frac{E_{g}}{c_{v}\tau}N_{0}(t)*_{t}G(\vec{r},2D_{t}t+2D_{c}\tau+\sigma_{p}^{2}).$
(11)

Note that we neglect additional initial heating from above-band absorption.

Given sufficiently small $|\delta n|$, the resulting linewidth-normalized
resonance shift

$\tilde{\Delta}(t)=\Delta(t)/\Gamma=-Q\int\frac{\delta
n}{n}(\vec{r},t)|\vec{E}(\vec{r})|^{2}{\rm d}^{3}\vec{r}$ (12)

for the electric field profile $\vec{E}(\vec{r})$ with normalization
($\int|\vec{E}(\vec{r})|^{2}{\rm d}^{3}\vec{r}=1$) is well-approximated by
first-order perturbation theory [11]. We consider a Gaussian-shaped mode
envelope $|\vec{E}|^{2}=G(\vec{r},\sigma_{0})$ fully-confined with uniform
transverse amplitude within the high-index slab.

Since Eqn. 11 must be evaluated numerically, we assume a constant temperature
change $T(\vec{r},t)=T(0,t)$ across the mode — valid for typical experimental
regimes of interest where $\sigma_{0}\ll 2D_{t}t+2D_{c}\tau+\sigma_{p}$ — to
avoid the additional integration in Eqn. 12. The overlap between the optical
mode and the static free carrier profile, on the other hand, can be
analytically evaluated to yield the combined result

$\displaystyle\tilde{\Delta}(t)=$
$\displaystyle\frac{\alpha_{c}}{n}N_{0}(t)\frac{\sigma_{0}^{2}}{2D_{c}\tau+\sigma_{p}^{2}+\sigma_{0}^{2}}\left(\frac{Q}{V}\right)$
$\displaystyle-Q\frac{\alpha_{t}}{n}\frac{E_{g}}{c_{v}\tau}N_{0}(t)*_{t}G(0,2D_{t}t+2D_{c}\tau+\sigma_{p}^{2})$
(13)

for the cavity mode volume $V=\int\epsilon|E|^{2}{\rm
d}^{3}\vec{r}/\max{\\{\epsilon|E|^{2}\\}}=2\pi w\sigma_{0}^{2}$. Since the
reflected signal directly tracks the cavity amplitude in cross-polarization,
the normalized reflectivity

$r(t)=\int_{0}^{t}{\rm
d}t^{\prime}e^{-\Gamma(t-t^{\prime})-i\int_{t}^{t^{\prime}}{\rm
d}t^{\prime\prime}2\Delta(t^{\prime\prime})}$ (14)

is finally found by numerically integrating the cavity evolution as dictated
by coupled mode theory [88].

Figure A1: Estimated normalized reflection coefficient
$r(t)=\sqrt{R(t)}e^{j\phi(t)}$ as a function of switching energy for the
parameters in Table A1. Insets: same results in polar form for comparison to
the Lorentzian reflection profile $r(\Delta)=1/(1+j\Delta)$ of a cavity under
cross-polarized excitation with static detuning $\Delta$ (black dashed line).

Fig. A1 plots the switching characteristics for the parameters in Table A1.
Free-carrier dispersion dominates the response for the nanosecond-order
timescales of interest followed by a slow ($\upmu$s-order), weak
($|\tilde{\Delta}|\ll 1$) thermal rebound. The three order-of-magnitude
timescale difference effectively decouples the two modulation mechanisms. Note
that the true reflection coefficient deviates from the Lorentzian response of
a quasi-static cavity due to the fast (relative to the cavity decay rate
$\Gamma$) carrier dynamics. These results indicate that a SLM with $10^{6}$
pixels operating with $\omega_{s}>2\pi\times 100$ MHz could be realized with
$\mathcal{O}(\text{watt})$ optical control power.

Parameter |  Value |  Source
---|---|---
$n_{\text{Si}}$ |  3.48 |  [107]
$E_{g}$ |  1.12 eV |  [107]
$\alpha_{t}$ |  $1.8\times 10^{-4}$ K-1 |  [107]
$c_{v}$ |  1.64 J/cm${}^{3}\cdot$K |  [108]
$D_{t}$ |  0.26 cm2/s |  [108]
$\alpha_{c}$ |  $8\times 10^{-9}$ $\upmu$m3 |  [109] (Linearized)
$D_{c}$ |  19 cm2/s |  [55]
$\tau$ |  1 ns |  [55]
$\lambda_{0}$ |  1550 nm |  Assumed
$Q$ |  200,000 |  Assumed
$\tilde{V}$ |  0.95 |  [108]
$w$ |  220 nm |  Measured
$2\sigma_{0}$ |  $0.66~{}\upmu\text{m}$ |  $2\sqrt{V/(2\pi w)}$
$\lambda_{p}$ |  $0.53~{}\upmu\text{m}$ |  Assumed
$T$ |  0.5 ns |  Assumed
$2\sigma_{p}$ |  $0.22~{}\upmu\text{m}$ |  $\lambda_{p}/2$
$\eta_{\text{abs}}$ |  0.6 |  FDTD
Table A1: Parameters used for the simulated switching results in Fig. A1. The
pulse parameters were selected to mimic the typical experimental conditions of
Section V in the main text.

##  Appendix B Performance Comparisons

Table A2 compares the PhC-SLM demonstrated here to other actively-controlled,
2D SLMs (Fig. 1b). Wavelength-steered devices and switch arrays are omitted to
restrict focus to the typical SLM architectures in Fig. 1. Notably, while
beamsteering with PhC waveguides [47, 110, 111] and laser arrays [112] has
recently been demonstrated, our device is the first (to our knowledge) to
feature simultaneous emission from a 2D array of individually controllable PhC
pixels.

Class [Year] |  Device |  $\bm{N_{x}\times N_{y}}$ |  $\bm{\Omega_{s}=\frac{\lambda}{\Lambda_{x}}\times\frac{\lambda}{\Lambda_{y}}}$ |  $\bm{\zeta}$ [$\bm{\%}$] |  $\bm{\omega_{s}/2\pi}$ [Hz]
---|---|---|---|---|---
EO [2022] |  PhC-SLM |  $\bm{8\times 8}$ |  $\bm{10.6^{\circ}\times 14.5^{\circ}}$ |  64 |  $\bm{1.4\times 10^{8}}$
EO [2021] |  $\chi^{(2)}$ polymer-coated grating [26] |  $2\times 2$ |  $0.2^{\circ}\times 0.2^{\circ}$ |  — |  $5.0\times 10^{7}$
EO [2019] |  $\chi^{(3)}$ thin-film plasmonic resonator [25] |  $4\times 4$ |  $0.8^{\circ}\times 1.1^{\circ}$ |  20* |  $1.0\times 10^{9}$
EO [2017] |  Bilayer guided resonators [23] |  $6\times 6$ |  $0.3^{\circ}\times 0.3^{\circ}$ |  40* |  $2\times 10^{8}$
EO [2011] |  $\chi^{(2)}$ polymer-coated grating [33] |  $4\times 4$ |  $0.1^{\circ}\times 0.1^{\circ}$ |  18* |  $8.0\times 10^{5}$
EO [2005] |  MQW micropillar modulators [24] |  $128\times 128$ |  $1.3^{\circ}\times 1.3^{\circ}$ |  50 |  $1.3\times 10^{7}$
Thermal [2018] |  Asymmetric Fabry-Perot cavity [17] |  $6\times 6$ |  $3.4^{\circ}\times 3.4^{\circ}$ |  59 |  $1.4\times 10^{4}$
Thermal [2013] |  Waveguided phased array [16, 113] |  $8\times 8$ |  $9.9^{\circ}\times 9.9^{\circ}$ |  10* |  $1.1\times 10^{5}$
MEMS [2019] |  Grating phase shifters [20] |  $160\times 160$ |  $4.4^{\circ}\times 4.1^{\circ}$ |  85* |  $5.5\times 10^{4}$
MEMS [2019] |  Piston mirrors [21] |  $960\times 540$ |  $3.4^{\circ}\times 3.4^{\circ}$ |  — |  $2.0\times 10^{4}$
MEMS [2014] |  High-contrast gratings [19] |  $8\times 8$ |  $2.7^{\circ}\times 2.7^{\circ}$ |  36* |  $5.0\times 10^{5}$
MEMS [2001] |  Piston mirrors [18] |  $256\times 256$ |  $0.9^{\circ}\times 0.9^{\circ}$ |  86 |  $5.0\times 10^{5}$
LC [2020] |  Plasmonic metasurface [14] |  $3\times 2$ |  $0.3^{\circ}\times 0.3^{\circ}$ |  — |  $2.5\times 10^{1}$
LC [2019] |  “MacroSLM” [3] |  $1536\times 1536$ |  $3.0^{\circ}\times 3.0^{\circ}$ |  95 |  $6.0\times 10^{2}$
LC [1994] |  Binary ferroelectric LC [13] |  $256\times 256$ |  $2.2^{\circ}\times 2.2^{\circ}$ |  79 |  $8.3\times 10^{3}$
Table A2: Performance comparison of selected active 2D spatial light
modulators from Fig. 1b. Estimated fill factors $\zeta$ are marked by a *.

##  Appendix C Inverse Design Strategy

We implement the inverse design strategy in Fig. 3b using the open-source
guided mode expansion (GME) package Legume [114]. GME approximates the cavity
eigenmode using the incomplete basis set of waveguide modes in an “effective”
unpatterned slab (in effect transforming the 3D eigenproblem to 2D) and
perturbatively computes the loss due to coupling to the radiative continuum
[77].

During each optimization step, we aggregate the losses of the fundamental slab
mode over four Bloch boundary conditions $\vec{k}_{i}$ at all wave vectors
$\vec{g}_{mn}=\vec{k}_{i}+\vec{G}_{mn}=\vec{k}_{i}+2\pi(\frac{m}{\Lambda_{x}},\frac{n}{\Lambda_{y}})$
satisfying $|\vec{g}_{mn}|<g_{\text{max}}=2.5\times 2\pi/a$ given the
reciprocal lattice vectors $\vec{G}_{mn}$. The objective function Eqn. 2
converges within tens of iterations, and the resulting design is then verified
with $g_{\text{max}}=3\times 2\pi/a$ using a $3\times 3$ $\vec{k}_{i}$ grid in
the Brillouin zone of the rectangular lattice of unit cells.

Exemplary GME-approximated far-field profiles for an $L3$ cavity with two
target quality factors $Q_{0}$ are shown in Fig. A2a,c for comparison to those
computed using near-to-far-field transformations of FDTD-simulated fields
[115, 82]. These results confirm that the perturbatively-computed GME coupling
coefficients can be used to accurately estimate a cavity’s far-field
scattering profile.

Figure A2: Comparison of resonant wavelengths $\lambda_{0}$, quality factors
$Q$, and far-field emission spectra computed from GME (left) and FDTD (right)
simulations for two target $Q_{0}$ (top, bottom).

The inverse design objective function (Eqn. 2) maximizes the directivity
$D=4\pi S(0)/\int_{\Omega}S(\vec{k})~{}{\rm d}\Omega$ (for the light cone
$\Omega$) of the emission profile $S(\vec{k})$ for any $Q_{0}$. The resulting
aperture efficiency

$\eta_{a}=\frac{D_{0}}{\max
D}=\frac{\lambda^{2}}{A}\frac{S(0)}{\int_{\Omega}S(\vec{k})~{}{\rm
d}\Omega}=\frac{A_{0}}{A}$ (15)

compares $D$ to the maximum directivity $4\pi A/\lambda^{2}$ of an area
$A=\Lambda_{x}\Lambda_{y}$ aperture at wavelength $\lambda$, and can therefore
be interpreted as the fill factor of light scattered from an effective area
$A_{0}$. Fig. A3 compares $\eta_{a}$ for grating-coupled and inverse-designed
$L3$ cavities. For most inverse designs, $\eta_{a}\approx 1$ regardless of
$Q_{0}$. Since GME assumes periodic boundary conditions (indicative of the
true array design), scattering from neighboring unit cells enables designs
with $\eta_{a}>1$. However, this “super-directive” performance is undesirable
since the steerable field-of-view is narrowed to $\Omega_{p}<\Omega_{s}$.

Figure A3: FDTD-computed aperture efficiency $\eta_{a}$ of vertically coupled
$L3$ cavities based on a grating perturbation $\delta r/r\in[0,0.05]$ or an
inverse-design target quality factor $Q_{0}\in[10^{2},10^{6}]$. Inset arrows
illustrate parameter trends.

##  Appendix D Experimental Setups

Fig. A4 schematically illustrates the major components of our experimental
setup. Here, we describe the design and function of each sub-assembly.

Figure A4: Overview of experimental setups for measuring and controlling the
photonic crystal SLM (PhC-SLM). A cross-polarized microscope (a) featuring
balanced homodyne measurement (b) enables near- and far-field characterization
of cavity arrays controlled by SLM-distributed coherent light (c) or high-
speed incoherent $\upmu$LED arrays (d). TL: tunable infrared laser (Santec
TSL-710), EOM: electro-optic amplitude modulator; $\lambda/2$: half-wave
plate, PBS: polarizing beamsplitter; L1: 250 mm back-focal-plane lens; DM:
long-pass dichroic mirror; OL: objective lens (Nikon Plan Fluor
40$\times$/0.60 NA or Nikon LU Plan 100$\times$/0.95 NA), L2: 250 mm back-
focal-plane lens; SF: spatial filter; L3: 200 mm tube lens; v-SWIR: visible-
short wave infrared camera (Xenics Cheetah 640); DAQ: data acquisition unit
(NI USB-6343); $\Delta t$: trigger delay generator (SRS DG645); LO: local
oscillator; PM: piezo mirror; BD: balanced detector (Thorlabs PDB480C-AC);
Phase Lock: TEM LaseLock; LPF: low-pass filter; CWTL: continuous-wave trimming
laser (Coherent Verdi V18); MLD: modulated laser diode (Hubner Cobolt or
PicoLAS LDP); BE: $5\times$ visible beam expander; LCOS: high-power liquid
crystal SLM (Santec SLM-300); L4: 300 mm; L5: 250 mm; PD: photo-detector; CL:
collection lens (Zeiss Fluar 5$\times$/0.25 NA); VBE: $0.5\times-2\times$
variable beam expander; DP: dove prism.

###  D.1 Near-Field Reflection Spectra

The wide-field, cross-polarized microscope in Fig. A4(a) allows us to
simultaneously measure the reflection from every cavity within a camera’s
field-of-view. A visible illumination path (not illustrated) is joined with
collimated infrared light from a tunable laser with a dichroic mirror and
focused onto the back-focal-plane (BFP) of an objective by lens L1. The angle-
of-incidence and spot size of the infrared beam on the sample are therefore
controlled by translating L1 and varying the collimated beam diameter,
respectively. In our typical wide-field configuration, a 7.2 mm beam diameter
focused to the center of a $40\times$ objective’s BFP yields a
$\mathord{\sim}150~{}\upmu$m waist-diameter, vertically-incident field that
quasi-uniformly illuminates $10\times 10$ PhC cavity arrays.

By orienting the input polarization at a $45^{\circ}$ angle relative to the
dominant cavity polarization axis (with a half-wave plate or by physically
rotating the sample), light coupled into and reflected by the PhC cavity is
polarization rotated and can be isolated from direct, specular reflections
with a polarizing beamsplitter. A kHz-rate free-running, dual-band (visible
and infrared) camera images this cross-polarized reflection signal through the
tube lens L3. For each frame collected during a laser sweep, the wavelength is
interpolated from the recorded camera and laser output triggers and each
cavity’s reflection is integrated over a fraction of pixels within its imaged
unit cell boundary. We use the resulting high-contrast reflection spectra
(across all devices within the field-of-view) to characterize device
performance and monitor the cavity trimming process.

The sample mount below the objective (OL) is temperature stabilized to within
$10$ mK with a Peltier plate and feedback controller. For trimming
experiments, the sample is placed in a high-pressure oxygen environment within
a custom chamber offering in-situ optical access through a glass window.

###  D.2 Calibrated Far-Field Measurement

Inserting a lens (L2) in the collection path one focal length from the
objective BFP allows us to measure the far-field profile $S(\vec{k})$ of
individual or multiple cavities using the same setup. We position an iris at
the intermediate image plane — located with a removable lens (not shown)
placed before L3 — to spatially filter the emission from desired devices. We
also calibrate the BFP scale using a reflective reference grating with known
pitch. Due to the cross-polarized configuration, only a single polarization
$\tilde{S}(\vec{k})\big{|}_{\theta}$ is imaged for any cavity-input
polarization angle difference $\theta$. The complete cavity emission profile

$S(\vec{k})=\tilde{S}(\vec{k})\big{|}_{\theta}+\tilde{S}(\vec{k})\big{|}_{\theta\pm\pi/2}$
(16)

can therefore be reconstructed by sequentially imaging both polarizations as
in Figs. A5a-c for $\theta=45^{\circ}$. For maximum accuracy, we used this
technique for the experimental results in Fig. 4.

Alternatively, the specific choice $\theta=45^{\circ}$ allows $S(\vec{k})$ to
be reconstructed from a single measurement. Due to mirror symmetry about the
cavity’s principal polarization axis $\hat{y}$, Fig. A5a-b show that
$\hat{\sigma}_{\hat{y}}\\{\tilde{S}(\vec{k})\big{|}_{\pm
45^{\circ}}\\}=\tilde{S}(\vec{k})\big{|}_{\mp 45^{\circ}}$ for the reflection
operator $\hat{\sigma}$. This alternative reconstruction

$S(\vec{k})=\left[1+\hat{\sigma}_{\hat{y}}\right]S(\vec{k})\big{|}_{\pm
45^{\circ}}$ (17)

is demonstrated experimentally in Fig. 4d, yielding excellent agreement with
Fig. 4c. This technique simplifies high-throughput far-field measurements
across cavity arrays (Fig. A10, for example).

Figure A5: Cross-polarized back-focal-plane (BFP) imaging techniques for a
grating-coupled $L3$ cavity. Two orthogonally polarized far-field profiles are
imaged by orienting the input polarization $E_{\text{in}}$ at a $+45^{\circ}$
(a) or $-45^{\circ}$ (b) angle from the dominant cavity polarization axis
(dashed line in inset). The complete cavity emission profile $S(\vec{k})$ can
be reconstructed by summing both images (c) or approximated from a single
polarized image (d), yielding near-identical images with quantitative
agreement between the extracted $\eta_{0}$.

###  D.3 Homodyne Measurement

The shot-noise-limited balanced homodyne detection setup in Fig. A4c enables
complex reflection coefficient measurements with greater than $>$3 dB shot-
noise clearance below $1$ GHz [108]. Signal light reflected from the cavity
combines with a path-length-matched (to within $\mathord{\sim}$mm based on
time-delay measurements with a picosecond-class pulsed laser) local oscillator
(LO), and both signals are coupled into a balanced detector using anti-
reflection coated fibers. The in-phase ($I(t)$) and quadrature ($Q(t)$)
components of the cavity reflection were sequentially measured by locking to
the first and second harmonics of the balanced output in the presence of a
piezo-driven LO phase dither. The resonant, cross-polarized cavity reflection
$R$ and phase shift $\phi$ are then reconstructed as

$\displaystyle R=\frac{\left[V_{p}-I(t)\right]^{2}+Q^{2}(t)}{V_{p}^{2}}$
$\displaystyle\phi=\arctan\frac{Q(t)}{V_{p}-I(t)}$ (18)

by normalizing to the measured peak voltage swing $V_{p}$ of the interference
signal.

###  D.4 Parallel Cavity Trimming

A liquid crystal on silicon (LCOS) SLM (Fig. A4c) actively distributes a high-
power, continuous-wave visible laser to target devices during the cavity
trimming procedure. The input laser was tunably attenuated with a motorized
half-wave plate (preceding a PBS) and subsequently expanded to overfill the
LCOS aperture. The LCOS SLM was re-imaged onto the objective BFP (as confirmed
by imaging with L2 in place) using two lenses (L4, L5) with focal lengths
chosen to optimally match the imaged SLM and objective pupil dimensions. Phase
retrieval-computed holograms then evenly distribute power to an array of
focused spots on the sample (Appendix I) when the mechanical, flip mirror
shutter is opened.

###  D.5 $\upmu$LED Imaging

The collection optics in Fig. A4d maximize the intensity of a $\upmu$LED array
projected onto the PhC membrane within the constraints dictated by the
constant radiance theorem of incoherent imaging. Assuming a Lambertian
emission profile, geometric optics gives the collection efficiency
$\eta_{c}=\alpha_{c}^{2}$ for an objective lens (CL) with numerical aperture
$\alpha_{c}$ focused on the $\upmu$LED array. The projection efficiency
$\eta_{p}$ through the projection objective (OL, with numerical aperture
$\alpha_{p}$) depends on the relative pupil sizes of both objectives and can
be similarly approximated from geometric optics. The resulting intensity
enhancement $\zeta=\eta_{c}\eta_{p}/M^{2}$ between the source and image (with
magnification $M$) reaches a maximum
$\zeta_{\text{max}}=\frac{1}{M^{2}+(1-\alpha_{p}^{2})/\alpha_{p}^{2}}$ when
the CL-collimated light overfills the back aperture of OL. The resulting
design criteria,

$\alpha_{c}>\sqrt{\frac{M^{2}\alpha_{p}^{2}}{(M^{2}-1)\alpha_{p}^{2}+1}},$
(19)

is achieved for our imaging setup with $\alpha_{c}=0.25$, $\alpha_{p}=0.95$,
and $M\approx 1/30$. After CL, The overall magnification and rotation are
fine-tuned with a variable beam expander and Dove prism, respectively.

##  Appendix E Optimum Gaussian Coupling

The far-field spatial overlap integral [116]

$O(w_{0})=R^{2}\iint_{\Omega}\vec{E}_{c}(\theta,\phi)\times\vec{H}_{g}^{*}(w_{0},\theta,\phi)\sin\theta
d\theta d\phi$ (20)

over the hemisphere $\Omega$ at distance $R$ yields the power coupling
$|O(w_{0})|^{2}$ between a cavity mode $c$ and fundamental Gaussian beam $g$
as a function of the Gaussian waist radius $w_{0}$. We compute the cavity
electric field profile $\vec{E}_{c}(\theta,\phi)$ by applying a near-to-far
field transformation to the FDTD-simulated cavity mode in a plane just above
and parallel to the PhC slab 444Flexcompute, Inc. Tidy3D.
https://simulation.cloud [118]. The resulting far-field profile is finely
discretized — relative to the diffraction-limited beamwidth
$\lambda/\Lambda\approx 14^{\circ}$ for typical cavity unit cell dimensions
$\Lambda$ and resonant wavelength $\lambda$ — on a 1∘ grid in zenith and
azimuth ($\theta$ and $\phi$, respectively). For the co-polarized Gaussian
mode, we instead convert the magnetic field [88]

$\displaystyle\vec{H}(x,y,z)=$
$\displaystyle\sqrt{\frac{\pi}{Z_{0}}}\frac{2w_{0}}{z+j\pi
w_{0}^{2}}\exp{\frac{-j2\pi(x^{2}+y^{2})}{2(z+j\pi w_{0}^{2})}}$
$\displaystyle\times$ $\displaystyle\left[\hat{x}+\left(\frac{-xz-jx\pi
w_{0}^{2}}{z^{2}+(\pi w_{0}^{2})^{2}}\right)\hat{z}\right]e^{-j2\pi z}$ (21)

derived from paraxial diffraction theory (for the free space impedance
$Z_{0}$) to spherical coordinates at a far-field distance $R$. Both modes are
normalized to carry unit power
$\int_{\Omega}\real\\{\vec{E}\times\vec{H}^{*}\\}/2$.

Figure A6: Comparison of coupling between a fundamental Gaussian beam with
waist $w_{0}$ and two different cavity designs with maximum unit cell
dimension $\Lambda$.

Fig. A6 compares the resulting power coupling as a function of normalized
waist $2w_{0}/\Lambda$ for grating-coupled and inverse-designed cavity designs
with maximum dimension $\Lambda$. In both cases, a backreflector is assumed
for unidirectional emission into $\Omega$. Besides the increase in maximum
coupling to 94%, the optimized waist diameter $2w_{0}\approx\Lambda$ indicates
that the inverse-designed cavity’s effective near-field scattering profile
fully fills the design unit cell.

The power coupling $|O|_{\text{max}}^{2}$ to a single desired free-space mode
also allows us to compute that mode’s amplitude reflection spectrum [88]

$r(\tilde{\Delta})=\frac{2|O|_{\text{max}}^{2}/p-j\tilde{\Delta}}{1+j\tilde{\Delta}}$
(22)

using temporal coupled mode theory assuming a normalized detuning
$\tilde{\Delta}=\Delta/\Gamma$ from the cavity resonance and $p$-directional
emission (i.e. $p=1$ for unidirectional emission with a backreflector or $p=2$
for symmetric emission). Fig. 4 of the main text plots these optimized
spectra. Whereas the grating-coupled cavity is undercoupled with an amplitude-
dominant reflection spectrum, the inverse-designed cavity is phase-dominant as
desired for high efficiency beamforming.

##  Appendix F Beamforming with a Coupled Amplitude-Phase Response

Phase retrieval algorithms (Appendix I) optimize unity-magnitude near-field
reflection coefficients $r=e^{j\phi}$ to generate a desired far-field
intensity pattern, but fail for the coupled amplitude and phase reflection
coefficients of Eqn. 22 [119, 120]. Here, we derive an alternative algorithm
to compensate for this coupling.

Assuming a spatially uniform field $E_{i}(\vec{r})=E_{i}$ incident on a
$\Lambda_{x}\times\Lambda_{y}$-pitch array, the reflected far-field defined in
Eqn. 3 is the product of the single-element far-field pattern $S(\vec{k})$ and
the array factor

$\mathcal{F}\\{r\\}=\sum_{m,n}\Gamma_{m,n}\exp{jk(m\Lambda_{x}u+n\Lambda_{y}v)},$
(23)

i.e., the 2D discrete Fourier transform ($\mathcal{F}$) of the reflection
coefficients $r$ with respect to the cosine-space coordinates $u=k_{x}/k$ and
$v=k_{y}/k$. Since $S(\vec{k})$ is determined by element design (Section II),
far-field pattern synthesis is achieved by optimizing the array factor. Given
Eqn. 23, we can represent constrained far-field synthesis with the nonlinear
program

$\min_{\Delta}f(\Delta)=\min_{\Delta}\frac{1}{2}\norm{\frac{|\mathcal{F}\\{r(\Delta)\\}|^{2}}{\norm{|\mathcal{F}\\{r(\Delta)\\}|^{2}}_{F}}-\frac{|\textbf{Y}|^{2}}{\norm{|\textbf{Y}|^{2}}_{F}}}_{F}^{2},$
(24)

where $r(\Delta)$ is the matrix of reflection coefficients as a function of
the detuning matrix $\Delta$, $|\cdot|$ represents the complex modulus,
$\norm{\cdot}_{F}$ is the Frobenius norm, and $Y$ is the goal image. By
normalizing the Fourier transforms, this program optimizes image appearance as
opposed to absolute intensity, which varies in the presence of reflection
loss.

We approximate the solution to the nonlinear program using the L-BFGS-B method
— a low-memory quasi-Newton method with simple box constraints to limit
detunings [121]. L-BFGS-B uses the gradient and a low-rank approximation of
the objective function’s Hessian to approximate Newton’s method, yielding
superlinear convergence.

To efficiently optimize Eqn. 24, we computed its analytic gradient

$\displaystyle\nabla f=$
$\displaystyle\frac{2\left(g(a_{1},b_{1})-\frac{\langle|\mathcal{F}\\{r(\Delta)\\}|^{2},h\rangle_{F}}{\norm{|\mathcal{F}\\{r(\Delta)\\}|^{2}}_{F}}g(a_{2},b_{2})\right)}{\norm{|\mathcal{F}\\{r(\Delta)\\}|^{2}}_{F}^{2}},$
(25)

for

$\displaystyle g(a,b)=\big{[}$
$\displaystyle(\Re{\mathcal{F}\\{a\\}}+\Im{\mathcal{F}\\{b\\}})\odot\Re{\nabla
r}+$
$\displaystyle(\Re{\mathcal{F}\\{b\\}}-\Im{\mathcal{F}\\{a\\}})\odot\Im{\nabla
r}\big{]},$
$h=\frac{|\mathcal{F}\\{r(\Delta)\\}|^{2}}{\norm{|\mathcal{F}\\{r(\Delta)\\}|^{2}}_{F}}-\frac{|\textbf{Y}|^{2}}{\norm{|\textbf{Y}|^{2}}_{F}},$
(26)

and

$\displaystyle a_{1}=h\odot\Re{\mathcal{F}\\{r(\Delta)\\}}$ $\displaystyle
a_{2}=|\mathcal{F}\\{r(\Delta)\\}|^{2}\odot\Re{\mathcal{F}\\{r(\Delta)\\}}$
$\displaystyle b_{1}=h\odot\Im{\mathcal{F}\\{r(\Delta)\\}}$ $\displaystyle
b_{2}=|\mathcal{F}\\{r(\Delta)\\}|^{2}\odot\Im{\mathcal{F}\\{r(\Delta)\\}},$

where $\odot$ and $\langle\rangle_{F}$ are the Hadamard/element-wise product
and Frobenius inner product, respectively. Since Eqn. 25 contains only Fourier
transforms and element-wise operations, it can be calculated in
$\mathcal{O}\left(N\log(N)\right)$ time.

We further accelerated L-BFGS-B by initializing $\Delta$ with a modified phase
retrieval algorithm that enforces near-field constraint $r(\Delta)$ [122]. The
described algorithm was implemented in python using the PyTorch package for
GPU acceleration. Using an RTX 2080, the generation of the far-field patterns
for Fig. 3 took only a few seconds. The success of the algorithm is
demonstrated by the lack of a conjugate image (even in the highly undercoupled
case).

##  Appendix G Effect of Substrate Reflections

Isolated slab PhC cavities feature symmetric, bi-directional emission due to
vertical reflection symmetry about the slab midplane. When placed above a
reflective substrate, however, interference between the (reflected) downwards
and upwards emission paths alters the cavity’s radiation pattern $S(\vec{k})$
and $Q$ [123]. The results are analogous to the modified spontaneous emission
from a quantum emitter placed above a mirror [124].

Figure A7: Quality factor ($Q$) trends for an $L3$ PhC cavity (resonant
wavelength $\lambda_{0}$) placed a distance $d$ above a silicon back-
reflector. Insets show the far-field emission pattern $S(\vec{k})$ at select
points.

Fig. A7 plots the FDTD-simulated quality factor trends of an inverse-designed
$L3$ cavity as a function of the membrane-substrate gap spacing $d$. Since the
optical thickness of the PhC slab is approximately $\lambda/2$, constructive
interference at $d\approx m\lambda/2$ ($m\in 1,2,...$) maximizes vertical
emission and minimizes $Q$. Destructive interference at
$d\approx(2m-1)\lambda/4$ has the opposite effects. The resulting variation in
$\eta_{0}$ explains the minor discrepancy between the simulated and measured
$\vec{S}(k)$ in Fig. 4.

We mitigated the impact of these effects through optimized sample preparation.
Compressive stress in our silicon-on-insulator (SOI) die buckles suspended
cavity arrays, yielding variations in $d$ — and therefore cavity reflectivity
— across the membrane. As an alternative to previously proposed stress-
engineered support structures [125], we flattened the suspended arrays by
mechanically bowing the die with a backside set pin in a custom sample mount
(Fig. A8). Unfortunately, the resulting (uniform) gap spacing
$d=2~{}\upmu\text{m}$ minimizes vertical coupling at the design wavelength
$\lambda_{0}=1.55~{}\upmu\text{m}$. We therefore added a back-side silicon
nitride anti-reflection coating (ARC) on the substrate and timed the release
etch to form a front-side ARC with the remaining oxide (Fig. A9).

Figure A8: Sample mount with set pin to flatten stress-buckled suspended
membranes. Figure A9: Comparison of simulated (dashed) and measured (solid)
silicon-on-insulator (SOI) sample reflection before (blue) and after (green)
anti-reflection optimization for normal incidence at $\lambda=1550$ nm.
Measured values were calibrated with a known reference mirror. The final layer
stack of silicon (Si, $n=3.48$), oxide (SiO2, $n=1.44$), and deposited silicon
nitride (SiNx, $n=1.90$) is shown in the inset with optimized layers
highlighted in green.

##  Appendix H Far-Field Uniformity

Fig. A10 demonstrates the far-field uniformity characteristic of inverse-
designed and grating coupled $L3$ cavity arrays. Averaged across the $8\times
8$ arrays, the former offers a $\mathord{\sim}3\times$ improvement in zero-
order diffraction and aperture efficiencies ($\langle\eta_{0}\rangle=0.86$,
$\langle\eta_{a}\rangle=0.99$).

Figure A10: Imaged far-field profiles $S(\vec{k})$ (over a 0.9 numerical
aperture) for each device in an $8\times 8$ array of inverse designed (top)
and grating-coupled (bottom) $L3$ PhC cavities. The extracted zero-order
efficiencies $\eta_{0}$ and standard deviations are also provided.

##  Appendix I The slm-suite Toolbox

We developed the open-source python package slm-suite to simplify the creation
of high-uniformity, arbitrary-geometry optical focus arrays using various
phase retrieval algorithms. The package features:

  1. 1.

Automated wavefront calibration routines that measure the Fourier-plane source
amplitude and phase using a super-pixel interference technique to compensate
for aberrations along the SLM imaging train [126]

  2. 2.

Various graphical processing unit (GPU)-accelerated Gerchberg-Saxton (GS)
algorithms that use the measured source constraints (1) to produce optimized
spot array phase masks [127, 128, 129]

  3. 3.

Automated affine transformations between grating wave vectors applied to the
SLM and image-space coordinates (i.e. camera pixels) by projecting and
detecting a GS-computed spot array

  4. 4.

Camera-based feedback of measured spot amplitudes at known (calibrated)
locations into phase retrieval algorithms to improve the uniformity of image-
space spot arrays

  5. 5.

Automated evaluation metrics to monitor diffraction efficiency, spot amplitude
and position tolerance, and spot quality.

  6. 6.

Simplified hardware interface and control.

After calibration, high-uniformity optical foci can be generated at arbitrary
image plane locations specified by the user. For example, Fig. A11 shows a
$10\times 10$ spot array with $\mathord{\sim}1\%$ power uniformity and sub-
micron placement accuracy formed on a $10\times 10$ cavity array during the
trimming procedure of Section IV.

Figure A11: Overlaid images of $10\times 10$ cavity (grey) and trimming spot
(color) arrays demonstrating the $\ll\upmu$m placement accuracy and percent-
order power uniformity of weighted Gerchberg-Saxton phase retrieval with
experimental camera feedback.

##  Appendix J Parallel Laser Oxidation

As illustrated by the wavelength trends as a function of incident laser power
and exposure time in Fig. A12, we found that laser-assisted thermal oxidation
could be accelerated in a high-pressure oxygen environment. We therefore mount
samples in a custom pressure chamber with a partial oxygen pressure ranging
from 5 to 10 atm.

Figure A12: Wavelength and quality factor trends as a function of incident visible laser power, demonstrating accelerated trimming with increased oxygen partial pressure $P_{\text{O}_{2}}$. Technique [Year] |  Cavity Type |  $\bm{N}$ |  $\bm{\Delta\lambda_{0}^{\textbf{p-p}}}$ [pm] |  $\bm{\langle Q\rangle}$ |  In situ? |  Parallel?
---|---|---|---|---|---|---
“Holographic” oxidation [2022] |  Si PhC |  64 |  13 |  $2\times 10^{5}$ |  Y |  Y
Germanium implantation [2021] |  Si ring [130] |  58 |  32 |  $4\times 10^{3}$ |  Y |  N
Laser-annealed cladding [2020] |  Si ring [131] |  2 |  20* |  $2\times 10^{4}$ |  Y |  N
Boron implantation [2019] |  Si ring [132] |  4 |  15 |  $5\times 10^{3}$ |  Y |  N
Electron-beam irradiation [2018] |  Si PhC [133] |  4 |  400 |  $3\times 10^{5}$ |  N |  N
Photo-electro-chemical etching [2017] |  GaAs disk [134] |  5 |  200* |  $2\times 10^{4}$ |  Y |  N
Annealed cladding [2016] |  Si ring [135] |  5 |  90* |  $3\times 10^{3}$ |  Y |  N
Ultraviolet irradiation [2014] |  a-Si ring [136] |  4 |  45 |  $8\times 10^{3}$ |  Y |  N
Post-fabrication etching [2013] |  GaAs PhC [137] |  18 |  100* |  $3\times 10^{4}$ |  N |  Y
Photochromatic thin-film [2011] |  GaAs PhC [138] |  3 |  340 |  $8\times 10^{3}$ |  Y |  N
Anodic oxidation [2006] |  GaAs PhC [139] |  2 |  100* |  $5\times 10^{3}$ |  N |  N
Table A3: Comparison of microcavity array trimming techniques. Estimated
values are marked with a *. $\Delta\lambda_{\text{p-p}}$ = peak-to-peak
wavelength error; $\langle Q\rangle$ = mean quality factor. Figure A13:
Flowchart of the holographic trimming algorithm. Trimming holograms are formed
with weighted Gerchberg-Saxton (GS) algorithms and projected onto desired
cavities for duration $\Delta t$ with power $P_{\text{trim}}$. Alternating
trimming and resonance readout periods continue until the instantaneous
wavelength $\lambda_{i}$ of any targeted cavity blueshifts past the target
wavelength $\lambda_{t}$. Thereafter, a new set of target cavities is selected
and trimmed. This selection and trimming sub-loop continues until all resonant
wavelengths $\\{\lambda_{0}\\}$ are below the “rest” wavelength
$\lambda_{\text{rest}}$, at which point trimming is halted and the resonances
are continuously monitored at readout interval $\Delta t_{\text{rest}}$. When
the resonances are sufficiently stable (redshifting from moisture adsorption
to the silicon membrane is arrested), the total “rehydration” redshift
$\Delta\lambda_{0}$ of each cavity is updated to better estimate the true
resonant wavelength $\lambda_{0}\approx\lambda_{i}+\Delta\lambda_{0}$ from the
instantaneous wavelengths $\\{\lambda_{i}\\}$ during trimming. The entire
process terminates when the peak-to-peak static resonant wavelength uniformity
$\Delta\lambda_{0}^{\text{p-p}}$ drops below the desired tolerance
$\Delta\lambda_{\text{tol}}.$

The main loop of the trimming process (Fig. 6) consists of device selection,
hologram setup, parallel laser oxidation, and resting intervals. The algorithm
monitors two resonant wavelengths: the instantaneous wavelength $\lambda_{i}$
and the steady-state wavelength $\lambda_{0}$. Initially
$\lambda_{i}=\lambda_{0}$; however, focusing high-power ($\mathord{\sim}10$
mW) visible light onto the cavity (as required to sufficiently heat the PhC
membrane for thermal oxidation) causes a temporary blueshift
$\Delta\lambda_{0}$ due to the desorption of moisture attached to hydrophilic
hydroxyl surface terminations. For any target rest wavelength $\lambda_{t}$,
we therefore trim devices to an instantaneous wavelength
$\lambda_{i}=\lambda_{t}-\Delta\lambda_{0}$ that relaxes over
$\mathcal{O}(\text{minute})$ timescales to
$\lambda_{i}=\lambda_{0}=\lambda_{t}$ as moisture re-adsorbs to the surface.
In practice, the stability and estimation of the “overtune”
$\Delta\lambda_{0}$ limit the uniformity and scale of the trimming process,
respectively.

After initializing the cavity locations, scanning the device resonances, and
calibrating the SLM (Section I), a spot array targeting every cavity (Fig.
A11, for example) is projected on the membrane for a short (few second)
duration. Monitoring the resonances at fixed intervals $\Delta t\approx 10$ s
until $\lambda_{i}$ stabilizes to the rest wavelength $\lambda_{0}$ gives an
initial estimate $\Delta\lambda_{0}=\lambda_{0}-\min\\{\lambda_{i}\\}$ for the
overtune parameter of each cavity. We also update the target wavelength
$\lambda_{t}=\min\\{\lambda_{0}\\}$ and rest wavelengths before continuing the
trimming procedure. To update $\Delta\lambda_{0}$, we periodically conduct
this same “rest loop” when $\lambda_{0}$ of each cavity is below an
algorithmically chosen “checkpoint” wavelength $\lambda_{\text{rest}}$.

As described in Section IV, a subset of $N$ cavities is then selected to
maximize the total possible trimming distance to $\lambda_{t}$. The number of
targeted cavities neighboring each untargeted cavity is also limited to reduce
crosstalk. A spot array is then formed to evenly distribute the trimming laser
to the selected devices. After confirming that the location accuracy and power
uniformity of the array are within tolerance, we alternate exposure and
readout intervals to grow thermal oxide with in situ monitoring. The laser
power is progressively increased to reach a desired, wavelength-uniformity-
dependent trimming rate. As evidenced by Fig. A12, the rate is relatively
power-independent until reaching a threshold power. We detect and save these
threshold powers for use when selecting the initial exposure power in each
trimming loop.

The trimming sub-loop continues until the estimated $\lambda_{0}$ of any
targeted cavity crosses $\lambda_{t}$. New cavities are then selected,
targeted, and trimmed until a rest period is triggered. When the peak-to-peak
wavelength uniformity at the end of a rest period is below the user-defined
tolerance $\lambda_{\text{tol}}$, the process is terminated. Table A3 compares
the demonstrated performance compared to other arrayed microcavity trimming
techniques.