# Ghost Line Waves

Massimo Moccia  Fields & Waves Lab, Department of Engineering, University of
Sannio, Benevento, I-82100, Italy  Giuseppe Castaldi  Fields & Waves Lab,
Department of Engineering, University of Sannio, Benevento, I-82100, Italy
Andrea Alù  Photonics Initiative, Advanced Science Research Center, City
University of New York, New York, NY 10031, USA  Physics Program, Graduate
Center, City University of New York, New York, NY 10016, USA  Vincenzo Galdi
<EMAIL_ADDRESS>Fields & Waves Lab, Department of Engineering, University
of Sannio, Benevento, I-82100, Italy<EMAIL_ADDRESS>
###### Abstract

Time-harmonic electromagnetic plane waves in anisotropic media can exhibit
complex-valued wavevectors (with nonzero real and imaginary parts) even in the
absence of material dissipation. These peculiar modes, usually referred to as
“ghost waves”, hybridize the typical traits of conventional propagating and
evanescent waves, displaying both phase accumulation and purely reactive
exponential decay away from the direction of power flow. Their existence has
been predicted in several scenarios, and has been recently observed
experimentally in the form of surface phonon polaritons with complex-valued
out-of-plane wavevectors propagating at the interface between air and a
natural uniaxial crystal with slanted optical axis. Here, we demonstrate that
ghost waves can arise also in lower-dimensional flat-optics scenarios, which
are becoming increasingly relevant in the context of metasurfaces and in the
field of polaritonics. Specifically, we show that planar junctions between
isotropic and anisotropic metasurfaces can support “ghost line waves” that
propagate unattenuated along the line interface, exhibiting phase oscillations
combined with evanescent decay both in the plane of the metasurface (away from
the interface) and out-of-plane n the surrounding medium. Our theoretical
results, validated by finite-element numerical simulations, demonstrate a
novel form of polaritonic waves with highly confined features, which may
provide new opportunities for the control of light at the nanoscale, and may
find potential applications in a variety of scenarios, including integrated
waveguides, nonlinear optics, optical sensing and sub-diffraction imaging.

###### keywords:

Metasurfaces, surface waves, line waves, ghost waves, anisotropy,
polaritonics.

IR,NMR,UV

## Introduction

In a transparent medium, the most common type of time-harmonic electromagnetic
wave is the propagating (or “uniform”) plane wave, characterized by a real-
valued wavevector, i.e., unattenuated oscillations and active power flow along
the propagation direction 1. In the case of total reflection at an interface,
plane waves can become evanescent (“nonuniform”), characterized by wavevectors
with either real-valued or purely imaginary components, resulting in phase
propagation and exponential attenuation (with purely reactive power flow)
along two orthogonal directions 1. Recent studies have theoretically predicted
the existence of a new type of plane waves in lossless biaxial anisotropic
materials, characterized by a complex-valued wavevector (with nonzero real and
imaginary parts) 2, 3. These modes, named “ghost waves” (GWs) in analogy with
ghost orbits in quantum mechanics 4, combine some typical traits of
propagating and evanescent waves, exhibiting both phase accumulation and
purely reactive power flow along the same direction (see also Refs. 5, 6 for
general theoretical considerations on nonuniform plane waves).

More recent studies have focused on the surface-wave (SW) realization of GWs.
In particular, ghost surface polaritons (GSP) have been theoretically
predicted in ionic-crystal (ZnS/SiO2) metamaterials 7 and antiferromagnets 8,
and have been experimentally observed in a bulk calcite crystal 9. Even in an
idealized lossless scenario, these wave modes are characterized by a complex-
valued out-of-plane wavevector, thereby combining the traits of conventional
bulk and surface polaritons, with both propagating (phase accumulation) and
evanescent (exponential decay) behavior inside the crystal. Their in-plane
propagation on the crystal surface is instead characterized by hyperbolic
dispersion, giving rise to long-range, directional and diffractionless
polariton propagation 9.

It is important to distinguish between GWs and leaky waves, as both exhibit
complex-valued wavevectors but have distinct properties. Leaky waves are
complex eigenmodes that can be supported by open waveguiding structures and
are compatible with radiation in terms of their momentum; they can effectively
model radiative phenomena from such systems 10. On the other hand, GWs are
complex bound modes that rely on broken symmetries for their realization, with
the necessity to operating with biaxial and slanted optical axes with respect
to an interface.

Far from being physical oddities of mere academic concern, GWs and GSPs
exhibit several unusual features of great interest, including the possibility
to exponentially enhance and modulate incident evanescent waves via resonant
coupling mechanisms 3, to attain exact frequency degeneracies in guided modes
11, and to control nanoscale light propagation via the optical-axis
orientation 9. These may find important applications in a variety of
scenarios, ranging from integrated waveguides and nonlinear optics 3, 11 in
optical sensing 12 and sub-diffraction imaging 9.

Expanding the aforementioned features and their associated applications to
flat-optics platforms becomes a fascinating development. As such, we
demonstrate here the possibility of exciting GWs in planar junctions of
metasurfaces. In analogy with the higher-dimensional concept of surface
polaritons, here we consider the recently introduced concept of line waves
(LWs), which can propagate along metasurface junctions characterized by
oppositely signed surface reactances 13, 14 or resistances 15. Conventional
LWs at isotropic metasurface junctions exhibit unattenuated propagation along
the interface and evanescent (monotonic) decay both out-of plane (away from
the metasurfaces, as conventional SWs) and in-plane (away from the interface),
thereby effectively channeling the power flow along a one-dimensional
waveguide (hence the name) 13, 14, with tremendous benefits for guiding in an
ultra-subdiffractive way light and polaritons.

Here we show that, in the presence of (lossless) anisotropy, a new kind of LW
can be supported exhibiting GW-type behavior (i.e., phase oscillations
combined with evanescent decay) both in-plane (along the metasurface) and out-
of-plane (in the surrounding medium). These ghost LWs (GLWs) can be viewed as
the one-dimensional version of GWs 2, 3 and GSPs 7, 8, 9, and they share
several intriguing features with their higher-dimensional equivalents, opening
up new perspectives in interface nano-optics for the advanced control of
surface polaritons in deeply sub-diffractive regimes 16, 17.

## Results and discussion

Figure 1: Problem schematic: A planar junction between an isotropic
metasurface and an anisotropic, reciprocal one with slanted optical axis. The
wavy arrows depict a GLW that propagates unattenuated along the interface, and
exhibits an oscillatory decay both in-plane and out-of-plane in the
anisotropic halfplane ($x>0$). In the isotropic region ($x<0$), a monotonic
decay (not shown) similar to conventional LWs is anticipated. Also shown are
the absolute ($x,y,z$) and principal ($\perp,y,\parallel$) reference systems.

Referring to Figure 1 for a conceptual illustration, we consider a junction in
the $x-z$ plane between an isotropic metasurface (characterized by a surface
conductivity $\sigma_{i}$) and an anisotropic, reciprocal one (characterized
by the parallel and orthogonal surface conductivities $\sigma_{\parallel}$ and
$\sigma_{\perp}$, respectively) with an optical axis rotated by an angle
$\phi$. Alternatively, the anisotropic metasurface can be described, in the
$(x,z)$ in-plane reference system, by a symmetric tensor
${\underline{\underline{\sigma}}}_{a}$ with components

$\displaystyle{\sigma_{xx}}$ $\displaystyle=$
$\displaystyle{\sigma_{\perp}}{\cos^{2}}\phi+{\sigma_{\parallel}}{\sin^{2}}\phi,$
(1a) $\displaystyle{\sigma_{zz}}$ $\displaystyle=$
$\displaystyle{\sigma_{\parallel}}{\cos^{2}}\phi+{\sigma_{\perp}}{\sin^{2}}\phi,$
(1b) $\displaystyle{\sigma_{xz}}$ $\displaystyle=$
$\displaystyle{\sigma_{zx}}=\left({{\sigma_{\perp}}-{\sigma_{\parallel}}}\right)\cos\phi\sin\phi.$
(1c)

This is a commonly utilized model for nanostructured metasurfaces 18 as well
as for natural van der Waals crystals 19. For now, we assume that the
metasurfaces are purely reactive (i.e., purely imaginary surface conductivity
values) and immersed in vacuum, so that the entire scenario is ideally
lossless; this is done to ascertain the genuine character of GW-type
solutions, since the presence of losses would naturally induce complex-valued
wavevectors. As conceptually sketched in Figure 1, we are interested in
finding GLW solutions that propagate unattenuated along the interface $x=0$
and, in the anisotropic region ($x>0$), decay with an oscillating character
both in-plane and out-of-plane. Conversely, in the isotropic region $x<0$, we
anticipate a monotonic decay (not shown) similar to conventional LWs.

The boundary-value problem in Figure 1 admits an analytical (Maliuzhinets-
type) solution for isotropic junctions 20, whereas the anisotropic case of
interest here is insofar largely unexplored. Previous studies 21 are
restricted to the investigation of “edge modes” in the particular case of a
truncated anisotropic metasurface (corresponding to $\sigma_{i}=0$ in Figure
1) via a Wiener-Hopf approach with a Fetter-type approximation. Here, in order
to consider the more general scenario in Figure 1, we follow a numerical
approach based on finite-element simulations. However, some useful physical
insight can be obtained via a preliminary analytical study of the SWs
supported by the anisotropic half assumed as of infinite extent (for the well-
known isotropic case, see, e.g., Ref. 22). Accordingly, we are interested in
time-harmonic modal solutions of the form
$\propto\exp\left[i\left(k_{x}x+k_{y}y+k_{z}z-\omega t\right)\right]$. These
modes are generally hybrid, and their dispersion equation can be written as
23, 24

$2\eta\left[{\sigma_{zz}}\left(k_{x}^{2}+k_{y}^{2}\right)+{\sigma_{xx}}\left(k_{y}^{2}+k_{z}^{2}\right)-2{\sigma_{xz}{k_{x}}{k_{z}}}\right]+kk_{y}\left[\eta^{2}\left(\sigma_{xx}\sigma_{zz}-\sigma_{xz}^{2}\right)+4\right]=0,$
(2a) with the constraint
$k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=k^{2},\quad\mbox{Im}\left(k_{y}\right)\geq 0,$
(2b)

where $\eta$ and $k=\omega/c=2\pi/\lambda$ are the vacuum intrinsic impedance
and wavenumber, respectively (with $c$ and $\lambda$ being the corresponding
wavespeed and wavelength, respectively), and the branch-cut is chosen so as to
ensure the physically feasible out-of-plane decay away from the metasurface.
From eqs 2b, we can calculate the iso-frequency contours, by fixing $\omega$,
varying one (real-valued) wavevector component (say $k_{z}$), and deriving the
remaining two ($k_{x},k_{y}$) by solving analytically a quartic equation (see
Supporting Information for details). Of the resulting four modal branches,
only some (typically two) exhibit the proper branch-cut $\mbox{Im}(k_{y})\geq
0$. In what follows, we focus on these proper solutions and, without loss of
generality, we consider the forward-propagating case ($k_{z}>0$) and the
angular range $0<\phi<90^{\circ}$; outside these ranges, the solutions follow
by symmetry considerations (see Supporting Information for details). Moreover,
for notational convenience, we introduce the (dimensionless) normalized
conductivities 19

$\alpha_{\nu}=2\pi\eta\sigma_{\nu},\quad\nu=i,\parallel,\perp.$ (3)

Figure 2: Representative iso-frequency contours for hyperbolic anisotropy.
(a,b) Real and imaginary parts of normalized wavenumbers
${\bar{k}}_{x}=k_{x}/k$ as a function of $\bar{k}_{z}=k_{z}/k$, for a
hyperbolic metasurface with $\alpha_{\perp}=-i0.07$,
$\alpha_{\parallel}=i0.26$, and $\phi=0$. Only the proper modal branches with
$\mbox{Im}(k_{y})\geq 0$ are shown, for $k_{z}>0$, with the insets displaying
some magnified views around ${\bar{k}}_{z}=1$. (c,d) Same as panels (a,b),
respectively, but for $\phi=27^{\circ}$. The pink-shaded region delimits the
GW range. (e,f) Same as panels (a,b), respectively, but for $\phi=45^{\circ}$.

Akin to their higher-dimensional counterparts, GLWs originate from broken
symmetries. For a representative illustration, we consider a hyperbolic
metasurface with normalized conductivities $\alpha_{\perp}=-i0.07$ and
$\alpha_{\parallel}=i0.26$, which are compatible with the values reported in
the literature19 for a thin layer of $\alpha$-MoO3. Figure 2 shows the
corresponding iso-frequency contours (in terms of normalized wavenumbers
${\bar{k}}_{x,z}=k_{x,z}/k$) for three representative values of the rotation
angle $\phi$. Specifically, Figures 2a,b pertain to the well-known 18
symmetric scenario in the absence of rotation (i.e., $\phi=0$), for which we
observe the expected presence of two propagating modes at small (see insets)
and large wavenumbers; these solutions are conventional SWs whose fields
extend in-plane and exponentially decay out-of-plane. The two propagating
regions are separated by a bandgap corresponding to evanescent modes (i.e.,
purely imaginary $k_{x}$ and $k_{y}$) that decay exponentially both in-plane
(along $x$) and out-of-plane. As it will be clearer hereafter, this spectral
region is compatible with the propagation of conventional LWs in junction-type
configurations as in Figure 1. However, by rotating the optical axis, as for
instance in Figures 2c,d ($\phi=27^{\circ}$), a symmetry-breaking deformation
of the iso-frequency contours occurs, resulting in a nonzero real part in
$k_{x}$ (and thus, from eq 2b, in $k_{y}$) over an otherwise evanescent region
$k<k_{z}\lesssim 39k$ (highlighted with pink shading). These modes exhibit the
aforementioned GW hallmarks (phase accumulation and exponential decay) both
in-plane and out-of-plane. It is interesting to note that, unlike the
volumetric scenario 2, 3, now the GW-type behavior is also induced in a
material (vacuum, in our example) that is not anisotropic; while this can be
expected for lossy, isotropic metasurfaces, it remains a remarkable feature in
our lossless scenario, which may provide new possibilities for applications.
We observe that the two GW branches are characterized by complex-conjugate
wavenumbers $k_{x1}=k_{x2}^{*}$ (and, from eq 2b, $k_{y1}=-k_{y2}^{*}$).
Clearly, for a metasurface of infinite extent, only the decaying branch
[$\mbox{Im}(k_{x})\geq 0$] is physically feasible but, in the presence of
spatial truncations, the other branch could resonantly couple with an incident
evanescent field, and amplify it through the anisotropic section, along the
lines of what predicted for conventional GWs 3. Beyond this GW spectral region
(i.e., $k_{z}\gtrsim 39k$), we recover conventional SWs as for the unrotated
scenario.

Larger rotation angles (e.g., $\phi={45}^{\circ}$) may cause even stronger
deformations in the iso-frequency contours (see Figures 2e,f), such as the
transition of propagating modes from improper to proper character (and
viceversa), leading to the vanishing of evanescent and GW regions, with the
presence of propagating SWs only.

Figure 3: Parametric study for hyperbolic anisotropy. False-color-scale map of
$\left|\mbox{Re}\left(k_{x}\right)/\mbox{Im}\left(k_{x}\right)\right|$, as a
function of rotation angle $\phi$ and normalized wavenumber ${\bar{k}}_{z}$,
for a hyperbolic metasurface with parameters as in Figure 2
($\alpha_{\perp}=-i0.07$, $\alpha_{\parallel}=i0.26$). Due to symmetry, only
the ranges $k_{z}>0$ and $0<\phi<90^{\circ}$ are displayed.

For a broader illustration, Figure 3 shows the parameter
$\left|\mbox{Re}\left(k_{x}\right)/\mbox{Im}\left(k_{x}\right)\right|$ as a
function of $k_{z}$ and the rotation angle $\phi$. This parameter quantifies
the oscillatory vs. damped character, with the asymptotic limits $0$ and
$\infty$ corresponding to purely evanescent and propagating modes,
respectively, and finite values indicating the intermediate GW regime. We find
that GW-type modes can exist within a limited interval of rotation angles and
wavenumbers, with the allowed spectral region progressively shrinking as
$\phi$ approaches a critical value of $\sim 44^{\circ}$.

Qualitatively similar results are observed for different parameter choices,
which mainly affect the rotation-angle and wavenumber ranges where GW-type
modes can exist.

Overall, the above results indicate that, in principle, generic hyperbolic
metasurfaces with slanted optical axes can support GW-type modal solutions
within certain spectral ranges. The question now arises on what physical
mechanism could be exploited to induce the desired (real-valued) in-plane
wavenumber $k_{z}$. A possible answer is provided by the metasurface junction
in Figure 1. The basic idea, in analogy with the GSP scenario 7, 8, 9, is to
tailor the surface reactance of the isotropic half so that a LW can propagate
along the metasurface junction with a real-valued propagation constant
$k_{z}^{(LW)}$. From analogies with the isotropic-junction scenario 13, 14,
20, we expect the field behavior at the interface to possibly exhibit a
singularity, and therefore be quite different from conventional SWs.
Nevertheless, at a sufficient distance (along $x$) from the interface, it is
reasonable to assume that the field is approximately ruled by the infinite-
metasurface dispersion relationship, with $k_{z}\approx k_{z}^{(LW)}$.

Figure 4: Eigenmode analysis for isotropic-hyperbolic junction. (a)
Numerically computed LW effective modal index for a metasurface junction as in
Figure 1 (see also inset), as a function of
$\mbox{Im}\left(\alpha_{i}\right)$, for a hyperbolic half with parameters as
in Figures 2c,d ($\alpha_{\perp}=-i0.07$, $\alpha_{\parallel}=i0.26$,
$\phi=27^{\circ}$). (b) Numerically computed field-map [$\mbox{Re}(E_{y})$]
for $\alpha_{i}=i0.344$ (i.e., ${\bar{k}}_{z}^{(LW)}=37$). (c,d) Corresponding
cuts (blue-solid) at $y=0.001\lambda$ and $x=0.0054\lambda$, respectively,
compared with theoretical predictions (red-dashed) from eqs 2b with
$k_{z}=k_{z}^{(LW)}$, yielding ${\bar{k}}_{x}=-39.88+i26.38$ and
${\bar{k}}_{y}=20.33+i51.73$. Fields are normalized with respect to maximum
value.

To verify the above reasoning and assumptions, we carry out a numerical
eigenmode analysis (see Supporting Information for details) of this structure.
Assuming a hyperbolic half with parameters as in Figures 2c,d, we show in
Figure 4a the numerically computed LW effective modal index
${\bar{k}}_{z}^{(LW)}=k_{z}^{(LW)}/k$ as a function of the (imaginary)
normalized conductivity $\alpha_{i}$ of the isotropic half. Indeed, by varying
$\alpha_{i}$ within the inductive range
[$\mbox{Im}\left(\alpha_{i}\right)>0$], the junction supports a LW mode with
real-valued $k_{z}$; notably, the range of achievable $k_{z}$ values aligns
well with our theoretical prediction of the GW spectral range (as shown in
Figures 2c,d).

The above results provide a key connection between our theoretical study in
Figure 2 and the actual physical structure in Figure 1, showing that this
latter can support GLW-type modes by suitably tailoring the surface
conductivity of the isotropic half. Interestingly, the required conductivity
values are in line with those attainable from two-dimensional materials such
as graphene at THz frequencies 25. We highlight that, given the hyperbolic
character of the anisotropic half, there is no clear-cut choice of a
“complementary” isotropic half in order to support a LW. For this specific
example, an inductive isotropic metasurface is needed, but different parameter
choices may require a capacitive one [i.e.,
$\mbox{Im}\left(\alpha_{i}\right)<0$].

For a specific value of $\alpha_{i}$ (corresponding to ${k}_{z}^{(LW)}=37k$),
Figure 4b shows a field-map of the numerically computed eigenmode
[$\mbox{Re}(E_{y})$; see Supporting Information for details on the numerical
simulations and the complete field maps], from which the typical GW hallmarks
(oblique wavefronts) are clearly observed in the anisotropic half. In the
isotropic half, instead, the in-plane and out-of-plane attenuation is
monotonic, as in conventional LWs 13, 14. For a more quantitative assessment,
Figures 4c and 4d show two representative cuts (at fixed $y$ and $x$,
respectively), from which we observe the expected oscillatory damping, both
in-plane and out-of-plane, in excellent agreement with our theoretical
predictions. As previously mentioned, these latter are complex exponentials,
with propagation constants $k_{x}$ and $k_{y}$ obtained from eqs 2b by
replacing $k_{z}$ with the numerically computed LW propagation constant
$k_{z}^{(LW)}$. Therefore, these modes can be viewed as the one-dimensional
version of GSPs and conventional GWs; it can be verified that the oscillatory
damping does not imply actual active power flow away from the junction (see
Supporting Information for details).

As anticipated, in the absence of rotation ($\phi=0$, i.e., symmetric case),
the modal solutions may take the form of conventional LWs, which decay
monotonically for both positive and negative values of $x$, as well as along
$y$ (see Supporting Information for a representative example). On the other
hand, for scenarios where an extended behavior is expected (such as for
$\phi={45}^{\circ}$, or for $\phi={27}^{\circ}$ within the spectral range
$k_{z}>40k$), a two-dimensional eigenmode analysis is not as meaningful.

The above results validate our theoretical reasoning and assumptions,
indicating that it is actually possible to design a planar junction that
supports a GLW, whose properties can be tailored and controlled by varying the
optical-axis slant and/or the conductivity of the isotropic half.

Figure 5: Dipole excitation and losses for isotropic-hyperbolic junction. (a)
Numerically computed in-plane field map $[\mbox{Re}(E_{y})]$ at
$y=0.003\lambda$, for a configuration with parameters as in Figure 3b but with
losses ($\alpha_{\perp}=0.0035-i0.07$, $\alpha_{\parallel}=0.0013+i0.26$,
$\phi=27^{\circ}$, $\alpha_{i}=0.0034+i0.344$), excited by a $y$-directed
elementary electric dipole located at $x=z=0$, $y=0.01\lambda$ (schematized by
a thick yellow arrow). (b,c) Magnified views in-plane ($y=0.003\lambda$) and
out-of-plane ($x=0.0057\lambda$), respectively. (d,e) Corresponding cuts
(blue-solid) at $z=0.3\lambda$, with $y=0.003\lambda$ and $x=0.0057\lambda$,
respectively, compared with theoretical predictions (red-dashed). Fields are
normalized with respect to maximum value.

As a last step, to verify the actual excitability of the above eigenmode
solutions, we carry out three-dimensional numerical simulations in the
presence of a physical excitation (see Supporting Information for details).
Moreover, we take into account the possible presence of losses, assuming for
the parameter configuration in Figure 4 a loss-tangent of 0.05 for the
hyperbolic half (compatible with the values reported 26 for $\alpha$-MoO3) and
0.01 for the isotropic one (compatible with typical values of graphene 25).

Figure 5a shows the in-plane field distribution for a dipole-excited
configuration. As expected, this source also excites (isotropic and
anisotropic) conventional SWs that are naturally supported by the two
metasurface halves. In principle, these SWs could be de-emphasized via
suitable optimization of the parameters and sources, but this was not a
concern here, since their presence does not overshadow the GLW phenomenon of
interest. Indeed, as evidenced by the magnified views in Figures 5b,c (in-
plane and out-of-plane, respectively), the GLW behavior remains quite visible
along the interface, in spite of the loss-induced attenuation. Figures 5d,e
shows two representative cuts (in-plane and out-of-plane, respectively), which
are in very good agreement with the theoretical predictions. These latter are
computed as for the eigenmode analysis in Figure 4, but now assuming lossy
constitutive parameters.

Overall, the above results confirm that GLW can be physically excited by
dipole sources, and can be observed in the presence of small losses, together
with the (isotropic and anisotropic) SWs naturally supported by the two
halves.

Although the results above are limited to the case of hyperbolic anisotropy,
GLWs can also be observed in junctions featuring elliptic-type anisotropy.
Such scenario is discussed in the Supporting Information, paralleling the
results in Figures 2–5. While most qualitative considerations still hold,
there are some important differences that should be emphasized. In particular,
GLW may exist for any rotation angle of the optical axis (with the exception
of $\phi=0$ and $\phi=90^{\circ}$), and over spectral regions that are
generally open intervals (i.e., for $k_{z}$ larger than a critical value).
Moreover, beyond a critical rotation angle, additional GLW spectral regions
may appear at lower wavenumbers. Finally, unlike the hyperbolic case, the
choice of the isotropic half is now clear-cut, based on the complementary
character (i.e., capacitive for inductive-elliptic anisotropy, and vice
versa).

## Conclusion

In summary, we have extended the concept of GW to flat-optics scenarios
featuring planar junctions between isotropic and anisotropic metasurfaces.
Specifically, we have demonstrated the existence of a new type of LW which
exhibits the typical GW hallmarks, by combining the phase accumulation and
evanescent decay, both in-plane (along the anisotropic surface) and out-of-
plane (in the surrounding medium).

Our theoretical predictions are in excellent agreement with (eigenmode and
dipole-excited) finite-element numerical simulations, also in the presence of
losses, and shed further light on the physics of GWs and LWs, two emerging
research areas that are still in their infancy. These outcomes may provide new
opportunities for the nanoscale tailoring of light, via advanced control of
surface polaritons, with potentially abundant applications ranging from
nonlinear optics to optical sensing and imaging.

Current and future studies are aimed at exploring more in detail possible
technological platforms for experimental observation of GLWs. Our results
indicate that they may exist within broad parameter ranges, which are
compatible with constitutive parameters exhibited by artificially engineered
metasurfaces as well as natural low-dimensional materials. Clearly, realizing
a planar isotropic-anisotropic junction (with precise control of the optical
axis) constitutes a major technological challenge. One technologically
feasible possibility could be nanopatterning two-dimensional materials such as
graphene 18. As for natural anisotropic materials, thanks to the recent
advances in the field of van der Waals junctions 27, 28, this possibility may
be within reach in the near future. It should also be mentioned that, while
for simplicity we have limited our study to isotropic-anisotropic junctions,
similar phenomena can occur interface between two anisotropic surfaces, with
additional degrees of freedom for their control. Within this framework, the
extension of the analysis to elliptic dual (capacitive/inductive)
configurations is relatively straightforward. However, in junctions involving
two hyperbolic metasurfaces, the interpretation of the involved phenomena
become more complex.

Also of great interest is the study of resonant excitation and coupling
mechanisms, as well as the possibility to attain exceptional points and other
singularities by coupling two GLWs 11, 12. Finally, it would be interesting to
explore to what extent some typical features of LWs (e.g., chiral coupling,
unidirectional excitation, robustness to bend/curvature) are extended to GLWs.

G. C. and V. G. acknowledge partial support from the University of Sannio via
the FRA 2021 Program. A. A. has been partially supported by the Air Force
Office of Scientific Research and the Simons Foundation.

Details on the analytical derivations and numerical simulations, and
additional results can be found in the Supporting Information.

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Vol.0 (20xx) No.0, 000–000

11institutetext: Department of Physics, Technion, Haifa, 3200003, Israel;
<EMAIL_ADDRESS>
Received 20xx month day; accepted 20xx month day

# Predicting gravitational waves from jittering-jets-driven core collapse
supernovae

Noam Soker

###### Abstract

I estimate the frequencies of gravitational waves from jittering jets that
explode core collapse supernovae (CCSNe) to crudely be 5-30 Hz, and with
strains that might allow detection of Galactic CCSNe. The jittering jets
explosion mechanism (JJEM) asserts that most CCSNe are exploded by jittering
jets that the newly born neutron star (NS) launches within few seconds.
According to the JJEM, instabilities in the accreted gas lead to the formation
of intermittent accretion disks that launch the jittering jets. Earlier
studies that did not include jets calculated the gravitational frequencies
that instabilities around the NS emit to have a peak in the crude frequency
range of 100-2000 Hz. Based on a recent study, I take the source of the
gravitational waves of jittering jets to be the turbulent bubbles (cocoons)
that the jets inflate as they interact with the outer layers of the core of
the star at thousands of kilometres from the NS. The lower frequencies and
larger strain than those of gravitational waves from instabilities in CCSNe
allow future, and maybe present, detectors to identify the gravitational wave
signals of jittering jets. Detection of gravitational waves from local CCSNe
might distinguish between the neutrino-driven explosion mechanism and the
JJEM.

###### keywords:

gravitational waves – stars: neutron – black holes – supernovae: general –
stars: jets

##  1 Introduction

Since the early days of gravitational wave detectors core collapse supernovae
(CCSNe) have been considered as potential sources of gravitational waves
(e.g., de Freitas Pacheco 2010), with intensified research in recent years
(e.g., Afle & Brown 2021; Gill et al. 2022; Saiz-Pérez, Torres-Forné, & Font
2022). Gravitational waves from CCSNe are yet to be detected (e.g.,
Szczepańczyk et al. 2023). Recent studies concentrate on the expected
gravitational waves from CCSNe during the explosion process (e.g., Powell &
Müller 2019; Lin et al. 2023; Mezzacappa et al. 2023; Wolfe et al. 2023;
Pastor-Marcos et al. 2023; for more on the results of some studies see section
3) and shortly after explosion, e.g., in relation to magnetar formation (e.g.,
Cheng et al. 2023; Menon, Guetta, & Dall’Osso 2023).

Studies that calculate the properties of gravitational waves from CCSN
explosions ignore the role of jittering jets. The goal of this exploratory
study is to estimate the expected contribution of jittering jets to
gravitational wave emission from CCSNe. The motivation for this study are
results from recent years that support the jittering jets explosion mechanism
(JJEM) of CCSNe (e.g., Soker 2022a, a, 2023; Shishkin & Soker 2023), and the
very recent study by Gottlieb et al. (2023) who found that the turbulent
cocoons that energetic relativistic jets form (e.g., Izzo et al. 2019) can be
a strong source of gravitational waves. A cocoon is the convective bubble that
a jet inflates and is filled with the shocked jet’s material and the shocked
ambient material. Gottlieb et al. (2023) simulated relativistic and very
energetic jets, $\approx 10^{52}-10^{53}{~{}\rm erg}$, that are relevant to
rare CCSNe where the pre-explosion core is rapidly rotating and the collapsing
core is likely to form a black hole. There are many studies of such rare CCSNe
(but not of the gravitational waves from jets) that have fixed-axis jets
(e.g., Khokhlov et al. 1999; Aloy et al. 2000; Maeda et al. 2012; López-Cámara
et al. 2013; Bromberg & Tchekhovskoy 2016; Nishimura et al. 2017; Wang, Wang,
& Dai 2019; Grimmett et al. 2021; Perley et al. 2021; Gottlieb et al. 2022;
Obergaulinger & Reichert 2023).

In this study, however, I deal with non-relativistic jets where each jet-pair
has a much lower-energy of $\approx 10^{50}{~{}\rm erg}$. The JJEM asserts
that such jets explode most CCSNe (e.g., Soker 2010; Papish & Soker 2011;
Soker 2020; Shishkin & Soker 2021; Soker 2023b). The newly-born neutron star
(NS), or in some cases a black hole, launches the jets as it accretes mass
through an accretion disk. There are two sources of the angular momentum of
the accretion disk (e.g., Soker 2023b). These are pre-collapse core rotation
that has a fixed angular momentum axis, and the convective motion in the pre-
collapse core (e.g., Papish & Soker 2014b; Gilkis & Soker 2015; Soker 2019a;
Shishkin & Soker 2022) or envelope (e.g., Quataert et al. 2019; Antoni &
Quataert 2022, 2023) that has a stochastically varying angular momentum axis.
When the pre-collapse core angular momentum is low the accretion disk has
rapidly varying axis direction. Each accretion episode through a given
accretion disk lasts for a limited period of time and leads to one jet-
launching episode of two opposite jets.

The convective fluctuations serve as seed perturbations that are amplified by
instabilities behind the stalled shock, which is at $\simeq 100-150{~{}\rm
km}$ from the newly born NS. Namely, the same instabilities that give rise to
gravitational waves in the frame of the neutrino-driven explosion mechanism
(e.g., Mezzacappa et al. 2020), which does not include jets, exist also in the
JJEM. The JJEM has in addition the jittering jets that inflate turbulent
bubbles (cocoons) that might emit gravitational waves according to the new
results of Gottlieb et al. (2023).

In the present, still exploratory, study I present the first prediction,
although very crude, for gravitational waves in the frame of the JJEM. I do
this by appropriately scaling the recent results that Gottlieb et al. (2023)
obtained for gravitational waves from much more energetic jets than the
jittering jets (section 2). I then present the general characteristic of the
strain of JJEM-driven CCSNe (section 3). I summarize the results (section 4)
and strongly encourage simulations of gravitational waves from jittering jets
in CCSNe.

##  2 Estimating gravitational waves from jittering jets

The calculation of gravitational waves by CCSNe as expected in the JJEM
requires very demanding three-dimensional hydrodynamical simulations. In this
preliminary study I make crude estimates by scaling the results of Gottlieb et
al. (2023) who conduct simulations of long-lived relativistic jets with
energies of $\simeq 10^{52}-10^{53}{~{}\rm erg}$.

In the JJEM the jets are relatively short-lived and have a typical velocity of
$0.3-0.5c$ (e.g., Papish & Soker 2014a; indeed, Guetta et al. 2020 claim that
neutrino observations limit the jets in most cases to be non-relativistic). In
an explosion process there are $\approx{\rm few}-30$ jet-launching episodes,
with a typical activity time of each episode of $\simeq 0.01-0.1{~{}\rm s}$,
and a typical energy of the two jets of $\approx 10^{50}{~{}\rm erg}$ to ${\rm
few}\times 10^{50}{~{}\rm erg}$ (Papish & Soker 2014a).

Gottlieb et al. (2023) estimate the range of frequencies of the gravitational
waves when the jets’ axis is at a large angle to the line of sight (off-axis)
to be between $f_{\rm min}\simeq 1/\Delta t_{\rm jc}$ and $f_{\rm max}\simeq
c_{s}/\Delta r_{\rm sh}$, where $\Delta t_{\rm jc}$ is the time the jets
energize the cocoons, $c_{s}$ is the sound speed, and $\Delta r_{\rm sh}$ is
the width of the shell formed by the shock. For their simulations this range
is $\approx 0.1-2000{~{}\rm Hz}$. The on-axis emission, i.e., when the jets’
axis is at a very small angle to the line of sight, has a strain amplitude
that is more than an order of magnitude smaller than for the off-axis emission
and the strain amplitude peaks at frequencies of $10-100{~{}\rm Hz}$.

To scale for one pair of jittering jets I consider the three-dimensional
simulations by Papish & Soker (2014b). They simulated three pairs of jittering
jets that have their axes on the same plane, each jet-launching episode
lasting for $0.05{~{}\rm s}$. In Figure 1 I present the density and
temperature maps in the jittering plane of these jets. In each jet-launching
episode the two opposite jets are seen as two opposite high density (red color
on the left column) strips touching the center. While the first jet-pair
inflates axisymmetric cocoons (bubbles), the second and third jet-pairs
inflate non-axisymmetric bubbles. This is seen by the compressed gas at the
head of the cocoon (bubble) that I point at with the double-lined arrows.

Figure 1: Density (left column with a colour coding in logarithmic scale and
units of ${~{}\rm g}{~{}\rm cm}^{-3}$) and temperature (right column in log
scale in units of K) maps at three times of three-dimensional hydrodynamical
simulation of jittering jets taken from Papish & Soker (2014b). There are
three jet-launching episodes, each composed of two opposite jets, one episode
after the other with activity times of $0-0.05{~{}\rm s}$ direction 1 in the
lower panel, $0.05-0.1{~{}\rm s}$ direction 2, and $0.1-0.15{~{}\rm s}$
direction 3. I added double-lined arrows to point at the two opposite masses
at the cocoon (bubble) head. While the first jet-pair inflates axisymmetric
cocoons, the following cocoons largely deviate from axisymmetry. Velocity is
proportional to the arrow length on the right column, with inset showing an
arrow for $30,000{~{}\rm km}{~{}\rm s}^{-1}$.

From the density maps of Figure 1 I estimate the width of the shells of the
bubbles/cocoons that the jets inflate to be $\Delta r_{\rm sh}\simeq
200-500{~{}\rm km}$, and from the temperature maps the sound speed is
$c_{s}\simeq 5000{~{}\rm km}{~{}\rm s}^{-1}$. This gives with the definition
of Gottlieb et al. (2023) $f_{\rm max}\simeq c_{s}/\Delta r_{\rm sh}\approx
10-25{~{}\rm Hz}$. In the JJEM the typical duration of a jet-launching episode
is $\Delta t_{\rm j}\approx 0.01-0.1{~{}\rm s}$. Even if the jets lasts for
$\simeq 0.01{~{}\rm s}$, the interaction with the core material lasts longer.
For that the interaction time is more likely to be $\Delta t_{\rm jc}\approx
0.05-0.2{~{}\rm s}$, which gives with the definition of Gottlieb et al. (2023)
$f_{\rm min}\simeq 1/\Delta t_{\rm jc}\approx 5-20{~{}\rm Hz}$ for typical
jittering jets, but with large uncertainties. The short-duration jets will
have small energy and therefore small strain amplitude. The longer-duration
jets have more energy. Therefore, waves with lower frequency are more likely
to be detected, i.e., $f_{\rm min}\simeq 5-10{~{}\rm Hz}$.

The relatively small ratio of $f_{\rm max}/f_{\rm min}\approx 1-5$ that I find
here shows that the typical spectrum of the gravitational waves of jittering
jets is qualitatively different from the case that Gottlieb et al. (2023)
study. In the case of the JJEM I expect the spectrum to be in narrow range of

$f_{\rm JJEM}\approx 5-30{~{}\rm Hz}.$ (1)

As seen in Fig. 1 the size of the cocoon is smaller than the typical
wavelength of $\approx 20,000{~{}\rm km}$, which makes phase cancellation very
small.

Scaling equation (2) of Gottlieb et al. (2023) for the strain amplitude for
one pair of jets out of many pairs in the JJEM, gives

$h\approx 4\times 10^{-22}\left(\frac{D}{10{~{}\rm
kpc}}\right)^{-1}\left(\frac{E_{\rm 2j}}{10^{50{~{}\rm erg}}}\right).$ (2)

I also consider the following quantity that is used in the study of
gravitational waves from CCSNe

$\begin{split}\frac{h}{\sqrt{f}}\approx 10^{-22}&\left(\frac{D}{10{~{}\rm
kpc}}\right)^{-1}\left(\frac{E_{\rm 2j}}{10^{50{~{}\rm erg}}}\right)\\\
\times&\left(\frac{f_{\rm JJEM}}{15~{}{\rm Hz}}\right)^{-1/2}{\rm
Hz}^{-1/2},\end{split}$ (3)

were I scaled with the expected frequency range for jittering jets from
equation (1).

I note that equations (2) and (3) treat each jet-launching episode as an
independent event. If several episodes are considered to inflate only two
opposite large bubbles (lower panel of Figure 1) then the energy in the
scaling of the equations should be the sum of several jet-launching episode.
Namely, the scaling energy should be $\simeq{\rm few}\times 10^{50}$ leading
to a strain larger by a factor of a few.

##  3 Identification of gravitational waves from jittering jets

Several papers calculated the gravitational waves properties from CCSNe when
jets are not included (e.g., Radice et al. 2019; Andresen, Glas, & Janka 2021;
Mezzacappa et al. 2023), i.e., in the frame of the delayed neutrino explosion
mechanism (e.g., Bethe & Wilson 1985; Heger et al. 2003; Janka 2012; Nordhaus
et al. 2012; Müller et al. 2019; Fujibayashi et al. 2021; Boccioli et al.
2022; Nakamura, Takiwaki, & Kotake 2022; Olejak et al. 2022). Mezzacappa et
al. (2020), for example, find that low-frequency emission, $\lesssim
200{~{}\rm Hz}$, is emitted by the neutrino-driven convection and the standing
accretion shock instability in the gain layer behind the stalled shock, while
high-frequency emission, $\lesssim 200{~{}\rm Hz}$, is emitted by convection
in the proto–NS. These studies find that the emission is mainly at frequencies
of $\approx 10-2000{~{}\rm Hz}$ with larger strain amplitudes at frequencies
of $\approx 100-1000{~{}\rm Hz}$ (e.g., Srivastava et al. 2019).

The gain region and the convection in the proto-NS exist also in the JJEM.
Neutrino heating play roles also in the JJEM (Soker 2022b). Therefore, the
contributions of the gain region and the proto-NS to gravitational waves in
the JJEM are similar to those in the delayed neutrino explosion mechanism. In
the JJEM there is the additional contribution of the cocoons that the jets
inflate in the core and envelope of the exploding star. In section 2 I crudely
estimated this contribution for jittering jets interacting with the core of
the exploding star. In Figure 2 I present results from Mezzacappa et al.
(2023). The results is of the characteristic gravitational wave strain from a
CCSN in the frame of the delayed neutrino explosion mechanism of a
$15M_{\odot}$ stellar model. I added my crude estimate of a typical
contribution of jittering jets (the horseshoe-shaped yellow region on the
graph).

Figure 2: A figure from Mezzacappa et al. (2023) to which I added a crude
estimate of the characteristic spectrum of $hf^{-1/2}$ from jittering jets in
a CCSN at a distance of $D=10{~{}\rm kpc}$ (the horseshoe-shaped yellow zone).
The signal in yellow is for one jet-launching episode. If several jet-
launching episodes are considered to inflate only two opposite large bubbles
(lower panel of Figure 1) then the strain will be larger, as it is about the
sum of these episodes. Other marks are as in the original figure. The blue
line is the calculation by Mezzacappa et al. (2023) of the characteristic
gravitational wave strain from a CCSN of a $15M_{\odot}$ stellar model. The
five other lines represents the sensitivity curves for gravitational wave
detectors: Advanced Laser Interferometer Gravitational Observatory (AdvLIGO),
Advanced VIRGO, and Kamioka Gravitational Wave Detector (KAGRA) that are
current-generation gravitational wave detectors, and the more sensitive next-
generation detectors, Cosmic Explorer and Einstein Telescope. The predicted
full gravitational wave spectrum includes both the contributions from the
regions near the NS that exist both in the JJEM and in the neutrino-driven
explosion mechanism (blue line), and the contribution of the jittering jets.

The peak of the contribution of the jittering jets is at much lower
frequencies than the peak of the other components of CCSNe. In addition, there
will be variations with time as the jittering jets are active intermittently.
As said, simulations of the JJEM are highly demanding because for the
calculation of gravitational waves high-resolution simulations are need to
resolve the convection in the cocoon and the head of the jet-core interaction.
At this point I only present the possible schematic behavior of the strain as
function of time due to the contribution of jittering jets. In the upper panel
of Figure 3 I schematically present such a gravitational wave signal due only
to jittering jets. I describe the distance times the strain of four jet-
launching episodes (but more are expected at later time until the star
explodes). Over the time period $0.2{~{}\rm s}-0.7{~{}\rm s}$ the average
frequency is $16{~{}\rm Hz}$. As commonly done I take $t=0$ at the bounce of
the shock wave from the newly born NS. There is some time delay until
instabilities start to feed the intermittent accretion disks that launch the
jets. These instabilities give rise to high-frequency-gravitational waves
(e.g., Radice et al. 2019; Andresen, Glas, & Janka 2021). In the lower panel
of Figure 3 I present one figure from Mezzacappa et al. (2023) that shows
their calculation for the gravitational wave of a CCSN of $15M_{\odot}$
stellar model. The expected signal is the sum of all contributions.

Figure 3: The gravitational wave strain times distance as function of time
during the early explosion process. The upper panel is a schematic
presentation of a possible wave form from jittering jets. The typical
amplitude and frequency are according to equations (2) and (1), respectively.
The double-headed arrows present the contributions of four jet-launching
episodes, E1 to E4. The lower panel is from Mezzacappa et al. (2023) for
calculations based on a simulation that do not include jets of an exploding
stellar model of $15M_{\odot}$.

My crude estimate of gravitational waves from jittering jets show that their
signal is qualitatively different than that of the other components that are
close to the NS, $\lesssim 100{~{}\rm km}$. The jittering jets add long period
modulations to the short-period waves from the other components. For a nearby
CCSN even the present Advanced LIGO detector might separate the signal of the
jittering jets from the other components. This depends on the signal-to-noise
ratio that should be calculated with future simulations of jittering jets.
Future detectors will be able to do so for CCSNe in the local group.

##  4 Summary

Based on the very recent results by Gottlieb et al. (2023), which I scaled
from long-lasting energetic relativistic jets in super-energetic CCSN to
short-lived low-energy non-relativistic jets in common CCSNe, I concluded that
jittering jets lead to detectable gravitational wave signals. The source of
the gravitational waves is the turbulence in the cocoons that the jets inflate
(Figure 1). Whether present detectors can reveal the gravitational wave
signals of jittering jets depend on the signal-to-noise ratio that simulations
of jittering jets should calculate, and of course on the distance to the CCSN.
Future detectors will be able to reveal the jittering jets signal from CCSNe
in the local group (Fig. 2).

The frequencies of the expected gravitational wave signals from jittering jets
are lower than the other components of CCSNe, as I mark by the yellow
horseshoe-shaped region in Figure 2. I schematically present a gravitational
wave signal from jittering jets in the upper panel of Figure 3, and compared
it with calculations from a CCSN simulation that include no jets from
Mezzacappa et al. (2023). The signal from jittering jets can be clearly
distinguished from the other gravitational waves sources in CCSNe (depending
on signal-to-noise ratio and the distance of the CCSN).

This, still exploratory, study calls for the performance of highly-demanding
simulations of jittering jets and the calculation of their gravitational wave
signals. The simulation must be of very high resolution as to resolve the
turbulence in the cocoon.

Because I expect jittering jets to explode most CCSNe, my prediction for the
gravitational wave signals from nearby CCSNe differ from the prediction of
studies that include no jets.

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