Also at ]Harvard Medical School and Wellman Center for Photomedicine,
Massachusetts General Hospital, 50 Blossom Street, Boston, MA 02114, USA Also
at ]Department of Physics, University of California, Santa Barbara, CA 93106,
USA

# Broadband Omni-resonant Coherent Perfect Absorption in Graphene

Ali K. Jahromi<EMAIL_ADDRESS>Massimo L. Villinger  Ahmed El
Halawany  Soroush Shabahang  [  H. Esat Kondakci  [  CREOL, The College of
Optics & Photonics, University of Central Florida, Orlando, FL 32816, USA
Joshua D. Perlstein  Department of Materials Science and Engineering,
University of Central Florida, Orlando, FL 32816, USA  Ayman F. Abouraddy
CREOL, The College of Optics & Photonics, University of Central Florida,
Orlando, FL 32816, USA

###### Abstract

Coherent perfect absorption (CPA) refers to interferometrically induced
complete absorption of incident light by a partial absorber independently of
its intrinsic absorption (which may be vanishingly small) or its thickness.
CPA is typically realized in a resonant device, and thus cannot be achieved
over a broad continuous spectrum, which thwarts its applicability to
photodetectors and solar cells, for example. Here, we demonstrate broadband
omni-resonant CPA by placing a thin weak absorber in a planar cavity and pre-
conditioning the incident optical field by introducing judicious angular
dispersion. We make use of monolayer graphene embedded in silica as the
absorber and boost its optical absorption from $\approx 1.6\%$ to $\sim 60\%$
over a bandwidth of $\sim 70$ nm in the visible. Crucially, an analytical
model demonstrates that placement of the graphene monolayer at a peak in the
cavity standing-wave field is not necessary to achieve CPA, contrary to
conventional wisdom.

††preprint: APS/123-QED

##  I Introduction

Coherent perfect absorption (CPA) refers to the complete absorption of light
incident on a partially absorbing structure [1, 2] (see also early
premonitions of this concept in [3]). The essence of CPA is that the
interaction of an incoming optical field with a partial absorber can be
maximized by engineering the field distribution [4]. Indeed, CPA suggests that
by arranging for two fields to interfere in a lossy planar structure, one can
always guarantee that 100% of the incident radiation is absorbed –
independently of the structure’s intrinsic absorption or its thickness. This
can be achieved by exploiting two counter-propagating fields incident on the
two ports of a planar Fabry-Pérot (FP) resonator containing a partially
absorbing layer between two mirrors M1 and M2 of reflectivities $R_{1}$ and
$R_{2}$, respectively [2, 5]. If the intrinsic single-pass absorption in a
layer of an optical material is $\mathcal{A}$, then providing symmetric
mirrors each having a reflectivity $R_{1}\\!=\\!R_{2}\\!=\\!1-\mathcal{A}$
guarantees that counter-propagating fields with prescribed relative amplitude
and phase are completely absorbed [5] – even when $\mathcal{A}$ is vanishingly
small. Alternatively, in the case of an ultrathin layer, placing it at a peak
in a standing-wave resulting from the interference of two counter-propagating
fields can yield complete absorption [6].

A variety of applications in photonics can benefit from maximizing optical
absorption in a given structure, ranging from optical detectors [7, 8] to
solar cells [9, 10]. Furthermore, a broad range of configurations can exploit
the large absorption modulation achievable in CPA, including optical logic
gates and switches [11, 12, 6, 13, 14, 15, 16, 17]. Despite the exciting
prospects afforded by CPA, several hurdles need to be first overcome,
especially in applications where broadband absorption is desired. First,
optical absorption in any medium over a finite bandwidth is wavelength
dependent $\mathcal{A}\\!=\\!\mathcal{A}(\lambda)$. As such, the cavity
mirrors are required to have a wavelength-dependent reflectivity
$R_{1}(\lambda)\\!=\\!R_{2}(\lambda)\\!=\\!1-\mathcal{A}(\lambda)$ that
counterbalances the dispersive absorption, which can be realized via aperiodic
multilayer dielectric mirrors [18, 19]. Therefore, CPA is in principle
materials-agnostic.

Second, practical settings in optical detection or solar cells necessitate
single-port incidence rather than a counter-propagating-field configuration.
This can be accommodated by placing the absorbing layer in an asymmetric
cavity having a back-reflector of reflectivity $R_{2}\\!=\\!1$ [20, 5, 21,
18]. In this case, light is incident on a front mirror that must have a
reflectivity $R_{1}(\lambda)\\!=\\!(1-\mathcal{A}(\lambda))^{2}$ to realize
complete absorption [5, 18]. Such a configuration is of course reminiscent of
critical coupling in micro-cavities [22, 23, 24]. Eliminating the need for
counter-propagating fields in turn eliminates the restrictions on the relative
amplitude and phase, thus making the arrangement amenable to broadband
incoherent light. Indeed, we recently demonstrated CPA in an asymmetric cavity
over a bandwidth of $\sim\\!60$ nm utilizing a fluorescent dye [19], and over
a full octave of bandwidth $\sim\\!800$-1600 nm using 2-$\mu$m-thick poly-
crystallince silicon layer [18].

The third and most serious hurdle is the resonant nature of this CPA
configuration. For example, a FP resonator can satisfy the CPA condition over
extremely broad bandwidths, but absorption is nevertheless harnessed only at
the discrete resonant wavelengths within this spectrum [18, 25]. This is
clearly inadequate for enhancing the performance of a photodetector or solar
cell, where absorption over a broad continuous bandwidth is imperative. We
have recently developed the concept of ‘omni-resonance’, which can help yield
continuous-wavelength CPA in a FP resonator. By introducing a specific angular
dispersion into the field such that each wavelength is assigned to a
prescribed angle, the entire incident spectrum – which can significantly
exceed the cavity free spectral range – resonates with a single longitudinal
cavity mode. The bandwidth of the resonant spectrum is thus decoupled from the
cavity photon lifetime and quality factor. Combining CPA with omni-resonance
in a FP cavity containing a partially absorbing layer can thus yield
resonantly enhanced absorption over a broad, continuous spectrum.

In this paper, we demonstrate that broadband CPA can be realized in an
ultrathin weakly absorbing layer (for instance, 2D materials such as graphene)
once embedded in the appropriate planar photonic environment. Besides their
remarkable electronic transport features[26], 2D materials strongly couple to
light, suggesting them as intriguing candidates for optoelectronics[27].
Indeed, monolayer graphene has an intrinsic single-pass absorption in free
space of $\mathcal{A}_{\mathrm{G}}\\!\approx\\!2.3\%$ despite being atomically
thin [28], which can make it potentially visible even to the naked eye.
Nevertheless, $\mathcal{A}_{\mathrm{G}}$ is too low for most optoelectronics
applications, including optical modulators and photovoltaics. Recent
strategies for enhancing optical absorption in graphene include exploiting the
extended interaction lengths afforded by waveguides [29, 30], tuning the
absorption via an electrostatic gate (also know as ‘chemical doping’) [31,
32], placing graphene in a standing wave formed by counterpropagating fields
[12, 33], or resonantly enhancing the absorption via guided-mode resonances
[14, 34, 35, 36], attenuated total internal reflection [37, 38], Salisbury
screens [39, 40, 41, 42, 43], or critical coupling [20, 21, 44]. [38]

Figure 1: Achieving broadband, spectrally continuous, omni-resonant CPA in a
planar FP resonator by engineering the angular dispersion introduced into the
incident field. (a) A collimated broadband field is normally incident on a
passive symmetric FP cavity (first row) that transmits light at resonant
wavelengths and reflects light otherwise (second and third rows). The resonant
wavelengths are those for which the cavity wave vector is an integer multiple
of $k_{0}\\!=\\!\pi/L$ (fourth row; ignoring the mirror reflection phases).
(b) Same as (a) except that the passive FP cavity is asymmetric: it is
provided with a back-reflector $R_{2}\\!=\\!1$ (first row); the cavity
therefore exhibits a spectrally flat, unity reflectivity (second row) and zero
transmission (third row). The resonant wavelengths are the same as those for
the symmetric cavity in (a). (c) After inserting a low-absorption material of
single-pass absorption $\mathcal{A}$ (first row) into the cavity in (b),
complete absorption is guaranteed when $R_{2}\\!=\\!1$ and
$R_{1}\\!=\\!(1-\mathcal{A})^{2}$, and CPA is thus realized at the resonant
wavelengths, resulting in spectral reflection dips (second row) and absorption
peaks (third row). (d) At oblique incidence, the resonances in (c) are blue-
shifted. (e) When the incident field is pre-conditioned such that each
wavelength $\lambda$ is assigned to an appropriate incidence angle
$\theta(\lambda)$ (first row), the wavelengths resonate continuously with the
cavity; reflection over the full spectrum is eliminated (second row) and
broadband CPA is achieved (third row); and the longitudinal wave-vector
components of the incident wavelengths correspond to a single resonant cavity
mode (fourth row).

Here, we demonstrate resonantly enhanced absorption ($\sim\\!20\\!-\\!60\%$)
in a monolayer of graphene placed within a FP cavity continuously across a
bandwidth of $\approx\\!70$ nm in the visible. The cavity is designed to
realize CPA at visible wavelengths and omni-resonance is simultanelously
achieved by pre-conditioning the incident broadband incoherent field.
Consequently, resonantly enhanced absorption is delivered over a broad,
continuous spectrum. We therefore establish a general framework for achieving
broadband perfect absorption in high-quality-factor (narrow-linewidth) optical
cavities. Our motivation for selecting monolayer graphene is that it is a
model for ultrathin absorbers with low intrinsic single-pass absorption.
Furthermore, because its absorption is spectrally flat, we can focus on the
juxtaposition of CPA with omni-resonance without the complications arising
from wavelength-dependent absorption (which we dealt with in [5, 18]).
Furthermore, graphene has been shown to be useful in a variety of device
architectures [27] that can benefit from maximizing absorption [45],
particularly when realizing strong coupling between excitons and cavity
photons [46, 47].

Realizing CPA by placing an ultrathin absorber such as graphene in a planar FP
cavity presents a new configuration that has not been carefully examined thus
far. Previous studies of CPA in planar cavities presumed ‘distributed’
absorption over a finite thickness rather than ‘localized’ absorption in a
deeply subwavelength absorber [5, 18]. Alternatively, absorption enhancement
in an ultrathin absorber has been realized by placing it at a particular
position within a standing wave formed of counterpropagating fields [48, 12,
33, 6, 11, 16]. It is commonly understood that maximizing the absorption
necessitates placing the absorber at a standing-wave peak. This apparently
sets stringent tolerances upon the position of the absorber within a FP cavity
to maximize the absorption because the standing wave cannot be adjusted by
modifying a field incident on a single port. We show here that the CPA-cavity
setting surprisingly provides more favorable conditions for resonant
absorption enhancement compared to the free-space configuration. Indeed, we
find that maximum absorption can be achieved even if the absorber is not
located at a standing-wave peak, which cannot be achieved in free space.
Furthermore, graphene and other 2D materials offer a unique opportunity for
post-fabrication tuning of the resonant absorption by adjusting the intrinsic
absorption through an electrical potential.

This paper is organized as follows: first, we briefly review the concept of
omni-resonance in the context of symmetric transparent FP cavities [49] and
asymmetric FP cavities incorporating a partially absorbing layer [50]. Next,
we describe theoretically the distinction between two scenarios of
‘distributed’ and ‘localized’ absorption. We then present our experimental
results on CPA in a graphene monolayer embedded in a planar FP cavity in
absence then in presence of omni-resonance, thereby confirming omni-resonance-
driven continuous broadband CPA in the visible.

##  II Concept of omni-resonance

The key to harnessing the resonantly enhanced optical absorption associated
with CPA over a broad, continuous bandwidth – rather than only at discrete
resonant wavelengths – is the notion of omniresonance [49, 51, 52, 53, 50],
which is illustrated in Fig. 1. Consider first a lossless symmetric planar FP
cavity comprising a transparent layer of thickness $L$ and refractive index
$n$ between a front mirror M1 and a back mirror M2 having equal reflectivities
$R_{1}\\!=\\!R_{2}\\!=\\!R$; Fig. 1(a). At normal incidence, light is
transmitted only at the cavity resonances, whereupon the roundtrip phase
$\varphi$ satisfies the condition [54]:

$\varphi(\lambda)=4\pi
n\frac{d}{\lambda}+\alpha_{1}(\lambda)+\alpha_{2}(\lambda)=2m\pi,$ (1)

where $m$ is an integer, and $\alpha_{1}$ and $\alpha_{2}$ are the reflection
phases from M1 and M2, respectively. Of course, if M2 is a back-reflector
having $R_{2}\\!=\\!1$ (i.e., a Gires-Tournois etalon [55]), then the entire
spectrum is reflected; Fig. 1(b). Introducing an absorbing layer of single-
pass absorption $\mathcal{A}(\lambda)$ into this cavity changes its response
notably; Fig. 1(c). Indeed, if
$R_{1}(\lambda)\\!=\\!(1\\!-\\!\mathcal{A}(\lambda))^{2}$, then light is fully
absorbed on resonance, $\mathcal{A}_{\mathrm{tot}}(\lambda)\\!=\\!1$, whenever
$\lambda$ satisfies Eq. 1 – even when $\mathcal{A}$ is vanishingly small [5,
18]. This is a manifestation of CPA, which is analogous here to critical
coupling [22, 23, 24].

The basis for the omni-resonance strategy can be appreciated by examining
oblique incidence at an angle $\theta$ with respect to the cavity normal; Fig.
1(d). The oblique-incidence resonant condition is determined by the
longitudinal component of the cavity wave vector:

$\varphi(\lambda,\theta)=4\pi
n\frac{d}{\lambda}\cos{\theta^{\prime}}+\alpha_{1}(\lambda,\theta^{\prime})+\alpha_{2}(\lambda,\theta^{\prime})=2m\pi,$
(2)

where $\alpha_{1}(\lambda,\theta^{\prime})$ and
$\alpha_{2}(\lambda,\theta^{\prime})$ are the reflection phases from M1 and
M2, respectively, at a free-space wavelength $\lambda$ and an internal angle
of incidence $\theta^{\prime}$, which is related to the external angle of
incidence $\theta$ through Snell’s law:
$\sin{\theta}\\!=\\!n\sin{\theta^{\prime}}$ (assuming incidence from free
space). As a result, all resonances blue-shift: if $\lambda$ corresponds to a
particular resonance at normal incidence, then the blue-shifted resonant
wavelength $\lambda(\theta)$ at oblique incidence is

$\lambda(\theta)=\frac{n}{\sqrt{n^{2}-\sin^{2}{\theta}}}\lambda,$ (3)

assuming that $\alpha_{1}(\lambda,\theta^{\prime})$ and
$\alpha_{2}(\lambda,\theta^{\prime})$ are independent of $\theta^{\prime}$ in
the spectral range of interest, which is typically the case for dielectric
multilayer Bragg mirrors within their bandgap.

Omni-resonance relies on arranging for each wavelength $\lambda$ to impinge on
the cavity at the appropriate angle $\theta(\lambda)$, such that it satisfies
the omni-resonance condition:

$\varphi(\lambda)\\!=\\!4\pi
n\frac{d}{\lambda}\cos{(\theta^{\prime}(\lambda))}+\alpha_{1}(\lambda,\theta^{\prime}(\lambda))+\alpha_{2}(\lambda,\theta^{\prime}(\lambda))\\!=\\!2m\pi.$
(4)

If this condition is satisfied over a certain bandwidth, it is guaranteed that
the incident spectrum will resonate continuously with the cavity.
Additionally, if the omni-resonance condition is met in conjunction with
$R_{2}\\!=\\!1$ and the proper design of $R_{1}(\lambda)$, then the entire
spectrum is absorbed and broadband omni-resonant CPA is achieved; Fig. 1(e).
This overall strategy is materials-agnostic, and is independent of the
physical nature of the absorbing layer. Furthermore, the design can be
adjusted in principle for any desired central wavelength or bandwidth. The
central challenge is to construct an optical pre-conditioning system that
introduces the particular wavelength-dependent angle of incidence
$\theta(\lambda)$ that satisfies the omni-resonance condition, which we
present below.

##  III Impact of graphene location inside the cavity

Our previous work on CPA using the configuration shown in Fig. 1(c) assumed
‘distributed’ absorption; that is, the single-pass absorption $\mathcal{A}$
results from a layer in which absorption extends uniformly over its thickness
$L$ [5, 18, 50, 19]. This assumption no longer holds for ‘localized’
absorption (such as a 2D material like graphene or a metasurface). It is
expected that the location of a deeply subwavelength-thick absorber with
respect to the peaks and minima of the cavity standing-wave field structure
will impact the net absorption. Intuitively, light will not interact with the
absorber when placed at a standing-wave null, whereas placing it at a peak
will likely maximize absorption $\mathcal{A}_{\mathrm{tot}}$. Indeed, previous
work on CPA in ultrathin layers has indicated that maximum absorption is
achieved only when the layer is placed at a standing-wave peak [12, 11].

In this Section we evaluate the impact of the position of an ultrathin
absorbing layer within the cavity on $\mathcal{A}_{\mathrm{tot}}$ and we
demonstrate that the cavity shown schematically in Fig. 2(a) modifies previous
predictions in a fundamental way. The incident and reflected field amplitudes
on the left at normal incidence are $A_{\mathrm{L}}$ and $B_{\mathrm{L}}$,
respectively, the corresponding quantities on the right are
$A_{\mathrm{R}}\\!=\\!0$ (the field is incident from the left only) and
$B_{\mathrm{R}}$, which are related through the cavity transfer matrix
$\mathbf{M}$,

$\begin{pmatrix}A_{\mathrm{L}}\\\
B_{\mathrm{L}}\end{pmatrix}=\mathbf{M}\begin{pmatrix}A_{\mathrm{R}}\\\
B_{\mathrm{R}}\end{pmatrix}.$ (5)

The cavity length is $L$, and the absorber is placed at a distance $d$ from
mirror M1. Therefore, from left to right, the cavity comprises mirror M1,
propagation a distance $d$, the absorbing layer, propagation a distance $L-d$,
and mirror M2, cascaded according to :

$\mathbf{M}=\mathbf{M}_{1}\cdot\mathbf{P}(\varphi_{1})\cdot\mathbf{M}_{\mathrm{G}}\cdot\mathbf{P}(\varphi_{2})\cdot\mathbf{M}_{2};$
(6)

here $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$ represent mirrors M1 and M2,
respectively, which have the form given in Eq. 12; $\mathbf{P}(\varphi_{1})$
and $\mathbf{P}(\varphi_{2})$ represent propagation for distances $d$ and
$L-d$, respectively, where $\varphi_{1}\\!=\\!nkd$,
$\varphi_{2}\\!=\\!nk(L-d)$, $k=\tfrac{2\pi}{\lambda}$, $\lambda$ is the free-
space wavelength, and $\mathbf{P}(\varphi)$ has the form given in Eq. 13; and
$\mathbf{M}_{\mathrm{G}}$ represents the ultrathin absorbing layer, taken to
be monolayer graphene for concreteness [56]:

$\mathbf{M}_{\mathrm{G}}\\!=\\!\begin{pmatrix}1\\!+\\!\delta&\delta\\\
-\delta&1\\!-\\!\delta\end{pmatrix},$ (7)

where $\delta\\!=\\!\tfrac{\sigma\mu_{\mathrm{o}}c}{2n}$, $\mu_{\mathrm{o}}$
is the free-space magnetic permeability, $c$ is the light speed in vacuum, $n$
is the refractive index of the surrounding medium, and the optical
conductivity of monolayer graphene is $\sigma\\!=\\!6.1\\!\times\\!10^{-5}$
$\Omega^{-1}$ [57]. Accordingly, the single-pass absorption is
$\mathcal{A}_{\mathrm{G}}\\!=\\!\frac{2\delta}{1+\delta^{2}}\\!\approx\\!2\delta\\!=\\!1.6\%$
($\delta\\!\ll\\!1$).

The cavity reflection amplitude
$r_{\mathrm{L}}\\!=\\!\tfrac{B_{\mathrm{L}}}{A_{\mathrm{L}}}$ is given by:

$r_{\mathrm{L}}\\!=\\!-e^{i(2\beta_{1}-\alpha_{1})}\frac{r_{1}\delta_{+}e^{-i\psi_{\mathrm{o}}}-\delta_{-}e^{i\psi_{\mathrm{o}}}+\delta
e^{-i\Delta\psi}+r_{1}\delta
e^{i\Delta\psi}}{\delta_{+}e^{-i\psi_{\mathrm{o}}}-r_{1}\delta_{-}e^{i\psi_{\mathrm{o}}}+r_{1}\delta
e^{-i\Delta\psi}+\delta e^{i\Delta\psi}},$ (8)

where $\delta_{+}\\!=\\!1\\!+\\!\delta$, $\delta_{-}\\!=\\!1\\!-\\!\delta$,
$\varphi_{\mathrm{o}}\\!=\\!nkL\\!=\\!\varphi_{1}\\!+\\!\varphi_{2}$,
$2\psi_{\mathrm{o}}\\!=\\!2\varphi_{\mathrm{o}}\\!+\\!\alpha_{1}\\!+\alpha_{2}$
is the cavity round-trip phase,
$\Delta\psi\\!=\\!\Delta\varphi+\tfrac{\alpha_{2}-\alpha_{1}}{2}$,
$\Delta\varphi\\!=\\!\varphi_{2}\\!-\\!\varphi_{1}$, $\beta_{1}$ and
$\alpha_{1}$ are the transmission and reflection phases for M1, respectively,
$\alpha_{2}$ is the reflection phase for M2, and $r_{1}$ is the reflection
amplitude from M1 ($R_{1}\\!=\\!|r_{1}|^{2}$) for incidence from within the
cavity (from a medium of refractive index $n$). The net cavity absorption is
$\mathcal{A}_{\mathrm{tot}}\\!=\\!1-|r_{\mathrm{L}}|^{2}$, and we maximize the
resonant absorption by finding $r_{1}$ that minimizes $|r_{\mathrm{L}}|$. On
resonance $\psi_{\mathrm{o}}\\!=\\!m\pi$, by setting
$\Re\\{r_{\mathrm{L}}\\}\\!=\\!0$ ($\Im\\{r_{\mathrm{L}}\\}\\!\ll\\!1$) we
find the optimal $R_{1}$ to realize $\mathcal{A}_{\mathrm{tot}}\\!\approx\\!1$
is:

$\displaystyle
R_{1}\approx\left(1-2\mathcal{A}_{\mathrm{G}}\sin^{2}\Big{(}\frac{m\pi
d}{L}-\frac{\alpha_{1}+\alpha_{2}}{4}\Big{)}\right)^{2}.$ (9)

Figure 2: Impact of the position of an ultrathin absorbing layer on the net
cavity absorption $\mathcal{A}_{\mathrm{tot}}$. (a) Schematic depiction of a
planar FP resonator containing a localized absorber having single-pass
absorption $\mathcal{A}_{\mathrm{G}}$. (b-d) Three scenarios in a FP cavity
having $R_{1}\\!=\\!(1-\mathcal{A}_{\mathrm{G}})^{2}\\!\approx\\!96.8\%$,
which corresponds to the CPA condition for a distributed absorber with
$\mathcal{A}\\!=\\!\mathcal{A}_{\mathrm{tot}}$. The thin absorber, which is
represented by a thick black line, is considered at three distinct locations
within the cavity standing-wave. (b) When the absorber is placed at the
standing-wave null $z_{1}\\!=\\!0$, $\mathcal{A}_{\mathrm{tot}}\\!=\\!0$. (c)
Placing the absorber midway between a null and peak
$z_{2}\\!=\\!\lambda^{\prime}/8$ results in perfect absorption
$\mathcal{A}_{\mathrm{tot}}\\!=\\!1$; $\lambda^{\prime}$ is the wavelength
inside the cavity. (d) When the absorber is placed at a standing-wave peak
$z_{3}\\!=\\!\lambda^{\prime}/4$, the net absorption is sub-optimal
$\mathcal{A}_{\mathrm{tot}}\\!\approx\\!0.9$. (e) Calculated
$\mathcal{A}_{\mathrm{tot}}$ as a function of the absorbing layer position $z$
and the front-mirror reflectivity $R_{1}$. The three red dots correspond to
the positions $z_{1}$, $z_{2}$, and $z_{3}$ in (b), (c) and (d), respectively.
The solid curve is the CPA contour $\mathcal{A}_{\mathrm{tot}}\\!=\\!1$. The
dashed curves are the contours for
$\mathcal{A}_{\mathrm{tot}}\\!\approx\\!0.9$, and the dotted curves are for
$\mathcal{A}_{\mathrm{tot}}\\!\approx\\!0.5$. Figure 3: Calculated cavity net
absorption $\mathcal{A}_{\mathrm{tot}}$ as of function of the graphene
position $z$ and the graphene single-pass absorption $\mathcal{A}$, for a
given mirror reflectivity, $R_{1}\\!=\\!0.95$. Compared to Fig. 2(e), this is
more applicable plot since the mirror reflectivities are not changeable after
the device fabrication, while the graphene absorption can be still varied in a
range via electrical injection. Depending on the graphene position, the graph
shows the cavity absorption can be quite modulated via tuning the graphene
single-pass absorption. The white and the yellow dashed curves determine the
areas within which the cavity absorption is higher than 50% and 70%,
respectively. Figure 4: (a) Measured single-pass-absorption spectrum
$\mathcal{A}(\lambda)$ for a graphene monolayer. The inset is a photograph of
a glass substrate on which a graphene layer is deposited. The graphene layer
is delimited by dashed white square. (b) The calculated reflectivities for
incidence from free space onto the back and front mirror M1 and M2,
respectively, via the transfer matrix method in the 500-600 nm spectral range.
Here M1 and M2 are constructed of 5 and 10 dielectric bilayers, respectively.
(c) Measured spectral reflectivities for the fabricated mirrors M1 and M2.

We can now establish a connection between the location of the absorber and the
total absorption by noting that the intensity of the cavity standing-wave in
absence of the absorber is $I(d)\\!\propto\\!\sin^{2}(\Delta\psi/2)$. The
phase $\Delta\psi$ therefore determines the location of the thin absorber with
respect to the cavity standing-wave: $\Delta\psi\\!=\\!0$ corresponds to a
null and $\Delta\psi\\!=\\!m\pi$ to a peak. When the absorber is placed at a
null, Eq. 9 indicates that the optimal reflectivity is
$R_{1}\\!\rightarrow\\!1$, and absorption is eliminated, as one expects; Fig.
2(b). Surprisingly, however, the absorber need not be placed at a peak to
realize CPA. Indeed, if the absorber is placed anywhere except for the null
$\Delta\psi\\!\neq\\!0$, then $\mathcal{A}_{\mathrm{tot}}\\!\approx\\!1$ can
always be maximized by selecting the reflectivity of M1 according to Eq. 9.
For example, if the absorber is placed midway between a null and a peak
$\Delta\psi\\!=\\!\tfrac{\pi}{2}$, then selecting
$R_{1}\\!=\\!(1-\mathcal{A}_{\mathrm{G}})^{2}$ yields complete absorption;
Fig. 2(c). Note that this is the optimal reflectivity for M1 when employing a
distributed absorber having single-pass absorption $\mathcal{A}_{\mathrm{G}}$.
If one makes use of this same mirror, but now places the absorber at a peak in
the standing wave, as shown in Fig. 2(d), then maximum absorption is not
attained; indeed, here $\mathcal{A}_{\mathrm{tot}}\\!=\\!90\%$. Placing the
absorber at this position requires instead a reflectivity
$R_{1}\\!=\\!(1-2\mathcal{A}_{\mathrm{G}})^{2}$ to achieve maximum absorption.

The presence of a cavity therefore modifies the dependence of absorption on
the location of the absorber in a fundamental way. Specifically, we find –
counter-intuitively – that complete absorption can be reached for any position
of the thin absorber (as long as it is not at a null) if the appropriate
reflectivity for M1 is utilized. This is emphasized in Fig. 2(e) where we plot
$\mathcal{A}_{\mathrm{tot}}$ as a function of the position $z$ of the absorber
over an optical wavelength ($\Delta\psi\\!=\\!2\pi z/\lambda$) and $R_{1}$.
From Fig. 2(e) it is clear that substantial tolerance is afforded by the
cavity design ($R_{1}$ and the absorber position) to approach
$\mathcal{A}_{\mathrm{tot}}\\!\approx\\!100\%$ for an ultrathin absorber. The
region between the two dashed contours in the parameter space in Fig. 2(e)
corresponds to $\mathcal{A}_{\mathrm{tot}}\\!<\\!90\%$ (an enhancement of
$\approx\\!55\times$ over $\mathcal{A}_{\mathrm{G}}$), whereas between the
dotted contours $\mathcal{A}_{\mathrm{tot}}\\!<\\!50\%$ (an enhancement of
$\approx\\!30\times$). In other words, large resonant absorption enhancement
can be achieved even when the absorber position deviates from the standing-
wave peak, or $R_{1}$ does not match the targeted reflectivity. For example,
taking $R_{1}\\!=\\!99.5\%$ results in $\mathcal{A}_{\mathrm{tot}}\\!>\\!50\%$
for $\sim\\!90\%$ of the positions of the graphene monolayer within the cavity
standing-wave. Indeed, Fig. 2(c,d) in which $R_{1}\\!=\\!96.8\%$ indicate that
$\mathcal{A}_{\mathrm{tot}}\\!>\\!90$ is reached for $\sim\\!50\%$ of the
locations.

A more realistic scenario involves a FP cavity having a fixed $R_{1}$ and a
graphene monolayer at a fixed position. In this case, by tuning $\mathcal{A}$
away from $\mathcal{A}_{\mathrm{G}}$ through chemical doping (by applying an
electrical potential) one may maximize $\mathcal{A}_{\mathrm{tot}}$. Indeed,
graphene and other 2D materials offer tremendous flexibility in this regard
[58]. In Fig. 3 we plot $\mathcal{A}_{\mathrm{tot}}$ for $R_{1}\\!=\\!0.95$
while varying the position of the graphene monolayer and its single-pass
absorption $\mathcal{A}$ from $\mathcal{A}_{\mathrm{G}}$ to
$2\mathcal{A}_{\mathrm{G}}$, which is accessible for graphene. It is clear
that tuning $\mathcal{A}$ in this range allows for increasing the resonant
absorption if the absorber is not placed at the targeted position within the
cavity. This all bodes well for constructing planar devices that resonantly
enhance the absorption in 2D materials without exorbitant fabrication
tolerances.

##  IV Design and characterization

We now proceed to our experiments validating resonantly enhanced absorption in
a graphene monolayer. The FP cavity we utilize comprises two multilayer
dielectric Bragg mirrors sandwiching a silica spacer in which a graphene
monolayer is embedded. In this Section we describe the components of the CPA
cavity and the performance of the cavity as a whole on resonance, and in the
next Section we will combine it with a light pre-conditioning system that
establishes omni-resonance over a broad continuous bandwidth.

Because the graphene’s intrinsic single-pass spectral absorption
$\mathcal{A}(\lambda)\\!\approx\\!\mathcal{A}_{\mathrm{G}}\\!\approx\\!1.6\%$
in silica is flat over a broad spectrum as shown in Fig. 4(a) [28], the mirror
reflectivity required to realize CPA at any wavelength within this range is
also spectrally flat
$R_{1}(\lambda)\\!\approx\\!R_{1}\\!=\\!(1-\mathcal{A}_{\mathrm{G}})^{2}\\!\approx\\!96.8\%$,
which can be realized with periodic dielectric Bragg mirrors. The light source
used in all our measurements is a halogen lamp (Thorlabs, QTH10/M), and the
spectrum is recorded with an optical spectrum analyzer (OSA; Advantest Q8383).
Light from the halogen lamp is spatially filtered by coupling it into a
1-m-long multimode, 50-$\mu$m-diameter optical fiber terminated with a fiber
collimator.

The targeted reflectivities for M1 and M2 are shown in Fig. 4(b):
$R_{2}\\!=\\!1$ for M2, and $R_{1}\\!\approx\\!96.8\%$ for M1. This
reflectivity is selected assuming the graphene monolayer is placed midway
between the peak and null of the cavity standing-wave field distrbution. Here
$R_{1}$ is the reflectivity of M1 for incidence from the silica spacer within
the cavity. In Fig. 4(b) we plot the corresponding reflectivity for incidence
from air to facilitate comparison with measurements. Both M1 and M2 are
constructed of bilayers of 94-nm-thick SiO2 and 61-nm-thick Ti2O3 that form a
broad reflection bandwidth $\sim\\!100$ nm centered at $\sim\\!550$ nm. The
mirrors M1 and M2 are formed of 5 and 10 bilayers, respectively, and their
measured normal-incidence reflectivities from free space are plotted in Fig.
4(c), displaying excellent agreement with their theoretical counterparts in
Fig. 4(b).

The overall FP cavity structure for CPA is depicted schematically in Fig.
5(a). Its construction starts with depositing the back mirror M2 onto a 1-mm-
thick, 25-mm-diameter glass slide, followed by a 4-$\mu$m-thick silica spacer.
The graphene monolayer (of nominal thickness 0.35 nm [59]) is deposited on the
spacer, and is then secured in place with a 100-nm-thick silica layer, before
depositing the front mirror M1. All deposition steps for the dielectric layers
are carried out via e-beam evaporation (Appendix A). Using the transfer-matrix
method, we calculate the net spectral absorption
$\mathcal{A}_{\mathrm{tot}}\\!=\\!1-|r_{\mathrm{L}}|^{2}-|t_{\mathrm{L}}|^{2}$
(throughout we have $t_{\mathrm{L}}\\!\approx\\!0$) in the cavity as a
function of the angle of incidence $\theta$. We plot
$\mathcal{A}_{\mathrm{tot}}(\lambda,\theta)$ in Fig. 5(b) where it is clear
that absorption enhancement is achieved at the cavity resonances. However,
$100\%$ absorption is not reached here because of the finite steps in
reflectivity achieved by adding each bilayer to M1. Indeed, 4 bilayers yield
$R_{1}\\!=\\!89\%$ for incidence from air and 5 bilayers yield
$R_{1}\\!=\\!96\%$, whereas the target reflectivity is $R_{1}\\!=\\!94\%$ (for
incidence from free space). The dashed curve in Fig. 5(b) is the theoretical
limit on $\mathcal{A}_{\mathrm{tot}}$ given our particular cavity design. The
corresponding measured absorption spectra are plotted in Fig. 5(c) for a
broadband field of diameter $\approx\\!10$ mm incident at angles in the range
$0^{\circ}$ to $60^{\circ}$ with respect to the cavity normal. The absorption
resonances do not reach the theoretical limit due to the imperfections in the
deposited structure (departures from the targeted layer thicknesses and from
an exact planar condition), and due to uncertainty in the exact location of
the graphene monolayer within the silica spacer as discussed above. The
absorption peaks extending over a bandwidth of 100 nm reach a maximum of
$\approx\\!60\%$.

Figure 5: Characterization of the CPA cavity containing the graphene
monolayer. The is practically no transmitted field through the back-reflector
M2. (a) Schematic of the CPA. (b) Calculated net cavity spectral absorption
for normally incident light is shown on the left, and as a function of the
angle of incidence $\mathcal{A}_{\mathrm{tot}}(\lambda,\theta)$ on the right.
(c) Measured absorption spectra corresponding to (b). The dashed curves in the
left panels in (b) and (c) represent the theoretical limit on absorption given
the experimentally realized value of the reflectivity $R_{1}$ for the front
mirror M1. Figure 6: Characterization of the CPA cavity in conjunction with
omni-resonance. (a) Schematic of the pre-conditioning optical system to
realize omni-resonance in the CPA cavity characterized in Fig. 5. (b)
Calculation of the absorption spectrum as a function of the cavity tilt angle
$\psi$. (c) Measured absorption spectra corresponding to theoretical plot in
(b). The measurements were carried out in increments of $1^{\circ}$ in the
angular range of $40^{\circ}\\!<\\!\psi\\!<\\!70^{\circ}$, corresponding to
the range highlighted with a white dashed rectangle in (b). The two cavity
tilt angles $\psi\\!=\\!50^{\circ}$ and $\psi\\!=\\!57^{\circ}$ correspond to
two distinct omni-resonances. (d) The omni-resonant spectral absorption at a
cavity tilt angle $\psi\\!=\\!57^{\circ}$. For comparison, we also plot the
single-pass absorption of the graphene monolayer, the theoretical limit on
cavity absorption, and the resonant CPA in absence of omni-resonance.

##  V Realization of broadband coherently enhanced omni-resonant absorption

It is clear from Fig. 5(b,c) that the spectral absorption in the cavity at any
incidence angle is resonantly enhanced only at discrete resonances rather than
across a spectrally continuous band. The latter is realized by exploiting the
concept of omni-resonance as illustrated in Fig. 1(e). This requires
introducing angular dispersion into the incident field to associate each
wavelength $\lambda$ with a specific angle of incidence $\theta(\lambda)$ via
an optical system that pre-conditions the incident field. The experimental
arrangement for demonstrating omni-resonance is shown in Fig. 6(a). Broadband
light from the halogen lamp is directed through a 1-mm-wide slit onto a
reflective diffraction grating (Thorlabs GR25-1850; 1800 lines/mm, area
$25\\!\times\\!25$ mm2), which is oriented at $50^{\circ}$ with respect to the
incident light. The spectrally resolved wave front in the first diffraction
order is focused onto the cavity by a lens L1, which is an aspheric condenser
(Thorlabs, ACL50832U) placed at a distance 12 cm from the grating. The cavity
is mounted on a rotational stage near the focal plane of L1, and is tilted an
angle $\psi$ with respect to an optical axis defined by a central wavelength
$\lambda_{\mathrm{c}}\\!=\\!550$ nm. This combination of grating and lens L1
provides the requisite linear angular dispersion over the wavelength range of
interest [49]; see Appendix B for further details. Light reflected from the
cavity is collimated by a 25-mm-focal-length lens L2 and directed to a
monochrome CCD sensor (Thorlabs DCC1545M). After calibration with a tunable
narrowband spectral filter ($\approx\\!3$-nm bandwidth), the spatial positions
on the CCD correspond to wavelengths, and the reflected spectrum is captured
as an image on the sensor for each $\psi$.

The computed absorption spectrum as we vary the cavity tilt angle $\psi$ is
plotted in Fig. 6(b). It is clear that at certain valuues of $\psi$ the
absorption is spectrally flat; i.e., broadband omni-resonance has been
achieved, and the attendant CPA is realized over a continuous broad spectrum.
Examples of omni-resonant CPA occur at $\psi\\!\approx\\!50^{\circ}$ and
$60^{\circ}$ in Fig. 6(b). Each of the angular settings for $\psi$ identifies
an ‘achromatic resonance’. This is a single resonance of the bare cavity that
is now spread over an extended, continuous spectral range rather than the
narrow linewidth at a fixed wavelength as shown in Fig. 5(b). The re-
orientation of the resonance trajectory in wavelength-angle $(\lambda.\psi)$
space is a consequence of the angular dispersion introduced into the incident
field. Moreover, note that each achromatic resonance extends in bandwidth
across multiple free spectral ranges of the bare cavity [49, 51, 50].

The measurement results for the spectral absorption while varying $\psi$ are
plotted in Fig. 6(c). Because of the physical constraints in the measurement
apparatus resulting from the size of the cavity and the focal lengths of the
lenses L1 and L2, measurements were collected for cavity tilt angles over the
range $45^{\circ}\\!<\\!\psi\\!<\\!60^{\circ}$. The measurements indicate that
a prominent wideband absorption occurs at the tilt angle
$\psi\\!=\\!57^{\circ}$ associated with an achromatic resonance. The
absorption spectrum at this tilt angle is plotted in Fig. 6(d), where it is
compared to the corresponding spectra of the graphene monolayer and that of
the bare cavity. We observe that the resonant enhancement in absorption is no
longer confined to the resonant linewidths separated by the FSR. Rather, a
continuous broadband spectrum of resonant enhancement is now realized as a
result of the juxtaposition of CPA with omni-resonance. Two points must be
noted here. First, the observed continuous absorption spectrum is not a
consequence of ‘filling in’ the intra-resonance gaps. Instead, the absorption
spectrum is associated with a single achromatic resonance. Second, this
achromatic resonance does not correspond to any of the bare cavity resonances
shown in Fig. 5(b,c). Instead, it corresponds to a higher-order bare-cavity
resonance that becomes spectrally flat in our wavelength range of interest;
see Fig. 6(b).

##  VI Conclusion

We have combined in the same device the physically independent phenomena of
CPA (resonantly enhanced complete absorption in a planar cavity containing a
partially absorbing layer) and omni-resonance (the continuous, broad resonance
spectrum realized when light incident on the cavity is pre-conditioned by
introducing the appropriate angular dispersion). By embedding a graphene
monolayer of a single-pass absorption $\mathcal{A}_{\mathrm{G}}$ in a planar
cavity provided with a back-reflector and an appropriately designed front
mirror, light on resonance can be fully absorbed, despite the low intrinsic
absorption of graphene. By adding a light pre-conditioning system, the
resonantly enhanced absorption is now realized over a broad continuous
spectrum and is associated with a single achromatic resonance. Our
measurements reveal at one such achromatic resonance enhanced absorption to
$\sim\\!60\%$ in monolayer graphene observed over a continuous bandwidth of
$\approx\\!70$ nm.

We have also elucidated the impact of the position of an ultrathin absorber
such as monolayer graphene on the net cavity absorption. We have shown that it
is no longer necessary to place the absorbing layer at a peak in the standing-
wave pattern within the cavity as is commonly understood to be the case in
free space. Instead, maximum absorption is guaranteed to reach its maximum
value if the absorber is placed anywhere except at a field null. This requires
only adjusting the reflectivity of the cavity front mirror. Furthermore, for
fixed front-mirror reflectivity, adjusting the intrinsic absorption in the
ultrathin layer (e.g., via an electric potential for graphene [58]) can
maximize the net cavity absorption.

The notion of omniresonance is a subclass of the more general framework of
classical entanglement, that refers to classical optical fields in which non-
separability is created by introducing deterministic correlations between
physically independent degrees of freedom [60, 61, 62, 63, 64, 65]. Here,
classical entanglement is introduced between two continuous degrees of
freedom: the wavelengths and their associated angles of incidence, whereby
each wavelength is assigned to a single prescribed angle [66], resulting
thereby in omni-resonance. In addition to omni-resonance, a similar conception
of classical entanglement has been instrumental in developing a new class of
propagation-invariant pulsed beams of wave packets that we have coined the
name ‘space-time’ (ST) wave packets [67, 68, 69, 70]. Moreover, we have
recently established the connection between omni-resonance and ST wave packets
and have demonstrated that a family of such pulsed beams are omni-resonant and
can thus traverse planar FP cavities without change even when the pulse
bandwidth exceeds the cavity resonant linewidth [52, 53].

The effect demonstrated here is an important step towards the utilization of
CPA in energy harvesting technologies despite the inherently resonant nature
of CPA. It is an open question whether the free-space arrangement we made use
of here (a diffraction grating and a lens) can be replaced by a single element
(such as a metasurface) that may be integrated with the multilayer cavity
structure into a single device. This may pave the way to future highly-
efficient ultrathin solar cell devices and photodetectors.

###### Acknowledgements.

This work was funded by the U.S. Air Force Office of Scientific Research
(AFOSR) under MURI award A9550-14-1-0037, and by the U.S. Office of Naval
Research (ONR) under MURI award N00014-20-1-2789.

##  Appendix A FP cavity design and construction

The Bragg mirrors used in constructing the FP resonator comprise bilayers of
$94$-nm-thick SiO2 ($n\\!=\\!1.46$ at $\lambda\\!=\\!550$ nm) and $61$-nm-
thick Ti2O3 ($n\\!=\\!2.3$ at $\lambda\\!=\\!550$ nm) deposited via e-beam
evaporation. The Ti2O3 films were formed by evaporating TiO2 source material
under O2 partial pressure. The cavity (total thickness $\approx\\!6.4$ $\mu$m)
is deposited monolithically on a $1$-mm-thick, $25$-mm-diameter glass slide.
The layered structure from the top (the incidence port on M1) is as following:
Air–(LH)5–[SiO2/Graphene/SiO2]–(LH)10–glass. The thickness of the two SiO2
layers are $\approx\\!0.1~{}\mu m$ (adjucent to M1) and $\approx\\!4~{}\mu m$
(adjucent to M2). Each bilayer (LH) consists of low-index SiO2 (L) and high-
index Ti2O3 (H) material. The result is a $100$-nm-wide spectral reflection
band with $\sim 94$% reflectivity at the center wavelength
$\lambda_{\mathrm{c}}\\!\approx\\!550$ nm at normal incidence from air ($\sim
96\%$ at normal incidence from SiO2).

##  Appendix B Omni-resonance pre-conditioning arrangement

Figure 7: Schematic depiction of the pre-conditioning optical system the
introduces the necessary angular dispersion for realizing omni-resonance. The
configuration comprises a diffraction grating and and a lens L1; $\alpha$ and
$\gamma(\lambda)$ are measured with respect to the grating normal. The optical
axis (shown in green) coincides with
$\gamma_{\mathrm{o}}\\!=\\!\gamma(\lambda_{\mathrm{c}}\\!=\\!550$ nm). Light
is then incident on the cavity, which is tilted an angle $\psi$ with respect
to the optical axis; $\varphi(\lambda)$ is measured with respect to the
optical axis, while $\theta(\lambda)$ is measured from the cavity normal:
$\theta(\lambda)\\!=\\!\varphi(\lambda)+\psi$.

A detailed configuration for the pre-conditioning system that modulates the
incident optical field before impinging on the FP cavity is depicted in Fig.
7; see also [49, 51, 50]. We assume an ideal grating, with linearly
polarization collimated white light from the halogen lamp directed at an
incidence angle $\alpha\\!=\\!50^{\circ}$ with respect to the grating normal.
The angularly dispersed light from the grating is directed to the FP cavity
via a lens L1. We take $\lambda\\!=\\!550$ nm as the central wavelength that
defines the optical axis. The tilt angle $\psi$ of the cavity is measured with
respect to this optical axis. We define the angle $\gamma(\lambda)$, which is
the diffraction angle of wavelength $\lambda$ with respect to the grating
normal. The central wavelength $\lambda_{\mathrm{c}}\\!=\\!550$ nm is
diffracted at $\gamma(\lambda_{\mathrm{c}}\\!=\\!550$
nm)$=\gamma_{\mathrm{o}}$ and coincides with the optical axis, and any other
wavelength $\lambda$ makes an angle
$\gamma(\lambda)\\!-\\!\gamma_{\mathrm{o}}$ with respect to the optical axis.
This angle is boosted via the lens L1 by a ratio $d_{1}/d_{2}$ , where $d_{1}$
and $d_{2}$ are the distances from the grating to L1 and from L1 to the
cavity, respectively. The incidence angle with respect to the optical axis
after L1 is

$\varphi(\lambda)=\tan^{-1}\left\\{\frac{d_{1}}{d_{2}}\tan[\gamma(\lambda)\\!-\\!\gamma_{\mathrm{o}}]\right\\},$
(10)

with
$\varphi_{\mathrm{o}}\\!=\\!\varphi(\lambda_{\mathrm{c}}\\!=\\!550\mathrm{nm})\\!=\\!0$.
The distances $d_{1}$ and $d_{2}$ are selected such that the illuminated spot
on the grating is imaged onto the cavity. If the focal length of L1 is $f$,
then $d_{2}\\!=\\!\frac{fd_{1}}{f-d_{1}}$. When the cavity is oriented such
that it is perpendicular to the optical axis, the angle of incidence of each
wavelength is $\varphi(\lambda)$. Upon tilting the cavity by $\psi$, the angle
of incidence with respect to the normal to the cavity is
$\theta(\lambda)\\!=\\!\varphi(\lambda)+\psi$. By adjusting $\psi$, one can
provide the optimal angular dispersion required to achieve the omni-resonance
condition.

##  Appendix C transfer matrix analysis for the CPA cavity

The scattering matrices associated with the front and back mirrors
($\mathbf{S}_{1}$ and $\mathbf{S}_{2}$, respectively) take the form:

$\mathbf{S}_{j}\\!=\\!\begin{pmatrix}t_{j}e^{i\beta_{j}}&r_{j}e^{i\alpha_{j}}\\\
-r_{j}e^{i(2\beta_{j}\\!-\\!\alpha_{j})}&t_{j}e^{i\beta_{j}}\end{pmatrix},$
(11)

where $t_{j}$ and $r_{j}$ ($j\\!=\\!1,2$) are the transmission and reflection
amplitudes of the mirrors, respectively, and $\beta_{j}$ and $\alpha_{j}$ are
the associated phases. The scattering matrix relates the right and left
outgoing waves ($A_{\mathrm{R}}$ and $B_{\mathrm{L}}$, respectively) to the
left and right incoming waves ($A_{\mathrm{L}}$ and $B_{\mathrm{R}}$,
respectively). The transfer matrix $\mathbf{M}$ corresponding to $\mathbf{S}$
is:

$\mathbf{M}_{j}=\frac{1}{t_{j}}\begin{pmatrix}e^{-i\beta_{j}}&-r_{j}e^{-i(\beta_{j}\\!-\\!\alpha_{j})}\\\
-r_{j}e^{i(\beta_{j}\\!-\\!\alpha_{j})}&e^{i\beta_{j}}\end{pmatrix}.$ (12)

Propagation for a distance $d$ in a medium of refractive index $n$ is
represented by a transfer matrix:

$\mathbf{M}(\varphi)=\begin{pmatrix}e^{-i\varphi}&0\\\
0&e^{i\varphi}\end{pmatrix}$ (13)

where $\varphi\\!=\\!nkd$ is the optical phase. This transfer matrix applies
to propagation for a distance $d$ from M1 to the graphene monolayer, and then
a distance $L-d$ to M2.

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# On Robustness and Generalization of ML-Based Congestion Predictors to Valid
and Imperceptible Perturbations

Chester Holtz<EMAIL_ADDRESS>University of California San Diego ,
Yucheng Wang<EMAIL_ADDRESS>University of California San Diego ,  Chung-Kuan
Cheng<EMAIL_ADDRESS>University of California San Diego and  Bill Lin
<EMAIL_ADDRESS>University of California San Diego

(2022)

###### Abstract.

There is substantial interest in the use of machine learning (ML)-based
techniques throughout the electronic computer-aided design (CAD) flow,
particularly methods based on deep learning. However, while deep learning
methods have achieved state-of-the-art performance in several applications
(e.g. image classification), recent work has demonstrated that neural networks
are generally vulnerable to small, carefully chosen perturbations of their
input (e.g. a single pixel change in an image). In this work, we investigate
robustness in the context of ML-based EDA tools—particularly for congestion
prediction. As far as we are aware, we are the first to explore this concept
in the context of ML-based EDA.

We first describe a novel notion of imperceptibility designed specifically for
VLSI layout problems defined on netlists and cell placements. Our definition
of imperceptibility is characterized by a guarantee that a perturbation to a
layout will not alter its global routing. We then demonstrate that state-of-
the-art CNN and GNN-based congestion models exhibit brittleness to
imperceptible perturbations. Namely, we show that when a small number of cells
(e.g. 1%—5% of cells) have their positions shifted _such that a measure of
global congestion is guaranteed to remain unaffected_ (e.g. 1% of the design
adversarially shifted by 0.001% of the layout space results in a predicted
decrease in congestion of up to 90%, while no change in congestion is implied
by the perturbation). In other words, the quality of a predictor can be made
arbitrarily poor (i.e. can be made to predict that a design is “congestion-
free”) for an arbitrary input layout. Next, we describe a simple technique to
train predictors that improves robustness to these perturbations. Our work
indicates that CAD engineers should be cautious when integrating neural
network-based mechanisms in EDA flows to ensure robust and high-quality
results.

††journalyear: 2022††copyright: rightsretained††conference: Proceedings of the
59th ACM/IEEE Design Automation Conference (DAC); July 10–14, 2022; San
Francisco, CA, USA††booktitle: Proceedings of the 59th ACM/IEEE Design
Automation Conference (DAC) (DAC ’22), July 10–14, 2022, San Francisco, CA,
USA††doi: 10.1145/3489517.3530620††isbn: 978-1-4503-9142-9/22/07††conference:
Make sure to enter the correct conference title from your rights confirmation
emai; June 03–05, 2018; Woodstock, NY

##  1\. Introduction

Electronic design automation (EDA) flows involve significant optimization and
verification challenges that continue to scale as the complexity of designs
increases. There is substantial interest in using machine learning techniques
for solving electronic computer-aided design (CAD) problems ranging from logic
synthesis to physical design and design for manufacturability (DFM) (Moore,
2018). Prior work has demonstrated that deep learning-enhanced design flows
are faster and more scalable, particularly when integrated to augment the
time-consuming stages of layout (Chhabria et al., 2021; Liu et al., 2021; Xie
et al., 2018), design space exploration (Greathouse and Loh, 2018), logic
optimization (Yu et al., 2018) and lithographic analysis (Yu et al., 2012).

However, although neural networks have been extremely successful in the
aforementioned EDA tasks, recent work (Szegedy et al., 2014; Goodfellow et
al., 2015) has demonstrated that image classifiers can be fooled by small,
carefully chosen perturbations of their input. Notably, Su et al. (2017)
demonstrated that neural network classifiers which can correctly classify
“clean” images may be vulnerable to targeted attacks, e.g., misclassify those
same images when only a single pixel is changed.

The question that we aim to explore in this work is the following:

> _To what degree are neural network-based congestion predictors vulnerable to
> small, but valid, changes in layout input?_

As the application of machine learning to production EDA tasks becomes more
widespread, understanding and addressing this question will become
increasingly critical. In this work, we provide evidence that supports an
affirmative answer:

> _Congestion predictors erroneously predict large changes to routing
> congestion with respect to changes to the layout that do not change the
> global routing._

Specifically, we investigate a novel notion of validity and design two
efficient methods for finding perturbations that demonstrate brittleness of
recently proposed congestion predictors. Furthermore, we describe one
potential approach to address the highlighted issues and demonstrate that
modifying the training procedure to promote robustness is one promising
direction to address brittleness to imperceptible changes. Although we focus
on congestion prediction, our work generalizes to arbitrary predictive models
integrated in EDA pipelines. More generally, our work motivates the need for
careful evaluation of the generalization of ML-based EDA tools—in excess of
typical performance metrics reported on a train-test split.

###  1.1. Contributions

The primary contribution of this work is to demonstrate that modern deep
learning-based EDA tools—specifically congestion predictors—are vulnerable to
valid perturbations to their inputs, i.e. may exhibit poor generalization to
perturbations of the cell layout.

Inspired by the perspective of _adversarial perturbations_ , given an input
design layout, we characterize _small_ perturbations as (1.) perturbations
that result in the adjustment of relatively few cell positions and (2.)
perturbations that maintain the global congestion structure (Sec. 3). We
describe a numerical algorithm to efficiently search the _feasible adversarial
neighborhood_ of an input to find small perturbations that maintain validity
of the input design while drastically reducing the efficacy of ML-based EDA
tool predictions (Sec. 3). We describe two variants of the proposed method:
while both rely on knowledge of the predictor weights—known as a white-box
perturbation model—one method requires knowledge of the underlying congestion
structure, while the second method does not necessitate such information. We
then demonstrate (1.) that congestion predictors are vulnerable to both models
and (2.) that _adversarial training_ significantly improves robustness with
only a modest performance trade-off. We emphasize that while we frame our
discussion in the context of robustness to perturbations, our findings
motivate a broader need to study implications of poor generalization when
integrating ML-based tools into design flows.

In summary, our contributions include the following:

  1. (1)

A novel formulation of the _feasible neighborhood_ of an input design—i.e.
given a layout, what small perturbations maintain the relevant measures of
congestion?

  2. (2)

Efficient supervised and unsupervised algorithms for computationally searching
the neighborhood of a layout.

  3. (3)

Exploration of adversarial training as a way to induce robustness and improve
generalization.

  4. (4)

Under a previously defined characterization of predictive quality for
congestion tasks (Alawieh et al., 2020), we show that the benchmark layouts we
evaluated can be perturbed such that the congestion predictions on the
perturbed layouts are poor.

##  2\. Preliminaries and Related Work

In this section, we provide an overview of ML-based EDA methods and
adversarially robust prediction.

Number of components $n\in\mathbb{R}_{+}$ Placement coordinates
$x,y\in\mathbb{R}^{n}$, $X=[x:y]\in\mathbb{R}^{n\times 2}$ Placement
perturbation $\delta_{x},\delta_{y}\in\mathbb{R}^{n}$,
$\Delta=[\delta_{x}:\delta_{y}]\in\mathbb{R}^{n\times 2}$ Neural network
parameters $\theta$ Early global routing bins $W\times H$ Feature map
$M\in\mathbb{R}^{W\times H}$ Predicted congestion map $f_{\theta}(M)$

Figure 1. Notation

Let $x,y\in\mathbb{R}^{n}$ be vectors corresponding to the coordinates of $n$
components such that the $i$-th component has coordinates encoded in the
$i$-th row of $X:=[x:y]$; $[x:y]_{i}$. We aim to find perturbations to the
layout so that the resulting layout satisfies certain constraints (i.e.
remains in the neighborhood of the original layout with respect to global
routing).

###  2.1. Global routing

The VLSI routing problem is usually solved in two steps: (1.) global routing
and (2.) detailed routing. The principle aim of the global routing step is to
generate a routing solution on a discretization of the layout space,
represented as a grid graph and provide a preferred routing region (i.e. a
route guide) for the detailed router.

A typical multi-commodity flow formulation of global routing partitions the
routing space into regular rectangles (G-Cells) and generates a grid graph
$G=(V,E)$ in which each vertex $v\in V$ represents a G-Cell and each edge
$e\in E$ represents the connection between adjacent G-Cells. The capacity of
an edge represents the maximum number of wires that can go through the edge
and the variable assignment of the edge corresponds the number of wires that
are currently using the edge, while the overflow is denoted by the number of
wires that exceeds the capacity. For each net, the routing problem is to find
a path that connects all the pins of a net in the given grid graph while
avoiding overflow on the edges. In other words, the global router maximizes a
measure of routability with respect to the detailed router while satisfying
certain constraints to manage design rule violations (DRVs), pin
accessibility, and irregular module geometries. An important concept is the
construction of the graph $G$. The graph, global routing solution, and
associated congestion metrics remain consistent as long as individual cells
remain within their G-Cells, regardless of their precise positions within each
G-Cell.

###  2.2. RUDY

Rectangular Uniform wire Density (RUDY) (Spindler and Johannes, 2007) is a
method to estimates the wirelength density by uniformly spreading the wire
volume of nets into its bounding box. It is very commonly used as a feature to
indicate the relative congestion of a region. The RUDY map of a net $e$
represents the average wirelength per unit area in the bounding box of the
net:
$\mu^{(e)}(\frac{1}{x_{\text{max}}-x_{\text{min}}}+\frac{1}{y_{\text{max}}-y_{\text{min}}})$
where the net-map $\mu^{(e)}\in\mathbb{R}^{W\times H}$ is

$\mu^{(e)}_{xy}=\begin{cases}1&x_{\text{min}}\leq x\leq x_{\text{max}}\text{
and }y_{\text{min}}\leq y\leq y_{\text{max}}\\\ 0&\text{otherwise}\end{cases}$

$x_{\min}$, and $x_{\max}$ correspond to the maximum and minimum $x$
coordinates of the associated net, and $y_{\min}$ and $y_{\max}$ correspond to
the maximum and minimum $y$-coordinate of the associated net. The RUDY score
assigned to a location $(x,y)$ is computed by aggregating RUDY scores over all
nets $e\in E$.

###  2.3. Machine learning and EDA

As previously mentioned, ML-based prediction has been explored for various
early-stage tasks in EDA flows including routability, DRC, and IR drop
prediction (Chhabria et al., 2021; Liu et al., 2021; Xie et al., 2018). For
the purposes of this work, we focus on the congestion prediction framework
proposed by Liu et al. (2021). Notably, Liu et al. (2021) use the Innovus
global router to obtain ground truth congestion hotspots, while an $W\times
H\times 3$ feature map $M$, comprised of RUDY scores (Spindler and Johannes,
2007), an associated pin-density variant PinRudy, and a macro placement map
MacroRegion is derived from the associated cell placement. A neural network is
used to learn a mapping from feature maps to congestion hotspots. The authors
of (Liu et al., 2021) also derive the gradients of the unsupervised congestion
penalty $\frac{1}{HW}||f_{\theta}(M)||_{F}^{2}$ with respect to cell
locations.

###  2.4. Machine learning and robustness

Consider the network $f:\mathbb{R}^{d}\to\mathbb{R}^{k}$, where the input is
$d$-dimensional and the output is a $k$-dimensional vector of likelihoods. For
example, the input could be a $d$-dimensional image and the $j$-th entry of
the output could correspond to the likelihood the image belongs to the $j$-th
class. The associated prediction is then
$c(x;\theta)=\arg\max_{j\in[k]}f_{j}(x;\theta)$.

Recently, machine learning practitioners have not just been concerned that the
prediction be correct, but also want robustness to random or adversarial
noise, i.e. small perturbations to the input which may change the prediction
to an incorrect class. We define the notion of $\epsilon$-robustness below:

###### Definition 2.1 ($\epsilon$-robust).

$f$ parameterized by $\theta$ is called $\epsilon$-robust with respect to norm
$p$ at $x$ if the prediction is consistent for a small ball of radius
$\epsilon$ around $x$:

(1)
$c(x+\delta;\theta)=c(x;\theta),\quad\forall\delta:||\delta||_{p}\leq\epsilon.$

The minimal $\ell_{p}$-norm perturbation $\delta_{p}^{*}$ required to switch
an sample’s label is given by the solution to the following problem:

$\delta^{*}_{p}=\arg\min||\delta||_{p}\quad\textrm{s.t.}\quad c(x;\theta)\neq
c(x+\delta;\theta).$

A significant amount of existing work relies on a first-order approximations
and Hölder’s inequality to recover $\delta^{*}$.

Projected Gradient Descent (PGD) is a first-order method that can be used to
find an approximation of $\delta^{*}_{p}$. PGD-type algorithms consist of a
descent step followed by a projection onto the feasible set $S$. Given the
current iterate $x^{(i)}$, the next iterate $x^{(i+1)}$ is computed via a
transformation $s$ applied to the gradient of the loss function $L$. For
example, if labels are available to the perturbation algorithm, $L$ could be
the original loss used to train the network $f$. Alternatively, $L$ can be
substituted for an unsupervised metric.

(2) $\displaystyle u^{(i+1)}=x^{(i)}+\eta^{(i)}\cdot s(\nabla L(x^{(i)}))$
$\displaystyle x^{(i+1)}=P_{S}(u^{(i+1)})$

$\eta^{(i)}>0$ the step size at iteration $i$,
$s:\mathbb{R}^{d}\to\mathbb{R}^{d}$ determines the descent direction as a
function of the gradient of the loss $L$ at $x^{(i)}$ and
$P_{S}:\mathbb{R}^{d}\to S$ is the projection on $S$. For example, an
$\ell_{1}$-perturbation model of radius $\epsilon$, we denote by
$B_{\infty}(x,\epsilon)=\\{z\in\mathbb{R}^{d}|\>\>||z-x||_{\infty}\leq\epsilon\\}$.
A crucial choice is that of the descent direction $s(\nabla L(x_{i}))$, a
mapping $s$ applied to a gradient. For example, the steepest descent direction
(Boyd and Vandenberghe, 2004):

(3) $\delta^{*}_{p}=\arg\max_{\delta\in\mathbb{R}^{d}}\langle
w,\delta\rangle\quad\text{s.t.}\quad||\delta||_{p}\leq\epsilon$

with $w=\nabla f(x_{i})\in\mathbb{R}^{d}$, the maximizer of a linear function
over a given $\ell_{p}$ ball constraint. Thus one gets
$\delta^{*}_{\infty}=\epsilon\text{sign}(w)$ and $\delta^{*}_{2}=\epsilon
w/||w||_{2}$ for $p=\infty$ and $p=2$ respectively, which define $s$.

##  3\. A novel notion of imperceptibility for congestion prediction

(a.) Congestion Prediction $\left[\begin{smallmatrix}x_{1}\\\
y_{1}\end{smallmatrix}\right]$$\left[\begin{smallmatrix}x_{2}\\\
y_{2}\end{smallmatrix}\right]$$\left[\begin{smallmatrix}x_{3}\\\
y_{3}\end{smallmatrix}\right]$$\left[\begin{smallmatrix}x_{4}\\\
y_{4}\end{smallmatrix}\right]$$\left[\begin{smallmatrix}x_{5}\\\
y_{5}\end{smallmatrix}\right]$$\left[\begin{smallmatrix}x_{6}\\\
y_{6}\end{smallmatrix}\right]$ netlist graph cell positions Preprocessing
(i.e. RUDY) GNN / MLP predictor Unperturbed Input Model predicted congestion
hotspots shown as lighter pixels.

(b.) Perturbed Congestion Prediction
$\left[\begin{smallmatrix}x_{1}^{\prime}\\\
y_{1}^{\prime}\end{smallmatrix}\right]$$\left[\begin{smallmatrix}x_{2}\\\
y_{2}\end{smallmatrix}\right]$$\left[\begin{smallmatrix}x_{3}\\\
y_{3}\end{smallmatrix}\right]$$\left[\begin{smallmatrix}x_{4}^{\prime}\\\
y_{4}^{\prime}\end{smallmatrix}\right]$$\left[\begin{smallmatrix}x_{5}\\\
y_{5}\end{smallmatrix}\right]$$\left[\begin{smallmatrix}x_{6}\\\
y_{6}\end{smallmatrix}\right]$ netlist graph cell positions Preprocessing
(i.e. RUDY) GNN / MLP predictor 1% Cell Perturbed Input Model predicts no
congestion hotspots.

Figure 2. General illustration of effect of imperceptible perturbations on EDA
predictions (a.): Vanilla prediction framework. The netlist-graph and cell
attributes (i.e. positions) are used to make predictions (e.g. DRC locations
or congestion hotspots) via the neural predictor. (b.): The attributes of a
subset of nodes are perturbed: $x^{\prime}=x+\delta_{x}$. The predictor is
vulnerable (i.e. can be made to predict that a design is congestion-free)—even
when both $\delta_{x}$ and the number of perturbed nodes are small.

In this section, we describe a method to compute layout perturbations that
guarantee consistency of the global routing. We present a general illustration
of our method in Fig. 2. In other words, given a trained model, we seek to
adjust a given layout (the coordinates of a subset of the cells in a design)
such that the adjusted layout remains valid, but spoils the congestion
predictions of the model.

Figure 3. Local constraints for each movable cell (highlighted in red) ensures
cells do not move G-Cells.

In particular, we define the feasible set of perturbations that we utilize in
the context of congestion predictions and an algorithm to perform a projection
onto this set. _Specifically, we discuss perturbations of coordinate-based
representations of chip layouts_. Inspired by earlier work on adversarial
attacks designed for image classifiers (Croce and Hein, 2020, 2021, 2019), we
impose a natural set of constraints on the perturbation to ensure that the
global routing does not change. Individually, these constraints prohibit cells
from moving outside of their G-Cell tiles.

Let $X:=[x:y]^{\top}\in\mathbb{R}^{n\times 2}$ be the coordinate assignments
to each cell and $\delta:=[\delta_{x}:\delta_{y}]^{\top}\in\mathbb{R}^{n\times
2}$ be a perturbation. For simplicity, and without loss of generality, let us
consider the 1st column of $X$ and $\delta$; $x,\delta_{x}$ (i.e. the
$x$-coordinate of each cell and associated perturbations such that the
perturbed cell $x$-coordinates of the layout is given by $x+\delta_{x}$).

We say that $\delta_{x}$ is an _imperceptible_ perturbation in the context of
physical design if the following conditions hold:

  1. (1)

Global imperceptibility: the maximum number of cells that can be moved is
bounded by $\epsilon_{0}$: $||\delta||_{0}\leq\epsilon_{0}$ (e.g. $1\%$ of the
total number of cells.

  2. (2)

Local imperceptibility: a cell can only be moved within it’s associated
G-Cell: $l_{i}\leq(x+\delta_{x})_{i}\leq u_{i}$

Given $x\in[0,1]^{d}$, we define the feasible set—the intersection of the
$\ell_{0}$-ball of radius $\epsilon_{0}$, the upper and lower-bound— _box_
—constraints on each entry of $x$, and the set $[0,1]^{d}$:

(4) $\displaystyle S(x)=\Bigl{\\{}$ $\displaystyle
z\in\mathbb{R}^{d}|\>\>\sum_{i=1}^{d}\mathbf{1}_{|z_{i}-x_{i}|>0}\leq\epsilon_{0},$
$\displaystyle l_{i}\leq z_{i}-x_{i}\leq u_{i},\>\>0\leq z_{i}\leq
1\Bigl{\\}}$

First, note that intersection of the two constraints $0\leq z_{i}\leq 1$ and
$l_{i}\leq z_{i}-x_{i}\leq u_{i}$ may be written:

$\max\\{0,l_{i}+x_{i}\\}\leq z_{i}\leq\min\\{1,u_{i}+x_{i}\\}$

So, the feasible set may be simplified and re-written

(5) $\displaystyle S(x)=\Bigl{\\{}$ $\displaystyle
z\in\mathbb{R}^{d}|\sum_{i=1}^{d}\mathbf{1}_{|z_{i}-x_{i}|>0}\leq\epsilon_{0},$
$\displaystyle\max\\{0,l_{i}+x_{i}\\}\leq
z_{i}\leq\min\\{1,u_{i}+x_{i}\\}\Bigl{\\}}$

The Euclidean projection onto $S(x)$; $P_{S}$ is then defined to be

(6) $\displaystyle\min_{z}||y-z||_{2}^{2}$ $\displaystyle\text{s.t.
}\sum_{i=1}^{d}\mathbf{1}_{|z_{i}-x_{i}|>0}\leq\epsilon_{0},$
$\displaystyle\max\\{0,l_{i}+x_{i}\\}\leq z_{i}\leq\min\\{1,u_{i}+x_{i}\\}$

Ignoring the combinatorial constraint, the solution is given by
$z_{i}^{*}=\max\\{l_{i}+x_{i},\min\\{y_{i},u_{i}+x_{i}\\}\\}$. We re-integrate
the $\ell_{0}$ constraint and resolve the projection by sorting according to
the _gain_ $\phi$:

$\phi_{i}=(y_{i}-x_{i})^{2}-(y_{i}-z_{i}^{*})^{2},\quad
z_{\pi_{i}}=\begin{cases}z^{*}_{\pi_{i}}\quad i=1,\ldots,k\\\
x_{\pi_{i}}\quad\text{otherwise}\end{cases}$

Thus, the final solution will have $k$ entries—those with the highest gain—and
that differ by no more than $l_{i}$ or $u_{i}$.

Importantly, the solution to this problem can be computed efficiently;
requiring a single backward pass through the trained model to compute the
perturbed layout $y$ and computation of the projection $P_{S}$ in linear
time—note that when ignoring the combinatorial constraint, Prob. 6 is a
separable problem (can be computed in parallel) over the cells. Finding the
subset of cells to shift (satisfying the $\ell_{0}$ constraint) involves a
linear-time scan over gains.

##  4\. Experiments

Table 1. Vulnerability of ML-based EDA predictors to imperceptible
perturbations. “*” denotes a perturbation that induces congestion-free
predictions. “${\dagger}$” denotes a perturbation that induces mispredictions
of hotspots.

| $\frac{1}{HW}||M||^{2}_{F}$ | NRMS | SSIM | $\frac{1}{HW}||M||^{2}_{F}$ | NRMS | SSIM
---|---|---|---|---|---|---
FCN Model | Vanilla training | Robust training
Vanilla / none | 0.010 | 0.0393 | 0.8044 | 0.010 | 0.0393 | 0.7970
Random noise | 0.012 | 0.0420 | 0.7970 | 0.012 | 0.0420 | 0.7123
1% cells perturbed | 0.0012 | 0.0791 | 0.6255 | 0.011 | 0.0561 | 0.6831
5% cells perturbed | 0.0011 | 0.0945 | 0.5152 | 0.012 | 0.0533 | 0.6181
†1% cells perturbed | 0.011 | 0.1055 | 0.4534 | 0.011 | 0.0431 | 0.7011
†5% cells perturbed | 0.011 | 0.1467 | 0.4334 | 0.011 | 0.0440 | 0.7193
GNN Model |  |  |  |  |  |
Vanilla / none | 0.011 | 0.0348 | 0.8130 | 0.011 | 0.0393 | 0.7970
Random noise | 0.013 | 0.0384 | 0.7974 | 0.011 | 0.0417 | 0.6933
1% cells perturbed | 0.0013 | 0.0743 | 0.6129 | 0.0098 | 0.0403 | 0.7643
5% cells perturbed | 0.0012 | 0.0892 | 0.5032 | 0.0097 | 0.0407 | 0.7392
†1% cells perturbed | 0.011 | 0.1744 | 0.4461 | 0.011 | 0.0411 | 0.694
†5% cells perturbed | 0.013 | 0.1835 | 0.4219 | 0.013 | 0.0417 | 0.695

Table 2. From (Chai et al., 2022). Statistics of designs and variations.

Design | Netlist Statistics | Synthesis Variations
---|---|---
#Cells | #Nets |  | Cell Area
---
($\mu m^{2}$)
#Macros |  | Frequency
---
(MHz)
RISCY-a | 44836 | 80287 | 65739 | 3/4/5 | 50/200/500
RISCY-FPU-a | 61677 | 106429 | 75985
zero-riscy-a | 35017 | 67472 | 58631
RISCY-b | 30207 | 58452 | 69779 | 13/14/15
RISCY-FPU-b | 47130 | 84676 | 80030
zero-riscy-b | 20350 | 45599 | 62648
Physical Design Variations
| Utilizations
---
(%)
| #Macro
---
Placement
| #Power Mesh
---
Setting
Filler Insertion
70/75/80/85/90 | 3 | 8 |  | After Placement
---
/After Routing

In this section we describe a set of comprehensive experiments on testcases
from the CircuitNet suite (Chai et al., 2022). Summary statistics of the
testcases are presented in Table 2. Our numerical experiments are aimed at
establishing the efficacy of our method with respect to spoiling congestion
predictions made by two trained architectures.

###  4.1. Experimental setup

We evaluate the impact of our small perturbations on the robustness of ML-
based congestion predictors. Moreover, we give illustrative examples of such
sparse and imperceivable perturbations. We utilize the CircuitNet benchmarks
(Chai et al., 2022) to validate our method. CircuitNet is an open-source
dataset consisting of more than $10$K samples from versatile runs of
commercial design tools based on open-source RISC-V designs with various
features for multiple ML for EDA applications. We summarize the designs and
generation procedure used to compose the CircuitNet dataset in Table 2.

Perturbations are generated using a momentum-based PGD algorithm with
restarts. We adapt the standard PGD iterations outlined in Eq. 2 in two ways:
(1.) we adjust the gradient-based update rule to incorporate a momentum term:

(7) $\displaystyle u^{(i+1)}=P_{S}(x^{(i)}+\eta^{(i)}\cdot s(\nabla
L(x^{(i)})))$ $\displaystyle
x^{(i)}=P_{S}(x^{(i)}+\alpha(u^{(i+1)}-x^{(i)})+(1-\alpha)(x^{(i)}-x^{(i-1)}))$

where $\alpha\in[0,1]$ regulates the influence of the previous update on the
current one and $P_{S}$ is described in Sec. 3. (2.) we introduce
“restarts”—i.e. we apply PGD to several random initializations and select the
best solution.

####  4.1.1. Algorithm parameters

We adopt the same $70/30$ train-test split described in the CircuitNet paper.
We perturb all samples in the test. In Table 1 we report several metrics for
each method including our congestion score, NRMS, and SSIM. We set
$\alpha=0.75$ and fix $\eta$ to be $2\cdot w$, where $w$ is the width of each
G-Cell. Each perturbed example is generated by running PGD for $100$
iterations with $5$ restarts, and $\eta$ is linearly decayed to $1/10\cdot w$.

###  4.2. On the robustness of congestion predictors

(a)

(b)

Figure 4. Performance metrics associated with a vanilla network evaluated on
unperturbed and perturbed layouts. (4(a)) Distribution shift in NRMS. (4(b))
Distribution shift in SSIM. Liu et al. (2021) characterize a good predictor as
achieving $NRMS<0.2$ and $SSIM>0.8$. Using our method, we are able create
valid inputs such that approximately $100\%$ of samples have $SSIM<0.8$ and
$60\%$ of samples have $NRMS>0.15$. $100\%$ of samples satisfy one of the two
conditions. Figure 5. Mean relative unsupervised error of PGD over iterations.
Shaded region denotes 1 standard deviation. Note the log-scale in blue implies
convergence in error.

In Table 1, we evaluate our method for producing valid and imperceptible
perturbations using the fully convolutional architecture proposed in Liu et
al. (2021) and a single-layer graph convolutional network (GNN Model) proposed
in (Kipf and Welling, 2017). Two variants of our algorithm are implemented.
For rows denoted by a *, the perturbation ascent direction is computed with
respect to the congestion score $\frac{1}{HW}||f_{\theta}(M)||_{F}^{2}$. For
rows denoted by a $\dagger$, the perturbation ascent direction is computed
with respect to the _supervised prediction loss_.

We demonstrate that neural network predictors fail to accurately predict
congestion of layouts produced by our method. Furthermore, when a larger
budget of cells are allowed to be shifted, performance is further degraded. In
particular, we first observe that vanilla models are relatively robust to a
random perturbation model. Namely, we uniformly at random select 1% of cells
and maximally perturb their associated positions such that they remain within
their associated G-Cell. When the layout is altered in this way, we see that
neural predictors generally maintain their performance with only a minor
degradation in NRMS and SSIM observed.

Next, we demonstrate that perturbations may be carefully chosen such that the
associated predictions correspond to congestion-free predictions, or even
adversarial predictions- i.e. the model predicts congestion in regions which
are congestion-free and predicts congestion-free regions in areas which are
highly congested (e.g. in regions with macros). We provide examples to
demonstrate these instances in Figure 7. Distributions of SSIM and NRMS scores
are provided in Figure 4. Notably, the predictor violates the conditions
necessary for good performance (NRMS $<0.2$, SSIM $>0.8$) (Chai et al., 2022;
Alawieh et al., 2020).

When a budget of $1\%$ of cells is prescribed, a degradation in SSIM of
$43.64\%$ and a degradation in NRMS of $168.45\%$ are observed for the FCNN-
based predictor. Likewise, when the budget is increased to $5\%$ of cells,
degradations in SSIM and NRMS amount to $46.12\%$ and $273.28\%$ are observed
respectively. As expected, the GNN-based model is also vulnerable to the
aforementioned issues. Interestingly, while the GNN-based method outperforms
the FCN-based method on unperturbed layouts, the GNN-based model is seemingly
_more_ vulnerable to our proposed method with degradations in NRMS and SSIM of
up to $427.3\%$ and $48.11\%$ for a budget of $5\%$ of cells.

###  4.3. Improving robustness of congestion predictors via momentum-based PGD

Figure 6. Robust congestion prediction statistic
($\frac{1}{HW}||f_{\theta}(M)||_{F}^{2}$) and percentage of cells that are
allowed to move (looseness of $\epsilon_{0}$ constraint).

(a)

(b)

(c)

(d)

Figure 7. (7(a)—7(b)): Predictions for robust and nonrobust estimators
(supervised loss). Both plots share the same gradient scale. (7(c)—7(d))
Predictions for robust and nonrobust estimators (unsupervised loss). Note the
difference in gradient scale.

A number of techniques (Madry et al., 2018; Carlini and Wagner, 2017; Wang et
al., 2020) have been proposed to mitigate the issue of robustness of deep
networks, with some of the most reliable being certified defenses (Cohen et
al., 2019) and methods based on the principle of adversarial training (Madry
et al., 2018). In this work, we forgo investigating provable defenses and
instead stick with PGD-based adversarial training. More concretely, each
iteration of SGD, a portion of each batch (i.e. $50\%$) is perturbed via our
method. Running PGD during training is expensive. One may exploit the Fast-
FGSM algorithm proposed in (Wang et al., 2020), which demonstrates that by
simply introducing random initialization points from which to compute
adversarial perturbations, one projected gradient step is as effective as
repeated steps during training while being significantly more efficient.

Using standard PGD, we train models on using the CircuitNet split and report
the _robust test statistics_ in the left column of Table 1. We see that models
trained using adversarial training are _significantly_ more robust with
respect to both unsupervised and supervised perturbations. More concretely, we
observe recovery of predictive quality primarily with respect to the metric
driving perturbations (i.e. when predictors are trained to be robust to the
unsupervised congestion metric, robustness is improved across all metrics, but
most significantly for unsupervised congestion). In Figure 6, we provide a
comparison between a vanilla network and a robust network trained via PGD. On
the $x$-axis, we plot the $\epsilon_{0}$ constraint, the percentage of cells
that are free to be adjusted. On the $y$-axis, we plot the mean congestion
metric $\frac{1}{HW}||f_{\theta}(M)||_{F}^{2}$ across the validation set. Note
that the robust network maintains good performance, even as the number of
cells increases.

The congestion value $0.01$ is from a baseline predictor—the blue bar at 0%
cells shifted. Note that the value may not be representative of prediction
quality (instead, see Figure 4). Ideally bars should be the same height across
perturbations. However, the vanilla model’s predictions on perturbed layouts
change significantly. In contrast, robust predictors generalize (orange bars
have similar height).

##  5\. Conclusion and Future Work

In this paper, we have demonstrated that CNN-based congestion detection models
are vulnerable to small perturbations. We have proposed and evaluated layout
perturbations that are _guaranteed_ to not alter a early global routing. To
address these issues, we have proposed to apply adversarial training,
demonstrating that such methods can improve robustness and generalization of
deep learning-based EDA systems. The implication of our work is that designers
should carefully evaluate deep learning-based models when employing ML-based
CAD systems in EDA pipelines. More broadly, we hope that our perturbation
methodology for evaluating vulnerabilities in congestion prediction and our
adversarial training approach to make congestion predictions more robust can
be adapted for evaluating ML-based EDA predictors.

###### Acknowledgements.

We acknowledge support from NSF CCF-2110419.

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